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<?xml version="1.0"?>
<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" >
<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_01_la_1588.xml</cvs_file>
<cvs_version/>
<locator>077.xml</locator>
</info> <text> <front> </front> <body>
<chap>
<pb id="p.0001"/>
<p type="head">
<s>GVIDIVBALDI <lb/>E MARCHIONIBVS <lb/>MONTIS <lb/>IN DVOS ARCHIMEDIS <lb/>ÆQVEPONDERANTIVM <lb/>LIBROS</s></p>
<p type="head">
<s>PARAPHRASIS <lb/>Scholijs illu&longs;trata.</s></p>
<figure></figure>
<p type="head">
<s>PISAVRI <lb/>Apud Hieronymum Concordiam; <lb/>M D LXXXVIII. <lb/><emph type="italics"/>Superiorum Conce&longs;&longs;<gap/>.<emph.end type="italics"/></s></p>
<pb/>
<p type="head">
<s>SERENISSIMO <lb/>FRANC.^{CO} MARIAE <lb/>II. VRBINI DVCI.</s></p>
<p type="head">
<s>GVIDVSVBALDVS <lb/>E' MARCHIONIBVS MONTIS S.</s></p>
<p type="main">
<s>I am decemnium elap&longs;um e&longs;t, DVX Sere­<lb/>ni&longs;&longs;ime, ex quo de rebus machanicis volu­<lb/>men, veras (ni fallor) mirabilium mechani­<lb/>corum effectuum cau&longs;as manife&longs;tans, in lu­<lb/>cem dedi; vbi non nulla an tiquiora, <expan abbr="præci-puaq;">præci­<lb/>puaque</expan> illu&longs;trium græcorum authorum pla­<lb/>cita ad &longs;u&longs;ceptum negotium pertinentia, <lb/>tanquam rect&ecedil; rationi magis con&longs;entanea amplexatus &longs;um. <lb/>quibus&longs;anè, tanquam &longs;olidi&longs;&longs;imis innixa fundamentis, theo­<lb/>remata multa, ac varia con&longs;truxi. quippe quæ, licet non inua­<lb/>lidis quoque demon&longs;trationum præ&longs;idijs à me ip&longs;o munita <lb/>fuerint; pleri&longs;què tamen, qui non admodum forta&longs;&longs;e in huiu&longs;­<lb/>modi rerum cau&longs;is inue&longs;tigan disver&longs;ati exi&longs;tunt, noua pror­<lb/>&longs;us (vt accepi) ac ferme inaudita, nec &longs;atis (vt opinor) apud eos <lb/>firma, atqueideo illis non omnino &longs;atisfeci&longs;&longs;e, vi&longs;a &longs;unt. Quo­<lb/>circa cogitanti mihi, qua ratione fieri po&longs;&longs;et, vtopusillud à <lb/>me editum, quàm plurimorum &longs;ibi gratiam in dies magis con <lb/>ciliaret, in men tem venit, non aliunde id mihi oportun iùs <expan abbr="cõ">com</expan> <lb/>tingere potui&longs;&longs;e, quàm &longs;i pri&longs;cosip&longs;os, & graui&longs;&longs;imos alioqui <lb/>authores de hac re elegan ti&longs;&longs;imè di&longs;&longs;erentes illis offerrem. ra­<lb/>tus, vt&longs;olidi&longs;&longs;imâ eorum doctrinâ, quæ à me propo&longs;ita, & ex­
<pb/>plicata fuere theoremata, firmiora redderentur. &longs;im ulquè alio <lb/>rum ambiguitati, ne dicam imbecillitau &longs;uccurreretur. vel&longs;al <lb/>tem ip&longs;i graui&longs;&longs;ima eorum authoritate non nullorum captiua­<lb/>rentintellectum, in ob&longs;equium meliùs, rectiù&longs;què <expan abbr="&longs;entientiũ">&longs;entientium</expan>, <lb/>atque intelligentium. Nihil enim tam, auta con&longs;uetudine, aut <lb/>ab opinione remotum e&longs;&longs;e &longs;olet, quod &longs;ola authoritate proba­<lb/>ri non po&longs;&longs;it. Verùm ne huiu&longs;modi negotium-in recen&longs;endis <lb/>multorum ad propo&longs;itam veritatem confirmandam te&longs;timo­<lb/>nijs la tiùs, quàm par e&longs;&longs;et, protraheretur; mihi con&longs;titui, ex mul <lb/>tis vnicum tantùm, eumquè reliquorum omnium hac in par <lb/>te facilè principem deligere: qui, & meam cau&longs;am tueretur: & <lb/>illis, &longs;i fieri po&longs;&longs;et, &longs;atisfaceret: vtquè coràm illis ip&longs;e &longs;e offerens, <lb/>tanquam meo quoque nomine mi&longs;&longs;us in telligeretur; quibu&longs;­<lb/>dam meis notis non in &longs;ignitum certè, &longs;ed a&longs;&longs;ociatum cundem <lb/>prodire volui. E&longs;t autem graui&longs;&longs;imus hic author Syracu&longs;iusille <lb/>Archimedes de mechanicis elementis con&longs;ulu&longs;&longs;imè di&longs;&longs;erens. <lb/>cuius nimirum dignitati, atque authoritati, vtomnes probè à <lb/>me con&longs;ultum in telligerent; decreui, vt quemadmodum inter <lb/>alios illius ordinis viros primatum obtinet, ita nulli alij, quàm <lb/>amplitudini tu&ecedil; DVX Sereni&longs;&longs;ime, hac no&longs;tra &ecedil;tate, doctrina, <lb/>rerumquè omnium cognitione &longs;ingulari, citra controuer&longs;iam <lb/>Principi &longs;upremo, &longs;uum in primis hoc tempore præ&longs;taret ob&longs;e <lb/>quium. quod incredibili &longs;anè animi mei iucunditate conti­<lb/>gi&longs;&longs;e fateor; non &longs;olùm, vt rur&longs;um aliquam &longs;ingularis meæ er­<lb/>ga amplitudinem tuam ob&longs;eruailtiæ, ac venerationis, tot, tan­<lb/>ti&longs;què nominibus iam pridem debit&ecedil; te&longs;tificationem ederem; <lb/>verùm etiam, vt munu&longs;culo illi meo tanto Principi audentiùs <lb/>forta&longs;&longs;e an tea oblato, ne pror&longs;us pr&ecedil; &longs;ua tenuitate de&longs;piceretur, <lb/>opem ferret. quanquam neque id quidem, pro eximia animi <lb/>tam excel&longs;i magnitudine, &longs;u&longs;picandum fuit. Per huncergo <expan abbr="tã">tam</expan> <lb/>celebrem authorem ad te Princeps optime, ac pr&ecedil;&longs;tanti&longs;&longs;ime <lb/>lætabundus accedo. Is enim mihi, quemadmodum & ego ip&longs;i, <lb/>ad te aditum patefeci&longs;&longs;e videtur; & &longs;icuteundem tibiloge gra­<lb/>ti&longs;&longs;imum futurum confido; ita me tui amanti&longs;&longs;imum, & ob&longs;er <lb/>uanti&longs;&longs;imum, vt eâdem, qua con&longs;ueui&longs;ti, benignitate pro&longs;e­<lb/>quaris, oro &longs;uplex, & ob&longs;ecro. Aueto dulce præ&longs;idium, ac &ecedil;tatis <lb/>no&longs;træ &longs;plendidum decus; & e&longs;to perpetuò f&ecedil;lix. </s></p>
<pb pagenum="1"/>
<p type="head">
<s>GVIDIVBALDI <lb/>E MARCHIONIBVS <lb/>MONTIS.</s></p>
<p type="head">
<s>PRAEFATIO:</s></p>
<p type="main">
<s>Mechanica facultas <expan abbr="nõ">non</expan> &longs;olùm ab imperitis, <lb/>verùm etiam ab eruditis admirabilis &longs;em­<lb/>per habita fuit; eorum enim, qu&ecedil; in admi­<lb/>rationem homines trahunt, duo e&longs;&longs;e gene­<lb/>ra Ari&longs;toteles in principio <expan abbr="&longs;uarũ">&longs;uarum</expan> <expan abbr="qu&ecedil;&longs;tionũ">qu&ecedil;&longs;tionum</expan> <lb/>Meehanicarum a&longs;&longs;eruit; quorum &longs;anè alte <lb/>rum ad ea pertinet, quæ natura quidem, <lb/>proximis tamen ip&longs;orum cau&longs;is latentibus in lucem <expan abbr="prodeũt">prodeunt</expan>; <lb/>alterum verò &longs;pectatad ea, qu&ecedil; preter naturam, & arte fiunt; <lb/>quibus natura &longs;uperari videtur (quamquam & ip&longs;a plurimùm <lb/>momenti ad &longs;e ip&longs;am euincendam tune quoque afferat) & <lb/>quod natur&ecedil; uiribus in lucem prodire nequit, id arte fieri con <lb/>tingat, obidquè maiorem adhuc admirationem excitat, quòd <lb/>ars natur&ecedil; çmula, qua&longs;i aduer&longs;us <expan abbr="naturamipugnãs">naturamipugnans</expan>, cam &longs;upe­<lb/>ret, & tan <expan abbr="quãvim">quanvim</expan> ip&longs;i in ferre videatur; cuius &longs;anè operationis <lb/>cau&longs;a quoque cognita admirationem parit; cùm exigua admo <lb/>dum ad tanti operisproductionem appareat. ad mirabilo e&longs;t &longs;a­<lb/>nèip&longs;ius artis magi&longs;terium, cùm adeò potens &longs;it, vt effectus na­<lb/>tur&ecedil; repugnantes producere tentet. quippè quibus, ni&longs;i ita &longs;en <lb/>&longs;ibus &longs;ub<gap/>jciàntur; vt tangi propemodum, & con&longs;pici po&longs;&longs;int, <lb/>vix fides adhibeatur; idquè <expan abbr="nõ">non</expan> &longs;ine admiratione adhuc cbgni <lb/>tum, acper&longs;ua&longs;um nobis e&longs;&longs;e po&longs;&longs;it. huiu&longs;modi autem mira­<lb/>bilium operum opifex e&longs;t ip&longs;a mechanica di&longs;ciplina, tam na­<lb/>tur&ecedil; emula, quàm oppugnatrix valida. H&ecedil;c enim grauia pro <lb/>prio fermè nutu &longs;ur&longs;um attolli, magnaquè pondera ab exigua
<pb pagenum="2"/>admodum virtute moueri, aliaquè id genus huiu&longs;modi &longs;pe­<lb/>ctanda proponit. vt tum imperitis exip&longs;orummet effectuum <lb/>intuitu, tum eruditis in cau&longs;arum varia contemplatione ad­<lb/>mirationem pariat. veluti &longs;i ea &longs;pectemus, qu&ecedil; neruis, vel ali­<lb/>quo mouétur in&longs;trumento; vel qu&ecedil; &longs;piritibus <expan abbr="cõcinnuntur">concinnuntur</expan>, & <lb/>fiunt; de quibus Heron, & alij pertractarunt; vel denique alijs <lb/>modis. quamquam nosinijs, quæ dicenda &longs;unt, de ea mecha­<lb/>nicæ facultatis parte, quæ ad pódera, <expan abbr="di&longs;tãtia&longs;que">di&longs;tantia&longs;que</expan> inter ip&longs;a exi­<lb/>&longs;tétes pertinet, <expan abbr="quorũ">quorum</expan> &longs;tatusad &ecedil;quilibrium reduci pote&longs;t, ver <lb/>ba faciemus. quæ quidé pars totius mechanic&ecedil; facultatis prin­<lb/>cepsexi&longs;tit. ea enim e&longs;t, in qua artem &longs;uperare naturam aper­<lb/>tiùs <expan abbr="cõ&longs;picitur">con&longs;picitur</expan>: quod quidem, qua ratione contingat, hincpla <lb/>num euadet. </s></p>
<p type="main">
<s>Ars quippe ex Ari&longs;totele phi&longs;icorum &longs;ecundo, & ex proæ­<lb/>mio quæ&longs;tionum mechanicarum triplici modo in &longs;uis opi&longs;i­<lb/>cijs &longs;e&longs;e habere videtur. Nam vel immitatur naturam; vel ea <lb/>perficit, quæ natura perficere non pote&longs;t; vel denique ea, quæ <lb/>pr&ecedil;ter naturam fiunt, operatur; in quibus tamen omnibus o­<lb/>perandi rationibus, &longs;i diligentereas con&longs;ideremus, artem &longs;em­<lb/>perimmitari naturam per&longs;piciemus. Primùm quidem multas <lb/>artes naturam immitari aperte videmus, vt &longs;culpturam, & hu­<lb/>iu&longs;modi alias. Quando autem arsea perficit, quæ&longs;ola natu­<lb/>ra per&longs;icere non pote&longs;t, vtin arte medica euenire &longs;olet; <expan abbr="naturã">naturam</expan> <lb/>ip&longs;am pariter emulatur, & naturæ a&longs;&longs;ociata, velut in&longs;trumen­<lb/>tum eius, naturalem effectum perficere dicitur: tuncquè eodé <lb/>modo operatur, ac &longs;i natura rem ip&longs;am ab&longs;que artis ope perfice <lb/>repo&longs;&longs;et, quod planè artis præ&longs;tantiam manife&longs;tat: quippè <lb/>cùm ni&longs;i ars ip&longs;inaturæ <expan abbr="manũ">manum</expan> porrigat, natura ip&longs;a proprios <lb/>effectus perficere ex &longs;e&longs;e minimè po&longs;&longs;it. At verò &longs;i ars <expan abbr="naturã">naturam</expan> <lb/>immitando ip&longs;am &longs;uperauerit; vtea, quæ ab arte fiunt, præter <lb/>naturam eueniant, longè adhuc præ&longs;tantiùs artis ingenium <lb/>apparebit. &longs;iquidem immitando naturam (paradoxum id for <lb/>tè videbitur, cùm tamen veri&longs;&longs;imum &longs;it) præter naturæ ordi­<lb/>nem operari dicatur. Ars. n. mirabili artificio naturam ipsa na <lb/>tura &longs;uperat; ita nimirum res di&longs;ponendo, vtip&longs;a efficeret na <lb/>tura, &longs;i eiu&longs;modi &longs;ibi producendos &longs;tatueret effectus. quod qui <lb/>dem &longs;ubiecto exemplo magis per&longs;picuum fiet. </s></p>
<pb pagenum="3"/>
<p type="main">
<s>Sint enim duo pondera <lb/>
<arrow.to.target n="fig1"></arrow.to.target><lb/>AB in aliquo vecte, A ma <lb/>ius, B minus; quorum &longs;i­<lb/>mulita in vecte di&longs;po&longs;ito. <lb/>rum &longs;it centrum grauitatis <lb/>C. &longs;it autem &longs;ub vecte in­<lb/>ter CA fulcimentum in D. <lb/>& quoniam pondera AB penes C grauitatis centrum inclinan <lb/>tur? tunc C deor&longs;um naturaliter mouebitur; ac per con&longs;equés <lb/><expan abbr="pōdus">pondus</expan> quoque B deor&longs;um tendet. Sed &longs;i B deor&longs;um mouetur, <lb/>A certè &longs;ur&longs;um eleuabitur. quippe quod, <expan abbr="quãuis">quanuis</expan>, vtgraue e&longs;t, <lb/>atque &longs;olutum ab&longs;que connexione ponderis B deor&longs;um tende <lb/>ret; attamen vtadnexum ponderi B, intercedente vecte AB, <lb/>&longs;ur&longs;um mouebitur: & (vt ita dicam) pondus A contra pro­<lb/>priam naturam naturaliter a&longs;cendet. Vndè <expan abbr="per&longs;picuũ">per&longs;picuum</expan> e&longs;t, hos <lb/>motus effectus e&longs;&longs;e naturales. Quid igitur efficit ars ip&longs;a? nil <lb/>fanè aliud, quàm quòd resita di&longs;ponit, & accomodat; vt &longs;imi­<lb/>les effectus inde prodeantatque &longs;i naturales omnino exi&longs;tant, <lb/>quare opus erit, ut Ars naturam immitetur, &longs;iquidem effectus <lb/>naturales prouenire debent. propterea vectem, fulcimentum­<lb/>què eodem modo di&longs;ponit; & loco ponderis B aliquam con­<lb/><gap/>tituit potentiam, quæ pr&ecedil;mendo parem vim habeat grauita­<lb/>ti ip&longs;ius B; atque tunc ip&longs;a potentia mouens, qu&ecedil; minore&longs;t gra <lb/>uitate ponderis A, ip&longs;um A grauius nihilominus attollet. <lb/>quod quamuis propriæ ip&longs;ius naturæ repugnet, naturaliter ta­<lb/>mé ab ip&longs;a potentia in B exi&longs;tente <expan abbr="&longs;ursũ">&longs;ursum</expan> feretur: res cnim ita di <lb/>&longs;po&longs;itæ talem habent naturam, vt A quidem &longs;ur&longs;um, B vero <lb/>deor&longs;um moueri debeant. qu&ecedil; &longs;anè ex no&longs;tro Mechanicorum <lb/>libro, & exijs, quæ in hoc pertractantur; comperti&longs;&longs;imè red­<lb/>dentur, & quod diximus devecte, de alijs quoque in &longs;trumen­<lb/>tis mechanicis in telligendum e&longs;t. quorum quidem apparatus <lb/>&longs;unt artis opera, effectus autem ip&longs;ius penè naturæ: cùm eius <lb/>momenta, inclinationes què lequantur, veluti præcipuas eiu&longs;­<lb/>modioperum effectrices cau&longs;as: quippè quæ &longs;unt omninoad­<lb/>mirabiles, acpr&ecedil;&longs;tanti&longs;&longs;ime; quemadmodum ex ip&longs;arum con <lb/>téplatione patere pote&longs;t. cuius rei <expan abbr="argumétũ">argumétum</expan> illud indica&longs;&longs;e &longs;at <lb/>e&longs;to, <expan abbr="nimirũ">nimirum</expan> eas à &longs;ummis uiris, Ari&longs;totele, & Archimede fui&longs;&longs;e
<pb pagenum="4"/>pertractatas. Ari&longs;toteles. n. in principio <expan abbr="Qu&ecedil;&longs;tionũ">Qu&ecedil;&longs;tionum</expan> <expan abbr="mechanica-rũ">mechanica­<lb/>rum</expan> multa, eaqué pr&ecedil;cipua ad cau&longs;as rei mechanicæ <expan abbr="digno&longs;c&etilde;das">digno&longs;cendas</expan> <lb/>aperuit; qué &longs;ecutus Archimedes in his libris mechanica prin­<lb/>cipia explicatiùs patefecit, eaquè planiora reddidit. Nec propte <lb/>rea Ari&longs;toteles diminutus extitit: etenim <expan abbr="eorũ">eorum</expan>, qu&ecedil; abip&longs;o pro <lb/>po&longs;ita, & explicata fuere, problematum cau&longs;as egregiè patefe­<lb/>cit. &longs;ed quoniam Archimedi &longs;copus fuit mechanic&ecedil; di&longs;ciplin&ecedil; <lb/>rudimenta explanare; propterea ad magis particularia <expan abbr="enucleã">enucleam</expan> <lb/>da de&longs;cendere voluit. Ari&longs;toteles. n. (gratia <expan abbr="ex&etilde;pli">exempli</expan>) <expan abbr="qu&ecedil;r&etilde;s">qu&ecedil;rens</expan> cur <lb/>vecte magna mouemus pondera? cau&longs;am e&longs;&longs;e ait <expan abbr="longitudin&etilde;">longitudinem</expan> <lb/>vectis maiorem ad partem potentiæ: & rectè quidem; cùm ex <lb/>principio ab ip&longs;o con&longs;tituto manife&longs;tum &longs;it, ea, qu&ecedil; &longs;untin <lb/>longiori à centro <expan abbr="di&longs;tãtia">di&longs;tantia</expan>, <expan abbr="maior&etilde;">maiorem</expan> quoque habere virtuté. Ar­<lb/>chimedes verò vltcriùs adhuc progredi voluit, hoc admi&longs;&longs;o, <expan abbr="n&etilde;">nem</expan> <lb/>pè quod e&longs;t in longiori di&longs;tantia maiorem uim habere, quàm <lb/>id, quod e&longs;t in breuiori, in quirere etiam voluit, quanta &longs;it vis <lb/>eius, quod e&longs;t in longiori di&longs;tantia ad id, quod e&longs;t in breuiori; <lb/>ita vt inter h&ecedil;c nota reddatur qualis, & qu&ecedil; &longs;iteorum propor­<lb/>tio determinata. atque ideo <expan abbr="fundam&etilde;tum">fundamentum</expan> illud mechanicum <lb/>pr&ecedil;&longs;tan ti&longs;&longs;imum manife&longs;tauit; videlicet ita &longs;e&longs;e habere pon­<lb/>dus ad pondus, vt di&longs;tantia ad in&longs;tantiam, vndepondera &longs;u­<lb/>&longs;penduntur, &longs;e&longs;e permutatim habet. quo ignoto, res mechani­<lb/>c&ecedil; nullo modo pertractari po&longs;&longs;e videntur. quandoquidem <lb/>huic tota mechanica facultas tanquam vnico, pr&ecedil;cipuoque <lb/><expan abbr="fundam&etilde;to">fundamento</expan> innititur. Quare Archimedes <expan abbr="Ari&longs;totel&etilde;">Ari&longs;totelem</expan> &longs;equi vide <lb/>tur; quod non &longs;olùm patet exijs, quæ dicta &longs;unt; verùm etiam <lb/>&longs;i Archimedis po&longs;tulata <expan abbr="cõ&longs;iderauerimus">con&longs;iderauerimus</expan>, quibus <expan abbr="cõ&longs;titu&etilde;dis">con&longs;tituendis</expan>, <lb/>ea, quæ de principijs mechanicis Ari&longs;toteles patefecit, Archi­<lb/>medé &longs;upponere <expan abbr="cõperiemus">comperiemus</expan>. vt deinceps &longs;uo loco <expan abbr="per&longs;picuũ">per&longs;picuum</expan> <lb/>fiet. In ratione pr&ecedil;terea, acmodo <expan abbr="cõ&longs;iderãdi">con&longs;iderandi</expan> mechanica, maxi­<lb/>ma ambo affinitate coniuncti in cedere vidétur. Ari&longs;toteles. n. <lb/>
<arrow.to.target n="marg1"></arrow.to.target> res mechanicas tum Mathematica, tú naturalia &longs;apere, acre&longs;pi <lb/>cerea&longs;&longs;e<gap/>uit: quod quidé & Archimedes optimè nouit: <expan abbr="nã">nam</expan> qu&ecedil; <lb/>Mathematicè &longs;unt con&longs;ideranda, geometricè demon&longs;trauit, <lb/>vt &longs;unt di&longs;tantiæ, proportiones, & alia huiu&longs;modi: quæ verò <lb/>funtnaturalia, naturaliter <expan abbr="quoq;">quoque</expan> <expan abbr="cõ&longs;iderauit">con&longs;iderauit</expan>; vtea, quæ ad gra <lb/>uitatis centrum &longs;pectant, & quæ &longs;ur&longs;um, & qu&ecedil; deor&longs;um moue
<pb pagenum="5"/>ri debent; & c&ecedil;tera huiu&longs;modi. Ex quibus <expan abbr="patetmaximũ">patetmaximum</expan> e&longs;&longs;e <lb/>inter tantos viros in his pertractandis con&longs;en&longs;um. Ambiget <lb/>forta&longs;&longs;e qui&longs;piam, nunquid h&ecedil;c principia rectè abillis fuerint <lb/>pertractata? &longs;ed &longs;tatim omnis ce&longs;&longs;at dubitandi occa&longs;io, &longs;i tan <lb/>torum virorum pr&ecedil;&longs;tantia ad memoriam reuocetur; quibus, <lb/>citra controuer&longs;iam in di&longs;ciplinis ab ip&longs;is traditis, omnes eru­<lb/>diti <expan abbr="palmã">palmam</expan> deferunt. vtquemadmodum <expan abbr="ab&longs;q;">ab&longs;que</expan> Ari&longs;totele duce, <lb/>atque doctore, nemo ad rectè <expan abbr="philo&longs;ophãdum">philo&longs;ophandum</expan>, ita neque <expan abbr="etiã">etiam</expan> <lb/>ad Mathematicam, <expan abbr="pr&ecedil;cipue&qacute;ue">pr&ecedil;cipueque</expan> Mechanicam di&longs;ciplinam <lb/><expan abbr="ab&longs;q;">ab&longs;que</expan> Archimede &longs;e&longs;e <expan abbr="qui&longs;piã">qui&longs;piam</expan> di&longs;ponere po&longs;&longs;it: quorum &longs;anè <lb/>apud peritiores authoritas meritò ob id &longs;uprema extat; quòd <lb/>abip&longs;is reseo meliori, <expan abbr="pr&ecedil;&longs;tantiori&qacute;">pr&ecedil;&longs;tantiorique</expan>; modo pertractat&ecedil; fuerút, <lb/>quo ip&longs;arum rerum natura, atque doctrin&ecedil; ratio po&longs;tulabat.s&s<lb/>qui &longs;cientiarum cupidi &longs;unt, illos &longs;equi, eorum què &longs;cripta &longs;&ecedil;pè <lb/>&longs;&ecedil;plus attentè perlegere debent. Pr&ecedil;terea philo&longs;ophi&ecedil;, ac Ma­<lb/>thematic&ecedil; profe&longs;&longs;ores in hoc conueniunt; quòd cùm aliqua ad <lb/>philo&longs;ophiam &longs;pectantia tractant; mirum in modum Ari&longs;to­<lb/>telem laudibus extollunt. qui verò Mathematicas pertractare <lb/>&longs;tudét, &longs;tatim ad Archimedis laudes pariter &longs;e <expan abbr="cōferũt">conferunt</expan>. tamet&longs;i <lb/>circa ea, qu&ecedil; nó &longs;unt Archimedis ver&longs;entur; vt <expan abbr="quã">quam</expan> plurimi fece <lb/>re, quod quidé optimo factum e&longs;t con&longs;ilio. etenim &longs;i ea, quæ <lb/>mathematica ope indigent, laudare volunt, ad Archimedem <lb/>confugiendum e&longs;t; vt&longs;i inuentionem, &longs;ubtili&longs;&longs;imum Archi­<lb/>medis inuentum afferant, quo modum adinuenit cogno&longs;cen­<lb/>d&ecedil; quantitatis argenti, quod eratin corona Regisaurea, vt Vi­<lb/>truuius te&longs;tatur; & alia huiu&longs;modi; &longs;i admirabilia, &longs;tatim affe­<lb/>rant Archimedis&longs;ph&ecedil;ram in globo vitreo elaboratam, in qua <lb/>omnes c&ecedil;le&longs;tis &longs;phæræ motus relucebant; ita utnatura potiùs <lb/>Archimedem immitata, quàm Archimedes naturam illu&longs;i&longs;&longs;e
<arrow.to.target n="marg2"></arrow.to.target><lb/>videatur; nauim præterea graui pondere oneratam è mari in <lb/>littus ab Archimedeeductam; aliaquèid genus plurima. De­<lb/>nique &longs;i res Mathematicas ciuitatibus e&longs;&longs;e vtiles o&longs;tendere vo­<lb/>lunt, ea, quæ ab Archimede contra Marcellum in defen&longs;io­<lb/>nepatriæ facta fuere, in medium afferant, quo tempore bellica <lb/>opera adeo mirabilia effecit, vt &longs;olus Archimedes contra bel <lb/>lico&longs;i&longs;&longs;imos Romanos pugnare &longs;ufficiens videretur. quæ qui­<lb/>dem omnia Mechanica di&longs;ciplina <expan abbr="cõfecta">confecta</expan> &longs;unt. Quid igitur
<pb pagenum="6"/>Mechanica admirabilius, & vtilius? è qua tot, tantaquè ad <lb/>humani generis vtilitatem conferentia prodeunt? eximia cer­<lb/>tè, & præclara admodum hæc Archimedisge&longs;ta fuerc; quæ ta­<lb/>men, &longs;iad alia quamplurima, quæ deip&longs;o dici, ac afferri po&longs; <lb/>&longs;unt, conferantur; exigua &longs;anè mihi videntur. Nam quæ ha­<lb/>ctenus commemorata &longs;unt, (quamquam forta&longs;&longs;e <expan abbr="nõ">non</expan> omnia) <lb/>multa tamen, huiu&longs;modiquè &longs;imilia alij quoque effecerunt, <lb/>& adhuc extant forta&longs;&longs;e viri co ingenij acumine pr&ecedil;diti, qui <lb/>talia aggredi non vererentur: &longs;ed <expan abbr="nõnulla">nonnulla</expan> egregia <expan abbr="extãt">extant</expan> ip&longs;ius <lb/>Archimedis opera, quorum &longs;imilia, nec antea, nec po&longs;t ipsú <lb/>facta fuere, neque in futurum facienda fore à nemine &longs;int ex­<lb/>pectanda. omnium enim admirabili&longs;&longs;ima, præ&longs;tanti&longs;&longs;ima­<lb/>què &longs;unteius &longs;eripta, in quibus, & ingenij acumen, inuentio­<lb/>nes &longs;ubtili&longs;&longs;imæ, perfectaquè doctrina planè con&longs;picitur. adeo <lb/>enim his omnibus Archimedis &longs;cripta aliorum &longs;cripta mathe <lb/>maticorum excellunt, &longs;uperantquè; vt quæ aliorum, facilè <lb/>quidem inter&longs;e&longs;e comparari, cum ijsverò, qu&ecedil; ab Archimede <lb/>nobis relicta fucrunt; nullo modo po&longs;&longs;int. ut aperti&longs;simè <lb/>(alijs interim omi&longs;sis) con&longs;picuum redditur ex ijs, quæ de <lb/>&longs;ph&ecedil;ra & cylindro, & exijs, qu&ecedil; de æqueponderantibus &longs;cri­<lb/>pta reliquit: quippè qu&ecedil; ob eorum <expan abbr="pr&ecedil;&longs;tãtiam">pr&ecedil;&longs;tantiam</expan>, ac dignitatem <lb/>meritò literis aureis e&longs;&longs;ent imprimenda. liber enim de &longs;ph&ecedil;ra, <lb/>& cylindro inter Archimedis &longs;cripta <expan abbr="excell&etilde;s">excellens</expan> adeò habit^{9} fuit; <lb/>vt ad eius &longs;epulcrú appofita fuerit &longs;ph&ecedil;ra, & cylindr^{9}: quib^{9} a <lb/>Cicerone con&longs;pectis; &longs;tatim illud Archimedis &longs;epulcrú e&longs;&longs;e in <lb/>tellexit: de ouius inuentione ob uiri excellen <expan abbr="tiã">tiam</expan> maximè glo­<lb/>riatur: Deindè qua ratione ip&longs;um à temerario van&ecedil; orationis <lb/>proferendæ au&longs;u, (dum &longs;icloquitur, da mihi vbi &longs;i&longs;tam, ter­<lb/>ramquè mouebo) vindicare po&longs;&longs;emus; ni&longs;ihec, quæ de æque­<lb/>ponde<gap/>antibus extant, &longs;cripta reliqui&longs;&longs;et<gap/>ex his enim habita <lb/>notitia proportionis ponderum, & di&longs;tantiarum, &longs;it manife­<lb/>ftum non e&longs;&longs;e à ratione, nequè à natura pror&longs;us alienum, po&longs;&longs;e <lb/>terram moueri, &longs;i daretur con&longs;iftendi locus. quod etiam ex <lb/>no&longs;tro volumine Mechanico annis ab hinc aliquot elap&longs;is e­<lb/>dito varijs quoquè in&longs;trumentis parere pote&longs;t. <expan abbr="quandoquid&etilde;">quandoquidem</expan> <lb/>multis modis, datum pondus à data potentia moueri, ibi <expan abbr="o&longs;t&etilde;">o&longs;tem</expan> <lb/>&longs;ume&longs;t. vbi demon&longs;trationes à nobis con&longs;titut&ecedil; ijs, quæ apud
<pb pagenum="7"/>Archimedem pre&longs;enti opere habentur, totam eorum vim fer­<lb/>ri voluntacceptam. Etne quidpiam, quod &longs;tudio&longs;is mecha­<lb/>nicæ facultatis prode&longs;&longs;e po&longs;&longs;it, pr&ecedil;termitteretur, ad horum <lb/>Archimedis librorum interprætationem aliquid operis con­<lb/>tuli&longs;&longs;e placuit; &longs;atisquè nobis feci&longs;&longs;e videbimur; &longs;i &longs;altem &longs;tu­<lb/>dio&longs;inos Archimedis ve&longs;tigia &longs;ecutos fui&longs;&longs;e cognouerint. <lb/>Et quamuis opus hoc fuerit ab Eutocio A&longs;calonita nonnullis <lb/>commentarijs illu&longs;tratum, quia tamen propter Archimedis <lb/>&longs;criptorú ob&longs;curitaté multa adhuc remanét ab&longs;tru&longs;a, nec pror <lb/>&longs;us omnibus peruia; pr&ecedil;&longs;ertim gr&ecedil;carum literarum experti­<lb/>bus; cùm liber hic in latinum ver&longs;us multis in locis ob&longs;curus, <lb/>alijsquè plerisque quodammodo mancus meritò &longs;u&longs;picetur; <lb/>ita vt adhucin tenebris iacere videatur; gr&ecedil;cusquè præterea <lb/>codex impre&longs;&longs;us, quem &longs;ecuti &longs;umus, multisin locis aliqua <lb/>correctione egere videatur; idcirco ab huiu&longs;modi munere <lb/>pr&ecedil;&longs;tando de&longs;i&longs;tere noluimus: quin &longs;imul hos libros in latinú <lb/>&longs;ermonem verteremus; commentarijsquè illu&longs;tratos redde­<lb/>remus. Cùm præ&longs;ertim hinc tutus ad mechanicam <expan abbr="di&longs;ciplinã">di&longs;ciplinam</expan> <lb/>pateat aditus. Quare vtmens huius pr&ecedil;clari&longs;&longs;imi Math ema <lb/>tici magis, atque magis, quàm fieri po&longs;sit, pro virili no&longs;tra <lb/>per&longs;picua reddatur; & huius &longs;cientiæ cupidi in adipi&longs;cendis <lb/>pulcherrimis hi&longs;ce theorematibus minùs laborent; à commu <lb/>ni genere interpr&ecedil;tandi aliquamnulum in præ&longs;entia di&longs;cedere <lb/>nobis vi&longs;um e&longs;t oportunum. Nam qui res mathematicas in­<lb/>terprætati &longs;unt, &longs;uos commentarios &longs;eor&longs;um à demon&longs;tratio­<lb/>nibus collocauere: nos verò, qu&ecedil; no&longs;tra &longs;unt, verbis ip&longs;ius
<arrow.to.target n="marg3"></arrow.to.target><lb/>Archimedis in&longs;eruimus, & hoc tantùm in ip&longs;is demon&longs;tra­<lb/>tionibus, non in propo&longs;itionibus, & huiu&longs;modi alijs, hac <lb/>planèhabita di&longs;tinctione, vt quæ &longs;unt Archimedis (his, vel <lb/><emph type="italics"/>his literarum notis<emph.end type="italics"/>) cogno&longs;cantur, ip&longs;iusquè tantùm Ar­<lb/>chimedis e&longs;&longs;e intelligantur. Qu&ecedil; verò alterius &longs;unt cha­<lb/>racteris, utqu&ecedil; huius exi&longs;tent formæ, no&longs;tra e&longs;&longs;e &longs;emper <lb/>&longs;int exi&longs;timanda. & quoad fieri potuit, verba omnia, qu&ecedil; <lb/>nobis declaratione aliqua, nec non correctione indigere vi&longs;a <lb/>&longs;unt (ijs tamen omi&longs;&longs;is, qu&ecedil; parui, imò nullius &longs;untmomenti, <lb/>vt e&longs;t literarum immutatio, & huiu&longs;modi alia) dilucidè expli­<lb/>care, atque emendare &longs;tuduimus. quibus etiam hanc adhibui
<pb pagenum="8"/>mus diligentiam, quod quamuis ea, quæ no&longs;tra, &longs;unt, verbis <lb/>&longs;int Archimedis in&longs;erta; &longs;iquis tamen verba tantùm Archi­<lb/>medislegere maluerit, rectè id a&longs;&longs;equi poterit; &longs;iquidem ne <lb/>verbum quidem Archimedis omi&longs;im us: quinnimo ea ita di­<lb/>&longs;po&longs;uimus, vt&longs;uum pror&longs;us retineant &longs;en&longs;um, po&longs;&longs;intquè <expan abbr="cō">com</expan> <lb/>tinuatè legi; ac &longs;i nihil inter ip&longs;a in&longs;ertum fuerit. quod qui­<lb/>dem &longs;tudio&longs;is non inutile fore iudicauimus; qui ab&longs;que no­<lb/>&longs;tris additionibus <expan abbr="Archimedē">Archimedem</expan> tantùm habebunt; cú no&longs;tris <lb/>verò additionibus Archimedis demon&longs;trationes continua­<lb/>tas, & explicatas habebunt. Huberionis autem doctrinæ gra <lb/>tia permulta adiunximus &longs;cholia, in quibus pa&longs;&longs;im ordinem, <lb/>Authori&longs;què artificium patefecimus; nec non multa lemma <lb/>ta ad Archimedis demon&longs;trationes nece&longs;&longs;aria <expan abbr="demõ&longs;traui-mus">demon&longs;traui­<lb/>mus</expan>; aliaquè nonnulla ad explicationem, &longs;ubiectamquè ma <lb/>reriam valde vtilia adiecimus. Vt etiam Archimedis dicta <lb/>magis eluce&longs;cant, antequam ad explicationem verborum <lb/>ip&longs;iusaccedamus, nonnulla prius declarare oportunum no­<lb/>bis vi&longs;um e&longs;t ad ea, quæ in his libris Archimedis &longs;upponit <lb/>tanquam cognita. Deinde con&longs;iderand us proponitur &longs;copus, <lb/>atque iiitentio Archimedis; diui&longs;io item librorum; huiu&longs;­<lb/>modiquè alia, quæ &longs;ummam afferent facilitatem ad intel <lb/>ligendam: mentem Archimedis. </s></p>
<p type="margin">
<s><margin.target id="marg1"></margin.target><emph type="italics"/>in princip. <lb/>que&longs;t. Me­<lb/>chan.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg2"></margin.target><emph type="italics"/>Claudianus<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg3"></margin.target><emph type="italics"/>declaratio <lb/>huius para <lb/>phra&longs;is.<emph.end type="italics"/></s></p>
<figure id="fig1"></figure>
<p type="main">
<s>Cùm itaquè &longs;upponat &longs;unos exqui&longs;itam habere no titiam <lb/>centri grauitatis; illius definitionem afferre libuit: pro cuius <lb/>to men faciliori notitia illud quoque in primis admonen­<lb/>
<arrow.to.target n="marg4"></arrow.to.target> dum duximus; nimirum quatuor reperiri centra. <expan abbr="Centrũ">Centrum</expan> ui­<lb/>delicet vniuer&longs;i, cen trum magnitudinis, centrum figuræ, & <lb/>centrum grauitatis, quod quidem grauitatis centrum rectè <lb/>definitur à Pappo Alexandrino in octauo libro mathemati­<lb/>carum collectio num hocpacto. </s></p>
<p type="margin">
<s><margin.target id="marg4"></margin.target><gap/></s></p>
<p type="head">
<s>DEFINITIO CENTRI GRAVITATIS</s></p>
<p type="main">
<s>Centrum grauitaris vniu&longs;cuiu&longs;que corporis e&longs;t pun ctum <lb/>quoddam intra po&longs;itum, à quo &longs;<gap/>graue appen&longs;um mente <lb/>conçipiatur, dum fertur, quie&longs;cit<gap/> & &longs;erua<gap/> eam, quam in <lb/>principio habebat po&longs;itionem, nequeinip&longs;a latione circum-
<pb pagenum="9"/>uertitur. </s></p>
<p type="head">
<s>EIVSDEM ALIA DEFINITIO.</s></p>
<p type="main">
<s>Centrum grauitatis vniu&longs;cuiu&longs;que &longs;olidæ figuræ e&longs;t <expan abbr="punctũ">punctum</expan> <lb/>illud in tra po&longs;itum, circa quod vndique partes &ecedil;qualium mo <lb/>mentorum con&longs;i&longs;tunt. &longs;i. n. per tale centrum ducatur <expan abbr="planũ">planum</expan> &longs;i <lb/>guram quomodo cunque &longs;ecans, &longs;emper in partes æqueponde <lb/>rantes ip&longs;am diuidet. </s></p>
<p type="main">
<s>Hanc po&longs;tremam definitionem, &longs;eu potiùs de&longs;eriptionem <lb/>tradidit Federicus Commandinus in libro de centro grauita­<lb/>tis &longs;olidorum. ex quipus &longs;anè definitionibus eluce&longs;cit natura, <lb/>
<arrow.to.target n="fig2"></arrow.to.target><lb/>atque facultas <expan abbr="c&etilde;tri">centri</expan> grauitatis. <lb/>vt &longs;i punctum A fuerit <expan abbr="centrũ">centrum</expan> <lb/>grauitatis corporis BC, tunc <lb/>ex Pappi&longs;ententia, &longs;i BC <expan abbr="&longs;u&longs;p&etilde;">&longs;u&longs;pem</expan> <lb/>datur ex A, magnitudo BC <lb/>eadem, qua reperitur, di&longs;po­<lb/>&longs;itionelocata manebit; neque <lb/>partes ullasip&longs;ius corporis, vt qu&ecedil; &longs;untad <lb/>
<arrow.to.target n="fig3"></arrow.to.target><lb/>BC, circumuerti, neque omnino &longs;uum <lb/>mutare &longs;itum depræhendetur. &longs;i verò vt <lb/><expan abbr="Cõmandino">Commandino</expan> placuit, A fuerit centrum <lb/>grauitatis magnitudinis BCD, eadem­<lb/>què per punctum A vtcunque &longs;ecúdùm <lb/>rectitudinem diuidatur, veluti per EAF. <lb/>tunc pars EBF ip&longs;i ECDF æqueponde­<lb/>rabit, quamuis EBF, & ED &longs;int magni <lb/>tudines inæquales. &longs;æpenumero enim e­<lb/>uenire &longs;olet, vt in diui&longs;ione figuræ per eius centrum graui­<lb/>tatis ip&longs;a aliquando in partes diuidatur æquales, ali­<lb/>quando in partes inæquales: vt &longs;uo loco o&longs;tendemus:
<arrow.to.target n="marg5"></arrow.to.target><lb/>&longs;emper tamen in partes diuiditur hinc inde æquepon­<lb/>derantes; non tamen &longs;eor&longs;um con&longs;titutas, ab inuicen <lb/>què &longs;eiunctas, & veluti ad æquilibrium examinatas; vt pu­<lb/>ta &longs;i EBF decem pondo ponderet; ED quoque totidem <lb/>pependiffe oporteat. res quippe non &longs;ic &longs;e habet, &longs;ed cas e&longs;&longs;e <lb/>in eo &longs;itu æqueponderantes, in quo reperiun tur; vt neutra
<pb pagenum="10"/>alteri pr&ecedil;ponderet. ex quibus colligipote&longs;t, &longs;i graue quidpiam <lb/>in centro mundi collo catum fuerit, oportere centrum graui <lb/>tatis illius in centro mundi con&longs;titutum e&longs;&longs;e: &longs;iquidem vt <lb/>graue illud tunc quie&longs;cat, partes vn dique ip&longs;um ambientes ç­<lb/>qualium momentorum exi&longs;tere, atque manere oporteat. <lb/>Quare dum a&longs;&longs;eritur, graue quod cumque naturali propen­<lb/>fione &longs;edem in mundi centro appetere, nil aliud &longs;ignifica­<lb/>tur, quàm quòd eiu&longs;modi graue proprium centrum grauitatis <lb/>cum centro vniuer&longs;i coaptare expetit, vt optimè quie&longs;cere va­<lb/>leat. Ex quo &longs;equitur motum deor&longs;um alicuius grauis fieri <lb/>per rectam lineam, quæ centrum grauitatis ip&longs;ius grauis, cen <lb/>trumquè mundi connectit. quandoquidem grauia deor&longs;um <lb/>rectà feruntur. Vnde manife&longs;tum e&longs;t, Grauia &longs;ecundum gra <lb/>uitatis centrum deor&longs;um tendere. quod nos in no&longs;tro Mecha <lb/>nicorum libro &longs;uppo&longs;uimus. </s></p>
<p type="margin">
<s><margin.target id="marg5"></margin.target><emph type="italics"/>in fine pri­<lb/>mi buius.<emph.end type="italics"/></s></p>
<figure id="fig2"></figure>
<figure id="fig3"></figure>
<p type="main">
<s>Ex ijs omnibus, quæ hactenus de centro grauitatis dicta <lb/>&longs;unt, per&longs;picuum e&longs;t, vnumquodque graue in eius centro <lb/>grauitatis propriè grauitare, veluti nomen ip&longs;um centri gra­<lb/>uitatis idip&longs;um manife&longs;tè præ&longs;eferre videtur. ita vt tota vis, <lb/>grauita&longs;què ponderisin ip&longs;o grauitatis centro coaceruata, col <lb/>lectaquè e&longs;&longs;e, ac tanquam in ip&longs;um vndiquè fluere videatur. <lb/>Nam ob <expan abbr="grauitat&etilde;">grauitatem</expan> pondus in <expan abbr="c&etilde;trum">centrum</expan> vniuer&longs;i naturaliter per <lb/>uenire cupit; centrum verò graui tatis (exdictis) e&longs;t id, quod <lb/>propriè in centrum mundi tendit. in centro igitur grauitatis <lb/>pondus propriè grauitat. Præterea quando aliquod pondus <lb/>ab aliqua potentia in centro grauitatis &longs;u&longs;tinetur; tunc pon­<lb/>dus &longs;tatim manet, to taquè ip&longs;ius ponderis grauitas &longs;en&longs;u per­<lb/>cipitur. quod etiam contingit, &longs;i &longs;u&longs;teneatur pondus in ali­<lb/>quo puncto, à quo per centrum grauitatis ducta recta linea <lb/>in centrum mundi tendat. hoc namque modo idem e&longs;t, ac <lb/>
<arrow.to.target n="marg6"></arrow.to.target> &longs;i <expan abbr="põdus">pondus</expan> in eius centro grauitatis propriè &longs;u&longs;tineretur. Quod <lb/>quidem non contingit, &longs;i &longs;u&longs;tineatur pondus in alio pun­<lb/>cto. neque enim pondus manet, quin potiùs <expan abbr="antequã">antequam</expan> ip&longs;ius <lb/>grauitas percipi po&longs;&longs;it, vertitur vtique pondus, donec &longs;imi <lb/>liter à &longs;u&longs;pen&longs;ionis puncto ad centrum grauitatis ducta re­<lb/>cta linea in vniuer&longs;i centrum recto tramite feratur. <lb/>quæ quidem ex prima no&longs;trorum Mechanicorum pro-
<pb pagenum="11"/>po&longs;itione &longs;unt manife&longs;ta, quando autem hæc linea e&longs;t hori­<lb/>zonti erecta, tuncidem pror&longs;us e&longs;t (vt mox diximus) perinde <lb/>ac &longs;i pondus in centro grauitatis ad vnguem &longs;u&longs;tineretur. <lb/>Quocirca &longs;i pònderis grauitas minimè percipi pote&longs;t, ni&longs;i in <lb/><expan abbr="c&etilde;tro">centro</expan> grauitatis ip&longs;ius, <expan abbr="põdus">pondus</expan> certè in ip&longs;o propriè grauitat. </s></p>
<p type="margin">
<s><margin.target id="marg6"></margin.target><gap/></s></p>
<p type="main">
<s>Centrum figuræ apud Mathematicos e&longs;t punctum, à quo <lb/>&longs;emidiametri exeunt; vel per quod <expan abbr="trã&longs;eunt">tran&longs;eunt</expan> diametri, vt circu <lb/>li centrum, & ellip&longs;is, necnon oppo&longs;itarum &longs;ectionum. </s></p>
<p type="main">
<s>Centrum verò magnitudinis e&longs;t id, quod medium figuræ <lb/>obtinet; vel quod &ecedil;qualiter ab exteriori &longs;uperficie di&longs;tat. vt <lb/>&longs;phær&ecedil; centrum. </s></p>
<p type="main">
<s>Centrum denique mundi e&longs;t punctum in medio vniuer&longs;i <lb/>&longs;itum, omniumquè rerum infimum. </s></p>
<p type="main">
<s>Cæterùm ad meliorem horum notitiam ob&longs;eruandum e&longs;t, <lb/>h&ecedil;c centra aliquando &longs;imul omnia inter&longs;e conuenire, <expan abbr="aliquã">aliquam</expan> <lb/>do nonnulla; aliquando autem minimè. &longs;imul verò omnia <lb/>conueniunt. vt centrum vniuer&longs;i, centrum magnitudinis ter <lb/>ræ (&longs;ph&ecedil;ræ &longs;cilicet ex aqua, terraquè compo&longs;it&ecedil;, quam nos bre <lb/>uitatis &longs;tudio terram tantùm nuncupabimus) centrum figu­<lb/>r&ecedil; terr&ecedil;; ac centrum grauitatis terr&ecedil;. Cùm enim terra &longs;it &longs;phæ­<lb/>rica (vt omnes fatentur.) eius medium erit centrum figur&ecedil;, à <lb/>quo &longs;emidiam etri exeunt. idip&longs;um què erit centrum magnitu <lb/>dinis, &longs;iquidem ip&longs;ius figur&ecedil; medium obtinet. Pr&ecedil;terea idem <lb/>punctum e&longs;t centrum grauitatis terr&ecedil;. & quoniam terra in me <lb/>dio <expan abbr="mūdi">mundi</expan> quie&longs;cit, erit hoc <expan abbr="centrũ">centrum</expan> grauitatis in centro vniuer&longs;i <lb/>collocatum. & hoc dun taxat modo centra omnia in <expan abbr="vnũ">vnum</expan> con <lb/>uenire po&longs;&longs;unt. quamquam verò &longs;ph&ecedil;ra, qu&ecedil; continet <expan abbr="terrā">terram</expan> & <lb/>aqu&acedil;, compo&longs;ita e&longs;t ex corporibus diuer&longs;&ecedil; &longs;peciei, <expan abbr="differ&etilde;ti&longs;què">differenti&longs;què</expan> <lb/>grauitatis, nimirum ex terra, & aqua; non <expan abbr="tam&etilde;">tamen</expan> efficitur, quin <lb/><expan abbr="mediũ">medium</expan> ip&longs;ius cum centro grauitatis con&longs;piret in vnum. <expan abbr="Nã">Nam</expan> ex <lb/>Ari&longs;to telis &longs;ententia terra circa mundi centrum vn dique <expan abbr="cõ&longs;i">con&longs;i</expan>
<arrow.to.target n="marg7"></arrow.to.target><lb/>&longs;tit; & Archimedes affirmat, <expan abbr="etiã">etiam</expan> <expan abbr="humidũ">humidum</expan> manens e&longs;&longs;e
<arrow.to.target n="marg8"></arrow.to.target> <expan abbr="&longs;ph&ecedil;ri-cũ">&longs;ph&ecedil;ri­<lb/>cum</expan>, cuius <expan abbr="c&etilde;trum">centrum</expan> e&longs;t <expan abbr="centrũ">centrum</expan> vniuer&longs;i. &longs;i ita que terra, & aqua ma <lb/><expan abbr="n&etilde;t">nent</expan>, <expan abbr="quie&longs;cũtquè">quie&longs;cuntquè</expan> circa <expan abbr="centrũ">centrum</expan> vniuer&longs;i, ergo <expan abbr="centrũ">centrum</expan> <expan abbr="mūdi">mundi</expan> <expan abbr="ip&longs;o-rũ">ip&longs;o­<lb/>rum</expan> &longs;imul <expan abbr="c&etilde;trũ">centrum</expan> grauitatis exi&longs;tit. atque adeo quatuorpr&ecedil;dicta <lb/>centra in <expan abbr="vnũ">vnum</expan> &longs;imul conueniunt punctum. Quod <expan abbr="aut&etilde;">autem</expan> tria &longs;i. <lb/>mul centra in vnum co<gap/>ant, &longs;atis <expan abbr="con&longs;picuū">con&longs;picuum</expan> e&longs;&longs;e poterit cuiquè
<pb pagenum="12"/>&longs;phæram aliquam, putà ligneam, vel al terius (&longs;imilaris <expan abbr="tam&etilde;">tamen</expan>) <lb/>naturæ intuenti; &longs;iquidem eius medium erit centrum magni­<lb/>tudinis, & centrum &longs;iguræ; idemquè punctum crit ip&longs;ius cen­<lb/>
<arrow.to.target n="marg9"></arrow.to.target> trum grauitatis; circa quod vndique partes æqueponderant. <lb/>& quoniam hæc &longs;phæra non e&longs;t in centro mundi; propterea <lb/>tria tantùm centra &longs;imul conuenient. &longs;i verò &longs;phçra non &longs;imi­<lb/>laris, &longs;ed di&longs;&longs;imilaris fuerit, veluti altera ip&longs;ius meditate plum­<lb/>bea, altera verò medietate lignea exi&longs;tente, tunc eius medium <lb/>erit quippe centrum magnitudinis, & figur&ecedil;, grauitatis verò <lb/>centrum nequaquam. Nam partes vndique circa medium æ­<lb/>queponderare non po&longs;&longs;ent; &longs;ed grauitatis centrum ad grauio­<lb/>rem partem, nimirum plumbeam declinabit. & hoc modo <lb/>duo tantùm centra inter &longs;e conuenient. vt etiam (modo ta­<lb/>men diuer&longs;o) accidit ellip&longs;i; cuius centrum e&longs;t centrum figu­<lb/>r&ecedil;, &longs;iquidem per ip&longs;um tran&longs;eunt diametri; idemquè <expan abbr="punctũ">punctum</expan> <lb/>
<arrow.to.target n="marg10"></arrow.to.target> e&longs;t ip&longs;ius centrum grauitatis. quod cùm non &longs;it propriè me­<lb/>dium figuræ, non erit quoque centrum magnitudinis. <expan abbr="mediū">medium</expan> <lb/>enim figuræ propriè circulo, ac &longs;phæræ tantùm competit. <lb/>Quare duo centra hoc quoque modo &longs;imul tantùm conue­<lb/>nient. In figura paraboles recta linea terminat&ecedil; centrum gra <lb/>
<arrow.to.target n="marg11"></arrow.to.target> uitatis intra figuram reperitur, quippè quod neque centrum <lb/>figuræ, neque centrum magnitudinis e&longs;&longs;e pote&longs;t. etenim in <lb/>hac figura non pote&longs;t dari medium, vnde neque centrum ma <lb/>gnitudinis dabitur, & quoniam in parabole diametri &longs;unt in <lb/>ter&longs;e &ecedil;quidi&longs;tantes, vt ex primo libro conicorum Apollonij <lb/>pergei con&longs;tat; neque etiam centrum figuræ dabitur. &longs;ic igi­<lb/>tur centra nullo modo conuenient. </s></p>
<p type="margin">
<s><margin.target id="marg7"></margin.target><emph type="italics"/>lib. de cælo<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg8"></margin.target><emph type="italics"/>lib. de iis <lb/>qu&ecedil; uehun <lb/>tur in aqua<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg9"></margin.target>16 <emph type="italics"/>Federi­<lb/>ci <expan abbr="cõm">comm</expan>. de <lb/>centro gra <lb/>uitatis &longs;oli <lb/>dorum.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg10"></margin.target>4. <emph type="italics"/>Fed. com <lb/>man. de cen <lb/>tro graui­<lb/>tatis &longs;olido <lb/>rum.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg11"></margin.target><emph type="italics"/>in &longs;ecundo <lb/>libro huius<emph.end type="italics"/></s></p>
<p type="main">
<s>Noui&longs;&longs;e quoque oportet centrum grauitatis communius <lb/>e&longs;&longs;e, in pluribu&longs;què reperiri, quàm centra magnitudinis, & fi­<lb/>guræ: centrum verò figuræ communius e&longs;&longs;e centro magnitu­<lb/>dinis. <expan abbr="Nã">Nam</expan> quodlibet corpus, & qu&ecedil;libet figura nece&longs;&longs;e e&longs;t, vt ha <lb/><expan abbr="beatc&etilde;trũ">beatcentrum</expan> grauitatis in trin&longs;ecùs, vel extrin&longs;ecùs. In trin&longs;ecùs vt <lb/><expan abbr="c&etilde;trũ">centrum</expan> grauitatis alicuius corporis regularis, quod e&longs;t in medio <lb/>figuræ, vel alicuius figuræ vt A; cuius centrum grauitatis &longs;it <lb/>in ambitu figuræ, vt in puncto B; extrin &longs;ecùs verò vt figura <lb/>C, cuius centrum grauitatis extrin&longs;ecus &longs;it, vt in D; quod <lb/>e&longs;t in telligendum, &longs;i graue C in centrum mundi ten deret,
<pb pagenum="13"/>tunc centrum D cum centro mundi <expan abbr="cõ-">con­<lb/></expan>
<arrow.to.target n="fig4"></arrow.to.target><lb/>ueniret; &longs;iguraquè C quie&longs;ceret circa cen <lb/>trum vniuer&longs;i, veluti &longs;e habetcirca <expan abbr="c&etilde;trum">centrum</expan> <lb/>D. partes enim figuræ talem po&longs;&longs;untha­<lb/>bere &longs;itum, vt inter&longs;e &ecedil;queponderare po&longs;­<lb/>&longs;int. vt ex &longs;ubiectis figuris per&longs;picuum e&longs;t. <lb/>& ad huc clariùs, &longs;i in telligatur figura, vt <lb/>E circulo tum exteriori, tum interiori ter <lb/>minata, cuius centrum grauitatis extra fi­<lb/>guram eritin F. quod quidem cum cir­<lb/>culorum centro conueniet. circa quod <lb/>(exi&longs;tente centro F in centro mundi) <lb/>partes vndique &ecedil;queponderabunt: cùm <lb/>omnes &ecedil;qualiter à centro grauitatis <expan abbr="di&longs;t&etilde;t">di&longs;tent</expan>. <lb/>præterea in hac figura E centrum graui­<lb/>tatis (quamuis &longs;it extra &longs;iguram) cum cen­<lb/>tro figuræ, <expan abbr="c&etilde;troquè">centroquè</expan> magnitudinis ip&longs;ius <lb/>figuræ conuenire, forta&longs;&longs;e non eritincon­<lb/>ueniens a&longs;&longs;erere. At verò figuræ AC nul <lb/>lo pacto figuræ, magnitudinisquè <expan abbr="centrũ">centrum</expan> <lb/>habebunt. & quamuis dictum &longs;it <expan abbr="centrũ">centrum</expan> <lb/>grauitatis corporum regularium e&longs;&longs;e me­<lb/>dium ip&longs;orum, non tamen propterea dicen dum e&longs;t, idem e&longs;&longs;e <lb/>centrum magnitudinis, atque figuræ, ni&longs;i impropriè; <expan abbr="mediũ">medium</expan> <lb/>enim his impropriè attribuitur, &longs;icuti etiam centrum figuræ; <lb/>cùm lineæ ex ip&longs;o prodeuntes non &longs;int ip&longs;orum corporum <lb/>(quatenus regularia &longs;unt) &longs;emidiametri. quare centrum gra­<lb/>uitatis reperiri pote&longs;t ab&longs;que alijs centris; at non è conuer&longs;o. <lb/>Rur&longs;us commune magis e&longs;t <expan abbr="c&etilde;trum">centrum</expan> figuræ centro magnitu­<lb/>dinis; quia præter circulum, & &longs;phæram, quæ tam figuræ, <expan abbr="quã">quam</expan> <lb/>magnitudinis centrum habent, nonnullæ figuræ &longs;uum ha­<lb/>bent figuræ centrum in ip&longs;is, & extra ip&longs;as; in ip&longs;is, vt ellip&longs;is, <lb/>cuius centrum in tùs habetur; &longs;emicirculus etiam, dimidia què <lb/>&longs;phæra centrum habent in limbo. extra figuram verò veluti <lb/>hyperbolæ centrum, quod extra figuram exi&longs;tit; vbi nempè <lb/>diametri concurrunt. Quæ quidem omnia &longs;unt figuræ cen­<lb/>tra; magnitudinis verò minimè. verùm obijciet hoc loco for
<pb pagenum="14"/>ta&longs;&longs;e qui&longs;piam, vel ambas, inquiens, centri grauitatis defini­<lb/>tiones allatas, diminutas e&longs;&longs;e; vel ijs, quæ modò à nobis de <expan abbr="c&etilde;">cem</expan> <lb/>tro grauitatis dicta &longs;unt, repugnare; cùm o&longs;tenderimus cen­<lb/>trum grauitatis aliquando e&longs;&longs;e, vel in ambitu figuræ, vel extra <lb/>figuram; definitiones verò allat&ecedil; &longs;emper &longs;upponunt illud e&longs;&longs;e <lb/>in ip&longs;is intra po&longs;it <expan abbr="ũ">um</expan>. <expan abbr="Cõfirmaturquè">Confirmaturquè</expan> difficultas, quandoqui­<lb/>dem, neque huiu&longs;modi centrum extra figuram con&longs;titutum, <lb/>fui&longs;&longs;e Archimedi pror&longs;usignotum, exi&longs;timare debemus; vt <lb/>colligere licet ex nono po&longs;tulato huius libri; cùm inquit. <lb/><emph type="italics"/>Omnis figuræ, cuius perimeter &longs;it ad eandem partem concauus, centrum <lb/>grauitatis intra ip&longs;am e&longs;&longs;e oportet.<emph.end type="italics"/> qua&longs;i non repugnet figur&ecedil; peri <lb/>metrum non ad eandem partem concauum habenti, extra <lb/>ip&longs;am grauitatis centrum obtinere. Cui obiectioni in hunc <lb/>modum occurri poterit, &longs;i dixerimus, quòd quamuis exempli <lb/>gratia in figura C dictum &longs;it centrum grauitatis D extra fi <lb/>guram exi&longs;tere, id ip&longs;um etiam intra figuram e&longs;&longs;e affirmati <lb/>poterit. &longs;iquidem ambitus figur&ecedil; C centrum D intra &longs;e <expan abbr="cõ">com</expan> <lb/>tinct; ita vt re&longs;pectu tötius &longs;it intra. idemquè dicen dum e&longs;t de <lb/>altera figura A. hoc autem euidenti&longs;&longs;imum e&longs;t in figura E. <lb/>& hic e&longs;t &longs;en&longs;us definitionum centri grauitatis. His itaque pri <lb/>mùm cognitis con&longs;ideranda e&longs;t intentio Archimedis in his li <lb/>bris, quç quidem vt plurimum à librorum in&longs;criptionibus e­<lb/>luce&longs;cere &longs;olet. </s></p>
<figure id="fig4"></figure>
<p type="head">
<s>DE SCOPO HORVM LIBR ORVM</s></p>
<p type="main">
<s>Si Archimedis propo&longs;itum in his libris ex ip&longs;a operis in­<lb/>&longs;criptione, vt in alijs quoque aliorum authorum volumini­<lb/>bus fieri vt plurimùm &longs;olet, inue&longs;tigandum erit, partim &longs;anè <lb/>con&longs;picuum illud e&longs;&longs;e videbitur, partim verò ignotum adeò, <lb/>vt potiùs nullius fermè rei &longs;e habiturum e&longs;&longs;e &longs;ermonem profi­<lb/>teatur Archimedes. quid enim (ob&longs;ecro) verbis illis &longs;ignificari <lb/>potuit, que primilibri initio ita &longs;e <expan abbr="hab&etilde;t">habent</expan>. A <foreign lang="greek"><10>ximh/dous e)w_iw_e/dwn i)so<10>­<lb/><10>o w_ixw_n, h_ ke(nt<10>a <32>a/<10>wn e)w_iw_e\dwn.</foreign> hoc e&longs;t. <emph type="italics"/>Archimedis planorum æquepon <lb/>derantium, pel centra grauitatum planornm.<emph.end type="italics"/> quando quidem vide­<lb/>tur Archimedes rem pror&longs;us <expan abbr="inutil&etilde;">inutilem</expan>, quinnimò natur&ecedil; repu­<lb/>gnantem &longs;ibi contemplandam proponere. dùm enim polli-
<pb pagenum="15"/>cetur&longs;e e&longs;&longs;e pertractaturum de planis æquæponderantibus, &longs;i <lb/>ue de centris grauitatum planorum; cùm ea, quæ æqueponde <lb/>rare debent, ponderare quoque oporteat; &longs;i plana æqueponde <lb/>rare <expan abbr="deb&etilde;t">debent</expan>, grauitate quadam illa prædita e&longs;&longs;e nece&longs;&longs;e e&longs;t. quod <lb/>valdè à planorum natura abhorret, cùm grauitas, nonn&longs;ii cor <lb/>poribus, neque tamen omnibus comperat. ip&longs;e tamen, dum <lb/>plana æqueponderantia, vel centra grauitatum planorum &longs;e <lb/>explicaturum pollicetur, apertè &longs;upponit plana, ac &longs;uperficies <lb/>graues exi&longs;tere, rem &longs;anè immaginariam pror&longs;us, ip&longs;iusquè rei <lb/>naturæ nuiiatenus re&longs;pondentem. ita vt Archimedes circa ea, <lb/>quæ omnino rei naturæ aduer&longs;antur, negotium &longs;ump&longs;i&longs;&longs;evi­<lb/>deatur. Verùm enimuero &longs;i Authoris <expan abbr="m&etilde;tem">mentem</expan> acuratiùs intuea <lb/>mur, rem planè egregiam, naturæquè rei apprimè con&longs;enta­<lb/>neam ip&longs;um pertra ctandam &longs;ump&longs;i&longs;&longs;e depræhendemus. Nam <lb/>quamuis plana, quatenus plana &longs;unt, nuiiam habeant graui­<lb/>tatem, non e&longs;t tamen à rei natura, neque à ratione alienum, <lb/>quin po&longs;&longs;imus planorum, &longs;uperficierum què centra grauitatis <lb/>depræhendere, ex quibus&longs;i &longs;u&longs;pendantur, planorum partes <lb/>vndiquè &ecedil;qualium momentorum confi&longs;ten tes maneant. <expan abbr="quã">quam</expan> <lb/>doquidem centrum grauitatis talis e&longs;t naturæ, vt &longs;i mente <expan abbr="cõ-cipiamus">con­<lb/>cipiamus</expan>, rem aliquam in eius centro grauitatis appen&longs;am e&longs;­<lb/>&longs;e, eo pror&longs;us modo, quo reperitur, quie&longs;cat, & maneat. vt <lb/>antea declarauimus. & quamuis re ip&longs;a, actùque plana <expan abbr="&longs;eorsũ">&longs;eorsum</expan> <lb/>à corporibus reperiri nequeant; in ip&longs;is tamen hæcip&longs;orum <lb/>circa centra grauitatis æqueponderatio ad actum facilè redigi <lb/>poterit. Vt &longs;it &longs;olidum AB pri&longs;­<lb/>
<arrow.to.target n="fig5"></arrow.to.target><lb/>ma, cui^{9} latera AE CF DB &longs;int <lb/>horizonti erecta, &longs;upetiorquè ba­<lb/>&longs;is ACD, quem ad modum & in­<lb/>ferior EFB &longs;it horizonti æquidi­<lb/>&longs;tans; &longs;it autem plani ACD cen­<lb/>trum grauitatis G, exquo G &longs;i <lb/>&longs;u&longs;pendatur totum AB patet <lb/>planum ACD horizonti æqui­<lb/>di&longs;tans permanere, ac plopterea <lb/>circa <expan abbr="c&etilde;trum">centrum</expan> grauitatis G æque­<lb/>ponderare. quod quidem, quamuis egeat demon&longs;tratione,
<pb pagenum="16"/>
<arrow.to.target n="marg12"></arrow.to.target> in præ&longs;entia omittatur; infraquè &longs;uo loco o&longs;ten den dum. &longs;at <lb/>autem nobis nunc &longs;it o&longs;tendi&longs;&longs;e, hæc ad praxim reduci, ma­<lb/>nibu&longs;què (vt dicitur.) contrectari po&longs;&longs;e. Quòd &longs;i hæc ita &longs;e ha <lb/>bent, huiu&longs;modi con&longs;ideratio non erit vana, neque vt inuti­<lb/>lis reijcienda. Sed vlteriùs adhuc progrediamur, dicamu&longs;­<lb/>què, quoniam planum ACD, quatenuse&longs;t corpori coniun­<lb/>ctum, horizonti æquidi&longs;tans permanere debet; &longs;i &longs;eor&longs;um à <lb/>corpore illud in telligamus, vt &longs;i ADC ex eius centro graui­<lb/>tatis G &longs;u&longs;pendatur, tunc quocunque modo reperiatur, hoc <lb/>e&longs;t &longs;iue horizonti &ecedil;quidi&longs;tans, &longs;iuè <lb/>minùs, idip&longs;um perman&longs;urum ni <lb/>
<arrow.to.target n="fig6"></arrow.to.target><lb/>hilominus in telligere po&longs;&longs;umus, <lb/>parte&longs;què vndique æqualium mo <lb/>men torum con&longs;i&longs;tentes. Neque <lb/>enim Ari&longs;to teles grauibus dunta­<lb/>xat, &longs;ed etiam leuibus momenta <lb/>tribuit, idip&longs;um què (vt Eutocius <lb/>in horum librorum comentarijs <lb/>refert) Ptolæmeo quoque placuit, vt habetur in líbro (à nobis <lb/>ramen de &longs;iderato) quem de momen tis &longs;crip&longs;it. Pr&ecedil;terea alij­<lb/>quoque Philo&longs;ophi id ip&longs;um &longs;en&longs;i&longs;&longs;evidentur. quod e&longs;t qui­<lb/>dem rationi con&longs;en taneum, &longs;uperuolant enim, quæ leuia &longs;unt, <lb/>& &longs;i mente concipiatur <expan abbr="ead&etilde;">eadem</expan> &longs;igura leuis cuiu&longs;piam e&longs;&longs;e, tunc <lb/>&longs;i detineatur in G, partes vndique &ecedil;qualium <expan abbr="momentorũ">momentorum</expan> <lb/>con&longs;i&longs;tent, e&longs;&longs;etquè G (vtita dicam) centrum leuitatis. Quo­<lb/>niam autem circa centrum grauitatis &ecedil;queponderationem <lb/>con&longs;ideramus, id circo plana, tanquam no bis apparentia gra­<lb/>uitatem habere, mente concipimus. Non e&longs;t igitur à ratio­<lb/>ne alienum, æqueponderantiam in planis, vt grauibus con&longs;i­<lb/>deratis intelligere, conciperequè. Nec quicquam nobis offi­<lb/>cit, quòd definitiones centri grauitatis priùs allatæ non pla­<lb/>norum, &longs;ed corporum centra explicarunt, ita vtgrauitatis <expan abbr="c&etilde;-trũ">cen­<lb/>trum</expan> ad corpora, <expan abbr="nõ">non</expan> ad plana &longs;it refe <gap/><expan abbr="ndũ">ndum</expan>. Hoc enim ideo fa <lb/><expan abbr="ctũ">ctum</expan> e&longs;t, quia propriè <expan abbr="centrũ">centrum</expan> grauitatis re&longs;picit corpora; non ta <lb/>men propterea impropriè re&longs;picit plana, &longs;ed quia primò re&longs;pi <lb/>cit corpora; in quib^{9} actu ine&longs;&longs;e <expan abbr="depræh&etilde;ditur">depræhenditur</expan>. propterea <expan abbr="e&ecedil;d&etilde;-met">e&ecedil;den­<lb/>met</expan> definitiones planis quoque in <expan abbr="hũc">hunc</expan> <expan abbr="modũ">modum</expan> aptari <expan abbr="poterũt">poterunt</expan>. </s></p>
<pb pagenum="17"/>
<p type="margin">
<s><margin.target id="marg12"></margin.target><emph type="italics"/>in fine pri­<lb/>milibri.<emph.end type="italics"/></s></p>
<figure id="fig5"></figure>
<figure id="fig6"></figure>
<p type="head">
<s>DEFINITIO CENTRI GRAVITATIS PLANORVM.</s></p>
<p type="main">
<s>Cen trum grauitatis vniu&longs;cuiu&longs;que plani e&longs;t punctum quod­<lb/>dam intra po&longs;itum, à quo &longs;i planum appen&longs;um mente con­<lb/>cipiatur, dum fertur, quie&longs;cit; & &longs;eruat eam, quam in princi­<lb/>pio habcbat po&longs;itionem, neque in ip&longs;a latione <expan abbr="circũuertitur">circunuertitur</expan>. </s></p>
<p type="head">
<s>EIVSDEM ALIA DEFINITIO.</s></p>
<p type="main">
<s>Centrum grauitatis vniu&longs;cuiu&longs;que plani e&longs;t punctum il­<lb/>lud intra po&longs;itum, circa quod vndique partes æqualium mo <lb/>mentorum con&longs;i&longs;tunt. &longs;i enim per tale centrum recta du­<lb/>catur linea figuram quomodocunque &longs;ecans, &longs;emper in par <lb/>tes<gap/>queponderantes ip&longs;am diuidet. </s></p>
<p type="main">
<s>Vtitaque in planis quoque centrum grauitatis con&longs;ide­<lb/>ratur, ita etiam plana grauitate prædita con&longs;iderare, non e­<lb/>rit ab&longs;urdum. &longs;i enim impo&longs;&longs;ibile e&longs;&longs;et con&longs;iderare plana gra <lb/>uitate prædita, centrum quoque grauitatis in ip&longs;is nullo mo­<lb/>do concipi po&longs;&longs;et; atque per&longs;picuum e&longs;t, centrum grauita tis in <lb/>ip&longs;is admitti, ac de&longs;ignari po&longs;&longs;e, igitur & plana grauitate in&longs;i <lb/>gnita. Et &longs;i mathematicus con&longs;i derat corpora &longs;eclu&longs;a interim <lb/>ip&longs;orum grauitate, & leuitate: & A&longs;tronomus corpora con&longs;i­<lb/>derans cæle&longs;tia, quæ neque grauia, neque leuia &longs;unt, non pro­<lb/>pterea <expan abbr="cõ&longs;iderat">con&longs;iderat</expan> ea ex propria <expan abbr="ip&longs;orũ">ip&longs;orum</expan> natura, neque grauia, ne <lb/>que leuia e&longs;&longs;e; etenim quamuis grauia, vel leuia e&longs;&longs;ent, nihilo <lb/>minus neque grauia, neque leuia e&longs;&longs;e ea con&longs;ideraret. quòd &longs;i <lb/>Mathematicus hoc pacto huiu&longs;modi corpora intelligere po­<lb/>te&longs;t; quid prohibet rur&longs;um <expan abbr="ead&etilde;">eadem</expan>, <expan abbr="quãuis">quanuis</expan> vt talia, neque grauia, <lb/>nequeleuia &longs;int; vel grauia, vel leuia e&longs;&longs;e concipere? <expan abbr="qu&etilde;ad-modum">quenad­<lb/>modum</expan> hoc quoque <expan abbr="ex&etilde;">exem</expan> <lb/>
<arrow.to.target n="fig7"></arrow.to.target><lb/>plo res magis eluce&longs;cet<gap/><lb/>veluti &longs;i intelligamus ex <lb/>AC appen&longs;a e&longs;&longs;e plana <lb/>DE, quæ &longs;int æqualia; &longs;u <lb/>&longs;pendaturquè AC in me <lb/>dio pror&longs;us in B; cur mente intelligere non po&longs;&longs;umus, quan <lb/><expan abbr="titat&etilde;">titatem</expan>, <expan abbr="&longs;paciũquè">&longs;paciunquè</expan> D <expan abbr="æquepõderare">æqueponderare</expan> &longs;pacio E; cùm &longs;int æqua <lb/>lia? <gap/> &longs;i planorum alterum, putà D, maius e&longs;&longs;etip&longs;o E; tunc
<pb pagenum="18"/>&longs;tatim non &longs;olùm &ecedil;queponderare non po&longs;&longs;e, verùm etiam pla <lb/>num D deor&longs;um tendere concipiemus. & hoc nulla alia de <lb/>cau&longs;a, quàm quòd cùm D maius &longs;it, quàm E, &longs;tatim <expan abbr="ipsũ">ipsum</expan> <lb/>D, quàm E grauius quoque e&longs;&longs;e concrpimus. Con&longs;iderare <lb/>igitur plana cum grauitate non e&longs;t omnino à ratione <expan abbr="alienũ">alienum</expan>. <lb/>Quare vtrum que titulum, nempe planorum æqueponderan <lb/>tium, vel centra grauita tis <expan abbr="planorũ">planorum</expan>, admittendum duximus. <lb/>Verùm quoniam Archimedes &longs;ecundum librum &longs;implicivo <lb/>cabulo, nimirum (qua&longs;i &longs;imul omnia complectens) <emph type="italics"/>œquepon­<lb/>derantium<emph.end type="italics"/> in &longs;crip&longs;it; idcirco tamprimum, quàm &longs;ecundum li <lb/>brum (æqueponderantium) in&longs;cribendum exi&longs;timamus. eo­<lb/>quèlibentiùs; quoniam ip&longs;emet Eutocius horum quoque li­<lb/>brorum explanator ho&longs;ce libros hoc tantùm nomine æque­<lb/>ponderantium nuncupauit: alijquè omnes, qui hos Archime <lb/>dis libros nominant; hoc titulo de æqueponderantibus nun <lb/>cupant. Præterea titulus hic magis operi congruere mihi vide <lb/>tur; quoniam nonnulla Archimedes in principio pertractat, <lb/>quæ tam &longs;olidis, quàm planis communia exi&longs;tunt; quamuis <lb/>cætera ad plana &longs;int tantu <expan abbr="refer&etilde;da">referenda</expan>. in quibus omnibus de re <lb/><expan abbr="admodũvtili">admodunvtili</expan>, & ad <expan abbr="quãplurima">quamplurima</expan> <expan abbr="cõdu&etilde;cti">conduencti</expan> pertractat. <expan abbr="quãdoqui">quandoqui</expan> <lb/><expan abbr="d&etilde;">dem</expan> ex ijs, quæ ab Archimede his libris docemur, in <expan abbr="multarũ">multarum</expan> <expan abbr="re-rũ">re­<lb/>rum</expan> <expan abbr="cognition&etilde;">cognitionem</expan> peruenire po&longs;&longs;umus. quod facilè con&longs;tat inpri <lb/>mis ip&longs;iu&longs;met Archimedis <expan abbr="ex&etilde;plo">exemplo</expan>. <expan abbr="&longs;iquid&etilde;">&longs;iquidem</expan> hac methodo ip&longs;e <lb/>in libro de quadratura paraboles <expan abbr="cõparãdo">comparando</expan> plana in libra <expan abbr="cõ">com</expan> <lb/>&longs;tituta, ip&longs;ius paraboles <expan abbr="quadraturã">quadraturam</expan> miro artificio adinuenit. <lb/>Deinceps ex cognitione <expan abbr="c&etilde;troiũ">centroium</expan> grauitatis planorum, nos in <lb/>cognitionem centrorum grauitatum &longs;olidorum deducimur. <lb/>Denique adeo pro&longs;icua e&longs;t hæc doctrina, quam nobis in his <lb/>libris Archimedes præ&longs;tat; vt affirmare non verear, nullum <lb/>e&longs;&longs;e Theorema, nullum què problema ad rem mechanicam <lb/>pertinens, quod in &longs;ui &longs;peculatione peculiare <expan abbr="nõ">non</expan> a&longs;&longs;umat fun <lb/><expan abbr="dam&etilde;tum">damentum</expan> ex ijs, quæ Archimedes in his libris edi&longs;&longs;erit. quem­<lb/>admodum (cæteris interim omi&longs;&longs;is) patet ex vulgata illa pro­<lb/>po&longs;itione enunciante, ita &longs;e habere pondusad pondus, vt di <lb/>&longs;tantia ad di&longs;tantiam permutatim &longs;e habet, ex quibus &longs;u&longs;pen <lb/>duntur. quæ præclari&longs;&longs;imè ab ip&longs;o in primo libro demon&longs;tra <lb/>tur. Et quamuis Iordanus Nemorarius (quem &longs;ecutus e&longs;t
<pb pagenum="19"/>Nicolaus Tartalea, & alij) in libello de ponderibus hanc <expan abbr="eã-dem">ean­<lb/>dem</expan> propo&longs;itionem quoque dem on&longs;trare conatus &longs;it; & ad <lb/><expan abbr="cã">cam</expan> o&longs;tendendam pluribus medijs fuerit v&longs;us; nulli tamen pro <lb/>bationi demon&longs;trationis nomen conuenire pote&longs;t. cùm vix <lb/>ex probabilibus, & ijs, quæ nullo modo nece&longs;&longs;itatem <expan abbr="afferũt">afferunt</expan>, <lb/>& forra&longs;&longs;e neque ex probabilibus &longs;uas componat rationes. <lb/>Cùm in mathematicis demon&longs;trationes requirantur exqui&longs;i­<lb/>ti&longs;&longs;imæ. acpropterea neque inter Mechanicos videtur mihi <lb/>Iordanus ille e&longs;&longs;e recen&longs;endus. Quapropter ad Archimedem <lb/>confugien dum e&longs;t, &longs;i fundamenta mechanica, veraquè huius <lb/>&longs;cientiæ principia perdi&longs;cere cupimus: qui (meoiudicio) ad <lb/>hoc poti&longs;&longs;imùm re&longs;pexit; vt elementa mechanica traderet. vt <lb/>ctiam Pappus in octauo Mathematicarum collectionum li­<lb/>bro &longs;entit; quod quidem ex diui&longs;ione, ac progre&longs;&longs;u horum li­<lb/>brorum facilè digno&longs;cetur. </s></p>
<figure id="fig7"></figure>
<p type="head">
<s>DE DIVISIONE HORVM LIBRORVM.</s></p>
<p type="main">
<s>Diuiditur enim in primis hic tractatus in duos libros diui­<lb/>&longs;us, in po&longs;tulata, & theoremata: theoremata verò &longs;ubdiui­<lb/>duntur in duas &longs;ectiones, quarum prima continet priora o­<lb/>cto theoremata; ad alteram verò reliqua theoremata <expan abbr="&longs;pectãt">&longs;pectant</expan>. <lb/>quæ quidem adhuc in alias duas partes diuidi pote&longs;t; nempè <lb/>in theoremata primo libro examina ta, & in ea, quæ &longs;ecun­<lb/>dus liber contemplatur. Hanc autem horum librorum con <lb/>&longs;tituimus diui&longs;ionem, quoniam imprimis Archimedes, (o­<lb/>mi&longs;&longs;is po&longs;tulatis, quæ primum locum obtinere debent) quæ­<lb/>dam tractauit communia in pricribus octo theorema tibus; <lb/>quorum &longs;copus e&longs;t inuenire fundamentum illud <expan abbr="præcipuũ">præcipuum</expan> <lb/>mechanicum, quòd &longs;cilicet ita &longs;e habet grauitas ad grauita­<lb/>tem, vt di&longs;tan tia ad di&longs;tantiam permutatim. ad quod demo <lb/>&longs;trandum quinque præmittit theoremata, quæ paulatim <lb/>deducunt nos in cognitionem demon&longs;tra tionis præfati fun <lb/>damenti. quo loco illud &longs;ummoperè notandum e&longs;t, nimi­<lb/>rum &longs;undamentum illud, nec non octo priora theorema­<lb/>ta communia e&longs;&longs;e tam planis, quàm &longs;olidis; atque promi&longs;­<lb/>cuè de vtri&longs;que <expan abbr="Archimed&etilde;">Archimedem</expan> demon&longs;trare. quòd &longs;i quis aliter
<pb pagenum="20"/>&longs;en&longs;erit, demon&longs;tratione&longs;què tan tùm de planis <expan abbr="cõcludere">concludere</expan> exi <lb/>&longs;timauerit, vel de &longs;olidis, non autem <expan abbr="quibu&longs;cũque">quibu&longs;cunque</expan>, &longs;ed vel de <lb/>rectilineis, vel de homogeneis tantùm, & de ijs, quæ inter&longs;e <lb/>&longs;unteiu&longs;dem &longs;peciei, longè aberrat à &longs;copo, & mente Archi­<lb/>medis. etenim in his &longs;emper loquitur. vel de grauibus &longs;impli <lb/>citer, veluti in primis tribus theorematibus; vel de magnitu <lb/>dinibus, vt in reliquis quinque quod quidem nomen tam <lb/>planis, quàm &longs;olidis quibu&longs;cunque e&longs;t <expan abbr="cõmune">commune</expan>, vt etiam ij, <lb/>qui parùm in Mathematicis ver&longs;ati &longs;unt, &longs;atis norunt. ficu­<lb/>ti etiam Euclides, dum quinti libri propo&longs;itiones pertracta­<lb/>uit, quantitatem continuam &longs;ub nomine magnitudinis <expan abbr="cõ">com</expan> <lb/>prehendit. quòd <expan abbr="aut&etilde;">autem</expan> nomen grauis &longs;it <expan abbr="cõmune">commune</expan>, iam &longs;atis <lb/>per &longs;e con&longs;tat. Per&longs;picuum e&longs;t igitur priora hæc octo Theo <lb/>remata communia e&longs;&longs;e, tam planis, quàm &longs;olidis. ac non &longs;o­<lb/>lùm &longs;olidis eiu&longs;dem &longs;peciei, & homogeneis, verùm etiam &longs;oli <lb/>dis diuer&longs;æ &longs;peciei, & hçterogeneis, vt&longs;uo loco manife&longs;tum <lb/>fiet. Iactoquè hoc fundamento, quod Archimedes in duob^{9} <lb/>propo&longs;itionibus, &longs;exta nempè, & &longs;eptima demon&longs;trauit; in o­<lb/>ctaua tanquam corrollarium colligit. Deinceps peculiariter <lb/>pertractat de centro grauitatis planorum, nec amplius plana <lb/>nominat magnitudinis nomine, &longs;ed proprijs cuiu&longs;cun que <lb/>nominibus; vt parallelogrammi, trianguli, & aliorum huiu&longs;­<lb/>modi. & in hac parte de&longs;cendit ad particularia. quippè cùm <lb/>& &longs;i non actu forta&longs;&longs;e, virture tamen cuiu&longs;libet particularis <lb/>plani centrum grauitatis nos doceat. in primo enim libro <lb/>&longs;at &longs;i bi vi&longs;um e&longs;t o&longs;tendi&longs;&longs;e centra grauitatum trianguloru, <lb/>ac parallelogrammorum, ex quibus cæterarum &longs;igurarum, <lb/>veluti pen tagoni, hexagoni, & aliorum &longs;imilium centra gra­<lb/>uita tis inue&longs;tigare non admodum erit difficile. &longs;iquidem hu <lb/>iu&longs;modi plana in triangula diuiduntur. vt in &longs;ine primi li­<lb/>bri attingemus. In &longs;ecundo autem libro altiùs &longs;e extollit, & <lb/>moro &longs;uo circa &longs;ubtili&longs;&longs;ima theoremata ver&longs;atur; nompè cir <lb/>ca centrum grauitatis conice &longs;ectionis, quæ parabole nun­<lb/>cupatur. nonnullaquè præmittit theorema ta, quæ &longs;unt tan­<lb/>quam præuie di&longs;po&longs;itiones ad inue&longs;tigandam demon&longs;tra­<lb/>tionem centri grauitatis in parabole. Itaque per&longs;picuum e&longs;t, <lb/>Archimedem propriè elementa mechanica tradere. quando-
<pb pagenum="21"/>quidem duo pertractat, quæ &longs;unt tanquam elementa huius <lb/>&longs;cientiæ. fundamentum nempè illud præ&longs;tanti&longs;&longs;imum iam <lb/>to ties præfatum, deinde centra grauitatis planorum o&longs;tendit. <lb/>& quamuis hi duo Archimedis libelli pauca continerevidean <lb/>tur, non tamen pauca docui&longs;&longs;e Archimedem exi&longs;timandum <lb/>e&longs;t. multa enim &longs;unt mole exigua, quæ tamen virtute maxima <lb/>habentur. quod planè Archimedis &longs;criptis accidit; hi&longs;què pr&ecedil; <lb/>&longs;ertim, ex quibus patet aditus ad multa, ac penè in&longs;inita theo­<lb/>remata, problemataquè mechanica. nihil enim in hoc gene­<lb/>re demon&longs;trari pote&longs;t, quod his non indigeat &longs;criptis. & <lb/>quod admirabilius e&longs;t, nos non &longs;olùm pro fundamento &longs;u­<lb/>&longs;cipere po&longs;&longs;e ad aliquod demon&longs;trandum theoremata in his <lb/>libris demon&longs;trata, verùm etiam ab his demon&longs;trationibus <lb/>perdi&longs;cerere ip&longs;um modum argumentandi, & demon&longs;trandi; <lb/>vt&longs;uis locis o&longs;tendemus. ita vt verè concludendum &longs;it, nemi­<lb/>nem pror&longs;us inter mechanicos connumerandum fore, qui <lb/>hæc Archimedis &longs;cripta ignorat. ignoratis enim principijs <lb/>nulla e&longs;t &longs;cientia, vt apud omnes &longs;apientes per&longs;picuum e&longs;t. <lb/>Ip&longs;um igitur Archimedem audiamus, eiu&longs;què &longs;cripta diligen <lb/>ti&longs;&longs;imè perpendamus. </s></p>
<pb pagenum="23"/>
<p type="head">
<s>GVIDIVBALDI <lb/>EMARCHIONIBVS <lb/>MONTIS. <lb/>IN PRIMVM ARCHIMEDIS <lb/>AEQVEPONDERANTIVM <lb/>LIBRVM <lb/>PARAPHRASIS <lb/>SCHOLIIS ILLVSTRATA.</s></p>
<p type="head">
<s>Archimedis tamen huius primi libri <lb/>titulus &longs;ic &longs;e habet.</s></p>
<p type="head">
<s><emph type="italics"/>ARCHIMEDIS PLANORVM AEQVEPONDERANTIVM, <lb/>VEL CENTRA GRAVIT ATVM PLANORVM.<emph.end type="italics"/></s></p>
<figure></figure>
<p type="head">
<s>AR CHIMEDIS POSTVLATA.</s></p>
<p type="head">
<s>I.</s></p>
<p type="main">
<s>Grauia æqualia ex æqualibus di&longs;tantijs æque­<lb/>ponderare. </s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Dvobvs modis grauia in di&longs;tantijs <lb/>collocata in telligi po&longs;&longs;unt. quod & <lb/>in cæteris po&longs;tulatis, & in propo&longs;i­<lb/>tionibus intelligendum e&longs;t. etenim <lb/>vel grauia <expan abbr="sũt">sunt</expan> appen&longs;a, vtin prima &longs;i­<lb/>gura æqualia grauia AB &longs;untin CD <lb/>appen&longs;a; ita vt di&longs;tantia EC &longs;it di­<lb/>&longs;tatiæ ED æqualis. intelligaturquè <lb/>CD tanquam libra, quæ &longs;u&longs;pendatur <lb/>in E. vel vtin &longs;ecunda figura grauia AB habent ip&longs;orum <lb/>centra grauitatis, quæ &longs;int CD, in ip&longs;a DC linea, in pun-
<pb pagenum="24"/>ctis <expan abbr="n&etilde;pè">nempè</expan> CD <lb/>
<arrow.to.target n="fig8"></arrow.to.target><lb/>con&longs;tituta. li­<lb/>braquè &longs;imili­<lb/>ter ex puncto <lb/>E &longs;u&longs;pendatur; <lb/>&longs;itquè di&longs;tátia <lb/>EC di&longs;tantiæ <lb/>ED æqualis. <lb/><expan abbr="erũt">erunt</expan> vtique in <lb/>vtraque figura <lb/>pondera AB <lb/>in di&longs;tantijs &ecedil;­<lb/>qualibus con­<lb/>&longs;tituta. ac pro­<lb/>pterea æquepondera bunt, atque manebunt. nulla enim ratio <lb/>afferri pote&longs;t, cur ex parte A, vel ex parte B deor&longs;um, vel &longs;ur <lb/>&longs;um fieri debeat motus; cùm omnia &longs;int paria. ea verò æque­<lb/>ponderare debere, aliqua ratione manife&longs;tari pote&longs;t ex eo, <lb/>quod o&longs;ten&longs;um e&longs;t à nobis in no&longs;tro mechanicorum libro, <lb/>tractatu de libra: quod quidem ab Ari&longs;to tele quoque in prin <lb/>cipio quæ&longs;tionum mechanicarum elici pote&longs;t: idem &longs;cilicet <lb/>pondus longius a centro grauius e&longs;&longs;e eodem pondere ip&longs;i cen <lb/>tro propinquiori. Vnde &longs;i duo e&longs;&longs;ent pondera æqualia alte­<lb/>rum altero propinquius centro, quod remotius e&longs;t, grauius al <lb/>tero appareret. &longs;i igitur grauia æqualia à centro æqualiter di­<lb/>&longs;tabunt, æque grauia erunt. ac propterea æqueponderabunt. <lb/>quod quidem &longs;upponit Archimedes. Punctum autem illud, <lb/>quod Archimedes accipit, vnde &longs;umuntur di&longs;tantiæ, ex qui­<lb/>bus grauia &longs;u&longs;penduntur, veluti punctum E, Ari&longs;toteles cen <lb/>rum appellat. & hæc quidem æqueponderatio tam ponderi­<lb/>bus in libra appen&longs;is, quàm in ip&longs;a (vt dictum e&longs;t) con&longs;titutis <lb/>competit: dummodo ea, quibus appenduntur pondera, libe­<lb/>re &longs;emper in centrum mundi tendere po&longs;&longs;int. vtroque enim <lb/>modo in punctis CD grauitant, vt diximus etiam in eodem <lb/>uactatu de libra. Noui&longs;&longs;e tamen oportet Archimedem in his <lb/>libris potiùs in tellexi&longs;&longs;e pondera e&longs;&longs;e in di&longs;tantijs collocata, vt <lb/>in &longs;ecunda &longs;igura, quàm appen&longs;a; vt ex quarta, & quinta
<pb pagenum="25"/>primi libri propo&longs;itione pater. demon&longs;trationes enim cla­<lb/>riores redduntur. </s></p>
<figure id="fig8"></figure>
<figure></figure>
<p type="main">
<s>Porrò non ignoran <lb/>dum hoc Archimedis <lb/>po&longs;tulatum verificari <lb/>deponderibus quocun <lb/>que &longs;itu di&longs;po&longs;itis, &longs;iuc <lb/>CED fuerit horizonti <lb/><expan abbr="æquidi&longs;tãs">æquidi&longs;tans</expan>, &longs;iuè minùs; <lb/>vtin hac prima figura, <lb/>codem modo femper <lb/>verum e&longs;&longs;e pondera æ­<lb/>qualia CD ex &ecedil;quali­<lb/>bus di&longs;tantijs EC ED <lb/>æqueponderare, vt in­<lb/>fra (po&longs;t &longs;cilicet <expan abbr="quartã">quartam</expan> <lb/>huius propo&longs;itionem) <lb/>per&longs;picuum erit. Qua­<lb/>re cùm Archimedes <expan abbr="tã">tam</expan> <lb/>in hoc po&longs;tulato, <expan abbr="quã">quam</expan> <lb/>in &longs;equentibus, &longs;uppo­<lb/>nit pondera in di&longs;tan­<lb/>tijs e&longs;&longs;e collocata, intel­<lb/>ligendum e&longs;t <expan abbr="di&longs;tãtias">di&longs;tantias</expan> <lb/>ex vtraque parte in ea­<lb/>dem recta linea exi&longs;te­<lb/>re. Nam &longs;i (vtin &longs;ecun <lb/>da figura) <expan abbr="di&longs;tãtia">di&longs;tantia</expan> AB <lb/>fuerit &ecedil;qualis di&longs;tanti&ecedil; BC, quæ non indirectum iaceant, <lb/>&longs;ed angulum con&longs;tituant; tunc pondera AB, quamuis &longs;int <lb/>&ecedil;qualia, non &ecedil;queponderabunt. ni&longs;i quando (vt in tertia fi­<lb/>gura) iuncta AC, bifariamquè diui&longs;a in D, ductaquè BD, <lb/>fuerit h&ecedil;c horizonti perpendicularis, vt in eodem tractatu <lb/>no&longs;tro expo&longs;uimus. Di&longs;tantias igitur in eadem recta linea <lb/>&longs;emper exi&longs;tere intelligendum e&longs;t. vt ex demon&longs;trationibus <lb/>Archimedis per&longs;picuum e&longs;t. </s></p>
<pb pagenum="26"/>
<p type="head">
<s>II.</s></p>
<p type="main">
<s>Aequalia verò grauia ex inæqualibus di&longs;tátijs <lb/>non æqu eponderare, &longs;ed præponderare ad gra­<lb/>ue ex maiori di&longs;tantia. </s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Si enim <expan abbr="di&longs;tã">di&longs;tam</expan> <lb/>
<arrow.to.target n="fig9"></arrow.to.target><lb/>tia EC maior <lb/>fuerit di&longs;tantia <lb/>ED, grauibus <lb/>AB &longs;imiliter æ­<lb/>qualibus <expan abbr="exi&longs;t&etilde;">exi&longs;tem</expan> <lb/>tibus, & in CD po&longs;itis, tunc concedendum videtur graue A <lb/>præponderareip&longs;i B, quandoquidem EC longior e&longs;t, quàm <lb/>ED. &longs;upponit autem Archimedes hoc po&longs;tulatum reipiciens <lb/>forta&longs;&longs;e ad ea, quæ Ari&longs;toteles in principio quæ&longs;tionum me­<lb/>chanicarum o&longs;tendit, vbi colligit Ari&longs;toteles idem pondus ce­<lb/>leriùs ferri, quò magis à centro di&longs;tat, vel quod idem e&longs;t, duo <lb/>pondera æqualia inæqualiter à centro di&longs;tantia, quod magis <lb/>di&longs;tat, celeriùs ferri. quod autem æqualium ponderum cele­<lb/>riùs fertur, grauius exi&longs;tit; erit igitur A grauius, quàm B. <lb/>quia EC longior e&longs;t, quàm ED. Nos quoque (vt diximus) <lb/>in libro no&longs;trorum Mechanicorum tractatu de libra, alijs <lb/>quoque rationibus o&longs;tendimus, quo pondus e&longs;t in longiori <lb/>di&longs;tantia grauius e&longs;&longs;e. ex quibus &longs;equitur propter longiorem <lb/>di&longs;tantiam EC pondus A præponderare ponderi B. acpro­<lb/>pterea deor&longs;um ferri. </s></p>
<figure id="fig9"></figure>
<p type="head">
<s>III.</s></p>
<p type="main">
<s>Grauibus ex aliquibus di&longs;tantijs æqueponderá <lb/>tibus, &longs;ialteri grauium aliquid adijciatur, non æ­<lb/>queponderare; &longs;ed ad graue, cui adiectum fuit, <lb/>deor&longs;um ferri. </s></p>
<pb pagenum="27"/>
<p type="head">
<s>SCHOLIVM</s></p>
<p type="main">
<s>Grauia enim <lb/>
<arrow.to.target n="fig10"></arrow.to.target><lb/>AB &longs;iuè æqua­<lb/>lia, &longs;iue in &ecedil;qua <lb/>lia æqueponde <lb/>rent ex di&longs;tan­<lb/>tijs AC CB, al­<lb/>teri verò gra­<lb/>uium, putà B, <lb/>adijciatur pon <lb/>dus D. per&longs;picuum e&longs;t pondera BD &longs;imul magis ponderare, <lb/>quàm A. &longs;i enim B &ecedil;queponderat ip&longs;i A; erit pondus B in <lb/>hoc &longs;itu æquegraue, vt A: pondera igitur BD in hoc &longs;itu <expan abbr="nõ">non</expan> <lb/>erunt æquegrauia, vt pondus A. &longs;ed grauiora exi&longs;tent, quàm <lb/>A. quare BD deor&longs;um tendent. </s></p>
<figure id="fig10"></figure>
<p type="head">
<s>IIII.</s></p>
<p type="main">
<s>Similiter autem, &longs;i ab altero grauium auferatur <lb/>aliquid, non æqueponderare; verùm ad graue, à <lb/>quo nil ablatum e&longs;t, deor&longs;um tendere. </s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Aequeponderent grauia BD &longs;imul, & A <expan abbr="&longs;ecundũm">&longs;ecundumm</expan>
<arrow.to.target n="marg13"></arrow.to.target> di­<lb/>&longs;tantias CB CA; vtin eadem figura, & ab altero eorum, putà <lb/>BD, auferatur D, remanenbunt grauia BA; eritquè A gra­<lb/>uius ip&longs;o B. Nam &longs;i BD &longs;imul æqueponderant ip&longs;i A, B <lb/>tantùm eidem A non æqueponderabit, &longs;edleuius erit. vnde <lb/>&longs;equitur ex parte A motum fieri deor&longs;um. </s></p>
<pb pagenum="28"/>
<p type="margin">
<s><margin.target id="marg13"></margin.target><emph type="italics"/>eadem figu <lb/>ra.<emph.end type="italics"/></s></p>
<p type="head">
<s>V</s></p>
<p type="main">
<s>Aequalibus, &longs;imilibu&longs;què figuris planis inter&longs;e <lb/>coaptatis, centra quoque grauitatum inter &longs;e coa­<lb/>ptati oportet. </s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Aequales, <expan abbr="&longs;imiles&qacute;">&longs;imilesque</expan>; &longs;int <lb/>
<arrow.to.target n="fig11"></arrow.to.target><lb/>figuræ ABC DEF, qua­<lb/>rum centra grauitatis &longs;int <lb/>GH; &longs;i ABC &longs;uperpona­<lb/>tur ip&longs;i DEF, & hoc <expan abbr="&longs;ecũ">&longs;ecum</expan> <lb/>dùm laterum <expan abbr="æqualitat&etilde;">æqualitatem</expan>, <lb/>hoc e&longs;t &longs;i latus AB fuerit <lb/>æquale lateri DE, tunc <lb/>ponatur AB &longs;uper DE; &longs;imiliter AC &longs;uper DF, & BC &longs;uper <lb/>EF; tunc manife&longs;tum e&longs;t centrum grauitatis G &longs;uper centro <lb/>grauitatis H ad unguem conuenire; ita vt &longs;int vnum tan <expan abbr="tũ">tum</expan> <lb/>punctum. Plana enim quæ &longs;e inuicem contingunt, non ef­<lb/>ficiunt, ni&longs;i vnum tantùm planum. Solius autem figuræ ex <lb/>planis ABC DEF inuicen coaptatis, vnum tantùm erit cen <lb/>trum grauitatis, vt nos in no&longs;tro mechanicorum libro &longs;up­<lb/>po&longs;uimus; centra igitur grauitatis inter&longs;e&longs;e conuenire nece&longs;­<lb/>&longs;e e&longs;t. &longs;i enim centra grauitatis inter &longs;e non conuenirent, v­<lb/>na tantùm figura duo po&longs;&longs;et centra grauitatis habere. quod <lb/>e&longs;&longs;et omnino <expan abbr="incõueniens">inconueniens</expan>. Dixit autem Archimedes oporte <lb/>re has figuras e&longs;&longs;e &longs;imiles, & æquales, nam figuræ æquales, <lb/>&longs;ed non &longs;imiles, item &longs;imiles, & <expan abbr="nõ">non</expan> æquales e&longs;&longs;e po&longs;&longs;unt. qua­<lb/>re, vtinter&longs;e&longs;e coaptari po&longs;&longs;int, & &longs;imiles, & æquales e&longs;&longs;e ne­<lb/>ce&longs;&longs;e e&longs;t. </s></p>
<figure id="fig11"></figure>
<p type="head">
<s>VI</s></p>
<p type="main">
<s>Inæ qualium autem, &longs;ed &longs;imilium centra graui­<lb/>tatum e&longs;&longs;e &longs;imiliter po&longs;ita. </s></p>
<pb pagenum="29"/>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Inæquales &longs;int figuræ, &longs;i­<lb/>
<arrow.to.target n="fig12"></arrow.to.target><lb/>miles verò ABCD EFGH, <lb/>quarum cétra grauitatis &longs;int <lb/>KL. &longs;upponit Archimedes <lb/>h&ecedil;e grauitatis centra KL e&longs;­<lb/>&longs;e in figuris ABCD EFGH <lb/>&longs;imiliter po&longs;ita. <expan abbr="cũm">cumm</expan> enim <lb/>&longs;imilium figurarum, & late­<lb/>ra, & &longs;pacia &longs;int &longs;imilia, nece&longs;&longs;e e&longs;t in ip&longs;is &longs;imili quo que mo­<lb/>do centra grauitatis e&longs;&longs;e po&longs;ita. vt in &longs;equenti clariùs apparebit. <lb/>quomodo autem Archimedes intelligathanc po&longs;itionis &longs;imi­<lb/>litudinem, hoc modo definit. </s></p>
<figure id="fig12"></figure>
<p type="head">
<s>VII.</s></p>
<p type="main">
<s>Dicimus quidem puncta in &longs;imilibus figuris e&longs;­<lb/>&longs;e &longs;imiliter po&longs;ita, à quibus ad æquales angulos <lb/>ductæ rectæ lineæ cum homologis lateribus angu <lb/>los æquales efficiunt. </s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>In &longs;imilibus figuris ABCD EFGH &longs;int homologa latera <lb/>AB EF, BCFG, CD GH, AD EH. anguli verò æquales, qui <lb/>ad AE, BF, CG, DH, primum quidem o&longs;tendendum e&longs;t fie <lb/>ri po&longs;&longs;e, ut à duobus punctis intra figuras con&longs;titutis, duci <lb/>po&longs;&longs;int rect&ecedil; line&ecedil; ad angulos æquales, qu&ecedil; cum lateribus an­<lb/>gulos &ecedil;quales efficiant. Qua&longs;i dicat Archimedes, quoniam <lb/>&longs;upponere po&longs;&longs;umus puncta in &longs;imilibus figuris e&longs;&longs;e &longs;imiliter <lb/>po&longs;ita, ideo &longs;upponere quoque po&longs;&longs;umus centra grauiratisin <lb/>ip&longs;is e&longs;&longs;e &longs;imiliter po&longs;ita. Itaque &longs;int figuræ ABCD EFGH &longs;i­<lb/>miles, vt dictum e&longs;t, &longs;umaturquè in ABCD vteumque pun­<lb/>ctum K à quo ducatur KA KB KC KD. deinde fiatan
<pb pagenum="30"/>
<arrow.to.target n="fig13"></arrow.to.target><lb/>gulus FEL angulo BAK æqualis; & EFL ip&longs;i ABK. Iun <lb/>ganturquè GL LH. Dico L e&longs;&longs;e &longs;imiliter po&longs;itum, vt K. <lb/>Quoniam enim anguli BAK ABK &longs;unt angulis FEL EFL <lb/>æquales, erit reliquus BKA ip&longs;i FLE æqualis, eritquè ob &longs;i­<lb/>
<arrow.to.target n="marg14"></arrow.to.target> militudinem triangulorum KA ad AB, vt LE ad EF. e&longs;t <lb/>verò AB ad AD, vt EF ad EH propter &longs;imilitudinem fi­<lb/>
<arrow.to.target n="marg15"></arrow.to.target> gurarum, erit igitur ex æquali AK ad AD, vt LE ad EH, <lb/>& quoniam angulus BAD angulo FEH e&longs;t æqualis, & BAK <lb/>ip&longs;i FEL æqualis; erit & reliquus angulus KAD angulo <lb/>
<arrow.to.target n="marg16"></arrow.to.target> LEH æqualis. Quare triangulum KAD triangulo LEH &longs;i <lb/>mile exi&longs;tit, eodemquè modo o&longs;tendetur BKG &longs;imile e&longs;&longs;e <lb/>FLG, & KCD ip&longs;i LGH. ex quibus con&longs;tat angulos KBC <lb/>LFG, KCB LGF, & huiu&longs;modi reliquos reliquis æquales e&longs;&longs;e. <lb/>& ob id puncta KL in figuris ABCD EFGH e&longs;&longs;e &longs;imili­<lb/>ter po&longs;ita. </s></p>
<p type="margin">
<s><margin.target id="marg14"></margin.target>4 <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg15"></margin.target>22 <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg16"></margin.target>6 <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<figure id="fig13"></figure>
<p type="main">
<s>Itaque demon&longs;trato dari po&longs;&longs;e puncta in figuris fimiliter <lb/>po&longs;ita, potuit &longs;anè Archimedes antecedens po&longs;tulatum &longs;up­<lb/>ponere, nempè inæqualium, &longs;ed &longs;imilium figurarum centra <lb/>grauitatis e&longs;&longs;e &longs;imiliter po&longs;ita. quod quidem po&longs;tulatum e&longs;t <lb/>rationivalde con&longs;entaneum. ex dictis enim (&longs;uppo&longs;itis KL <lb/>centris grauitatum) triangulum ABK triangulo EFL &longs;imi­<lb/>
<arrow.to.target n="marg17"></arrow.to.target> le exi&longs;tit; veluti BKC ip&longs;i FLG. & reliqua reliquis. Quarevt <lb/>AK ad KB, &longs;ic EL ad LF, ac permutando vt AK ad EL, <lb/>ita BK ad FL. &longs;imiliter o&longs;tendetur ita e&longs;&longs;e BK ad FL, vt <lb/>KC ad LG, & KD ad LH. quare centra grauitatis KL
<pb pagenum="31"/>proportionaliter ab angulis di&longs;tant. </s></p>
<p type="margin">
<s><margin.target id="marg17"></margin.target>4 <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>16 <emph type="italics"/>quinti<emph.end type="italics"/></s></p>
<p type="main">
<s><expan abbr="Ducãtur">Ducantur</expan> pr&ecedil;terea à punctis KL ad latera perpendiculares <lb/>KM KN KO KP, LQ LR LS LT. & quoniam anguli <lb/>KMA LQE &longs;unt recti, ac propterea æquales, & KAM LEQ <lb/>&longs;unt æquales, ut o&longs;ten&longs;um e&longs;t; erit reliquus MKA reliquo <lb/>QLE &ecedil;qualis, triangulumquè AKM triangulo ELQ &longs;imile. <lb/>vtigitur AK ad KM; &longs;ic EL ad <expan abbr="Lq.">Lque</expan> & permutando AK
<arrow.to.target n="marg18"></arrow.to.target><lb/>ad EL, vt KM ad <expan abbr="Lq.">Lque</expan> pariquè ratione o&longs;tendetur triangu <lb/>lum BKM triangulo FLQ &longs;imile exi&longs;tere; e&longs;&longs;equè BK ad <lb/>FL, vt KM ad <expan abbr="Lq.">Lque</expan> &longs;imiliterquè in alijs triangulis o&longs;ten­<lb/>detur, ita e&longs;&longs;e Bk ad FL, vt KN ad LR; & Ck ad GL e&longs;&longs;e, vt <lb/>kO ad LS; atque kD ad LH, vt kP ad LT. quia verò AK <lb/>EL, Bk FL, Ck GL, Dk HL in eadem &longs;untproportione, vt <lb/>proximè demon&longs;tratum fuit; in eadem quoque proportione <lb/>erit kM ad LQ, & KN ad LR; & KO ad LS, atque kP ad <lb/>LT. ex quibus &longs;equitur centra grauitatis KL, non &longs;olùm ab <lb/>angulis in cadem proportione di&longs;tare; verùm etiam à lateri­<lb/>ribus in eadem quoque proportione di&longs;tare. Itaque cognito, <lb/>quomodo intelligar Archimedes centra grauitatis in &longs;imili­<lb/>bus figuris e&longs;&longs;e &longs;imiliter po&longs;ita; nunc con&longs;iderandum e&longs;t præ <lb/>cedens po&longs;tulatum, quatenus nimirum oporteat grauitatis <expan abbr="c&etilde;">cem</expan> <lb/>tra in &longs;imilibus figuris &longs;imiliter e&longs;&longs;e con&longs;tituta. Nam inti­<lb/>miùs con&longs;iderando hanc &longs;imilem horum grauitatis <expan abbr="centrorũ">centrorum</expan> <lb/>po&longs;itionem, congruum, & nece&longs;&longs;arium videtur, &longs;imiles &longs;igu­<lb/>ras &longs;ecundùm eandem proportionem e&longs;&longs;e æquepon <expan abbr="derãtes">derantes</expan>; <lb/>eademquè ratione (ob earum &longs;imilitudinem) circa grauita­<lb/>tis centra æqueponderare, veluti &longs;i figuræ: AC EG (quarum <lb/>centra grauitatis &longs;int KL) à rectis lineis PN TR vrcumquè <lb/>diuidantur, quæ percentra KL tran&longs;eant; dummodo in figu <lb/>ris &longs;int &longs;imiliter ductæ; hoc e&longs;t, vellatera, vel angulos in <expan abbr="ead&etilde;">eadem</expan> <lb/>proportione di&longs;pe&longs;cant: vt &longs;it AP ad PD, vt ET ad TH. æ­<lb/>queponderabunt vtique partes PABN PNCD, veluti partes <lb/>TEFR TRGH. & hæc non e&longs;t &longs;implex æqueponderatio; ve­<lb/>rùm etiam (vtita dicam) &longs;imilis, & æqualis æqueponderatio. <lb/>cùm &longs;it &longs;ecundùm eandem proportionem, quandoquidem <lb/>e&longs;t PB ip&longs;i TF &longs;imilis, cùm triangula AKB ELF, AKP ELT, <lb/>BKN FLR, &longs;intinter&longs;e &longs;imilia, quæ quidem efficiunt, figuras
<pb pagenum="32"/>PB TF inter&longs;e &longs;imiles e&longs;&longs;e. ob eademquè cau&longs;am e&longs;t PC &longs;i­<lb/>milis TG. quod quidem ex dem on&longs;tratis etiam facilè con­<lb/>&longs;tat. cùm anguli &longs;int &ecedil;quales, & latera proportionalia. Vtau­<lb/>tem clariùs intelligatur hæc &longs;imilis, & æqualis æquepondera <lb/>rio, adducerelibuit nonnulla ex ijs, quæ po&longs;teriùs tractanda <lb/>&longs;umentur. Itaque intelligatur punctum V centrum e&longs;&longs;e gra­<lb/>
<arrow.to.target n="fig14"></arrow.to.target><lb/>uitatis figuræ PB, X verò centrum grauitatis figure TF. &longs;i <lb/>militer punctum Y centrum e&longs;&longs;e grauitatis figuræ PC, Z <lb/>verò figur&ecedil; TG. Iunganturquè VY XZ. quæ quidem per <lb/>centra grauitatis KL tran&longs;ibunt. quòd ex ijs, qu&ecedil; dicenda <lb/>&longs;unt, manife&longs;tum erit, percipuèque ex octaua proportione <lb/>primi huius. quod tamen interim &longs;upponatur. At verò quo­<lb/>niam PB PC &ecedil;queponderant &longs;ecundùm proportionem, <lb/>quam habet YK ad KV; TF verò & TG &ecedil;queponderant <lb/>&longs;ecundùm proportionem, quam habet ZL ad LX. e&longs;t. n. <lb/>ac &longs;i AN e&longs;&longs;et appen&longs;a in V, & PC in Y; ER in X, & <lb/>TG in Z. vt in &longs;equentibus manife&longs;ta erunt. Atverò quo­<lb/>
<arrow.to.target n="marg19"></arrow.to.target> niam AN &longs;imilis e&longs;t ip&longs;i ER, habebit AN ad ER <expan abbr="duplã">duplam</expan> <lb/>proportionem eius, quam habet latus PN ad TR. pariquè <lb/>ratione quoniam PC &longs;imilis e&longs;t TG, habebit PC ad TG <lb/>duplam proportionem eius, quam habet idem latus PN ad <lb/>
<arrow.to.target n="marg20"></arrow.to.target> TR. quare ita &longs;e habet AN ad ER, ut PC ad TG. & per­<lb/>
<arrow.to.target n="marg21"></arrow.to.target> mutando AN ad PC, vt ER ad TG. Sed vt AN ad PC, ita e&longs;t <lb/>Y K ad KV, & vt ER ad TG. &longs;ic ZL ad LX. eandem igitur
<pb pagenum="33"/><expan abbr="proportion&etilde;">proportionem</expan> habebit YK ad KV, quam ZL ad LX. Quare <lb/>AN PC, & ER TG &longs;ecundùm eandem proportionem æ­<lb/>queponderabunt. quod quidem contingit ex &longs;imilitudine fi­<lb/>gurarum, & ex centris grauitatum KL &longs;imiliter po&longs;itis, qu&ecedil; <lb/>quidem magnitudines, &longs;i non e&longs;&longs;ent &longs;imiles, diui&longs;&ecedil; quide per <lb/>centrum grauitatis, partes vtique &ecedil;queponderarent; non ta­<lb/>men &longs;emper &longs;ecundùm eandem proportionem. quod tamen <lb/>&longs;emper figuris &longs;imilibus (cùm in ip&longs;is grauitatis centra &longs;int &longs;i <lb/>militer po&longs;ita) contingit; dummodo (vt dictum e&longs;t) diui­<lb/>dantur. Vnde con&longs;tat, quam &longs;it conueniens grauitatis centra <lb/>in figuris hac ratione e&longs;&longs;e con&longs;tituta. ex quibusomnibus per <lb/>&longs;picuum e&longs;t, centra grauitatis debere in figuris &longs;imilibus e&longs;&longs;e &longs;i <lb/>militer po&longs;ita. vt Archimedes in pr&ecedil;cedeti po&longs;tulato pr&ecedil;mi&longs;it. </s></p>
<p type="margin">
<s><margin.target id="marg18"></margin.target>4 <emph type="italics"/>&longs;exti<emph.end type="italics"/><lb/>16 <emph type="italics"/>quinti<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg19"></margin.target>20 <emph type="italics"/>&longs;exti<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg20"></margin.target>11 <emph type="italics"/>quinti<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg21"></margin.target>16 <emph type="italics"/>quinti<emph.end type="italics"/></s></p>
<figure id="fig14"></figure>
<p type="head">
<s>VIII.</s></p>
<p type="main">
<s>Simagnitudines ex æqualibus di&longs;tantijs æque­<lb/>ponderant, & ip&longs;is æquales ex ij&longs;dem di&longs;tantijs æ­<lb/>queponderabunt. </s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Hoc e&longs;t per&longs;picuum, <expan abbr="nã">nam</expan> <lb/>
<arrow.to.target n="fig15"></arrow.to.target><lb/>&longs;i magnitudines AB ex di­<lb/>&longs;tantijs CA CB &ecedil;quepon­<lb/>derant: &longs;itautem D ip&longs;i A <lb/>&ecedil;qualis, & E ip&longs;i B. <expan abbr="auferã">auferam</expan> <lb/>turquè magnitudines AB à <lb/>linea AB, ip&longs;arumquè loco ponatur D in A, & E in B, ma <lb/>gnitudines DE fimiliter <expan abbr="&ecedil;quepond&ecedil;rabũt">&ecedil;quepond&ecedil;rabunt</expan>. qua ratione enim <lb/>magnitudines AB inter&longs;e&longs;e &ecedil;queponderare dicuntur; eadem <lb/>pror&longs;us, & magnitudines DE ex ij&longs;dem di&longs;tantijs &ecedil;quepon <lb/>derabunt. quandoquidem omnia data &longs;unt paria. illud ta­<lb/>men non e&longs;t pretereundum, nimirum non oportere DE ip&longs;is <lb/>AB &ecedil;quales e&longs;&longs;e in magnitudine, &longs;ed in grauitate. pote&longs;t enim
<pb pagenum="34"/>magnitudinum in&ecedil;qualium minor maiore grauior exi&longs;tere, <lb/>ob naturæ diuer&longs;itatem, ac propterea cùm inquit Archimedes <lb/><emph type="italics"/>& ip&longs;is aquales<emph.end type="italics"/>, &longs;iue &longs;int magnitudine æquales, velinæquales, in <lb/>telligendum e&longs;t e&longs;&longs;e omnino æquales in grauitate. grauitas. n. <lb/>cau&longs;a e&longs;t, vt magnitudines æqueponderare debeant. </s></p>
<figure id="fig15"></figure>
<p type="head">
<s>VIIII,</s></p>
<p type="main">
<s>Omnis figuræ, cuiusperimeter &longs;it ad <expan abbr="eand&etilde;">eandem</expan> par <lb/>tem concauus, centrum grauitatis intra figuram <lb/>e&longs;&longs;e oportet. </s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<figure></figure>
<p type="main">
<s>Quid intelligat Ar­<lb/>chimedes per has figu­<lb/>ras ad eandem partem <lb/>concauas, apertiùs &longs;i­<lb/>gnificauit initio libro­<lb/>rum de&longs;ph&ecedil;ra, & cylin­<lb/>dro. vbi primùm vult <lb/>has figuras e&longs;&longs;e termina <lb/>tas; quod non &longs;olùm in <lb/>telligendum e&longs;t decur­<lb/>uilineis, verùm etiam <lb/>de rectilineis, & de mi­<lb/>xtis. rectiline&ecedil; quidem <lb/>erunt trium, quattuor, <lb/>quinque & plurium la­<lb/>terum; quamuis latera <lb/>non &longs;int æqualia, ne­<lb/>que anguli &ecedil;quales, vt
<pb pagenum="35"/>ABCDE, cuiusom nes ang uli&longs;unt flexi ad interiorem figuræ <lb/>partem. & hocmodo perimeter huius figuræ erit ad eandom <lb/>partem con cauus. vnde excludun tur figuræ, exempli gratia <lb/>FGHKL; cùm angulus K non &longs;it &longs;inuo&longs;us, & con oauus ad <lb/>eandem partem, vt reliquidnguli; qui &longs;unt &longs;in uo&longs;<gap/> ver&longs;us lifte <lb/>riorem pamem &longs;igur&ecedil; K vero bd exterioitem. &longs;imili modo <lb/>intelligen dum e&longs;t ded<gap/>lineis, vt dir<gap/>lis ellip&longs;es, vel alteri us <lb/>generis&longs;igræ, vt &longs;unt MN, quæ &longs;uam habent conqau tatem <lb/>adiean dem partem: &longs;ed curuline¸ OP ilnon &longs;unt ad ea n dem <lb/>partem concau&ecedil;. Mixtæ quoque figuræ, ut&longs;unt portiones eil <lb/>culi, hyperbab&ecedil; ac para bod&ecedil; rectis linen <gap/>eminat&ecedil;; vel <gap/><lb/>rius gen erisfigur&ecedil;, vt &longs;pnt QR. h&ecedil; quidemom nes&longs;unt ad ea­<lb/>dem partem concauç Mixcæ verò ST minimè Regulgm au­<lb/>tem qua<gap/> vniuer&longs;alemper verbis Archimedislodo qitato <lb/>elicere po&longs;&longs;unus, vtoog nofcere valeam us, an figu<gap/> &longs;int ad <lb/>eandem partem concauæ, vel minùs vt fcilicet inboblata figu <lb/>ra vbicum que duo &longs;umi po&longs;&longs;int puncta, quæ &longs;i rectal<gap/><lb/>nectantur, tota recta li <lb/>
<arrow.to.target n="fig16"></arrow.to.target><lb/>nea, velip&longs;ius pars ali­<lb/>qua extra figuram non <lb/>cadat. vt in figuris A, <lb/>quæ &longs;unt ad <expan abbr="eand&etilde;">eandem</expan> par <lb/>tem concauæ, vtcum­<lb/>que duo &longs;umantur <expan abbr="pũ-cta">pun­<lb/>cta</expan> BC, quæ conne­<lb/>ctantur, tota utique re­<lb/>cta linea inter puncta <lb/>BC exi&longs;tens, extra figu <lb/>ram non cadet. Quòd <lb/>&longs;i hæclinea cum termino, hoc e&longs;t eum latere figur&ecedil; conueni­<lb/>ret, vt &longs;i &longs;iguræ latus fueritrectum, in quo duo &longs;umantur pun <lb/>cta, nihilominus recta linea inter hæc puncta extra figuram <lb/>non cadei: quandoquidem figuræ terminus extra figuram mi <lb/>nimè roperitur atque hac ratione quomodocunque, & vbicú <lb/>que in his figuris duo &longs;um a ntur puncta, idem &longs;emper con tin <lb/>get. Quod tamen figuris D &longs;emper euenite non pote&longs;t in qui <lb/>bus (cùm non &longs;int ad eandem partem concau&ecedil;) duo &longs;umero
<pb pagenum="36"/>po&longs;&longs;umus puncta EG, inter quç tota recta linea EG extra <lb/>&longs;iguram cadet. vel fumerepo&longs;&longs;umus puncta FG, ita vt rect&ecedil; <lb/>line&ecedil; FG pars EG extra figuram cadat. figur&ecedil; igitur, quæ <lb/>ad ean dem partem &longs;unt concauæ, ill&ecedil; &longs;unt, qu&ecedil; &longs;inuo&longs;itatem, <lb/>concauitatemquè &longs;uam habent &longs;emper interiorem ip&longs;ius fi­<lb/>gur&ecedil; partem re&longs;picientem. Harum què rectè &longs;upponit Archi­<lb/>medes centrum grauitatis &longs;emperle&longs;&longs;e intra ip&longs;am figuram. <lb/>ita vt neque centrum e&longs;&longs;e po&longs;&longs;icin ambitu ip&longs;ius figur&ecedil; ete­<lb/>nim &longs;i extra figuram, &longs;iue in ambitu ip&longs;ius e&longs;&longs;e po&longs;&longs;et, num­<lb/>quam circa centrum grauitatis partes figur&ecedil; vndiquè <expan abbr="&ecedil;quepõ">&ecedil;quepom</expan> <lb/>
<arrow.to.target n="marg22"></arrow.to.target> derarent: neque facta ex grauitatis centro &longs;u&longs;pen&longs;ione figura <lb/>vbicumque, & in omni &longs;itu maneret. quod ramen ex ratione <lb/>centri grauitatis efficere deberet. to ta nimirum figura ex vna <lb/>e&longs;&longs;et parte, & ex altera nihil e&longs;&longs;et, quod ip&longs;i figur&ecedil; &ecedil;queponde <lb/>rare po&longs;&longs;et. Nece&longs;&longs;e e&longs;t igitur centrum grauitatis cuiu&longs;libet fi­<lb/>gur&ecedil; ad ean dem partem concau&ecedil; e&longs;&longs;ein &longs;pacio à figur&ecedil; ambi <lb/>tu contento. vt figur&ecedil; AB <lb/>
<arrow.to.target n="fig17"></arrow.to.target><lb/>centrum grauitatis erit in­<lb/>tra ip&longs;am, putà in C. quod <lb/>quidem non euenit &longs;emper <lb/>in alijs figuris, qu&ecedil; &longs;uum <expan abbr="cõ">com</expan> <lb/>cauitatis ambitum interio­<lb/>rem figur&ecedil; partem <expan abbr="nõ">non</expan> re&longs;pi­<lb/>cientem habent. cùm varijs <lb/>modis po&longs;&longs;itcentrum graui <lb/>tatisin figuris e&longs;&longs;e <expan abbr="collocatũ">collocatum</expan>. <lb/>vt &longs;uperius quoque diximus. <lb/>Nam &longs;igur&ecedil; D <expan abbr="centrũ">centrum</expan> gra­<lb/>uitatis erit extra ambitum fi <lb/>gur&ecedil;, vt in E. figura verò F <lb/>ita &longs;e habere poterit, vt cen­<lb/>trum grauitatis &longs;it in perime <lb/>tro, vt in G. <expan abbr="euenitaut&etilde;">euenitautem</expan> aliquando vt in figura HK <expan abbr="centrũ">centrum</expan> <lb/>grauitatis L intra ip&longs;am figuram reperiatur; quamuis conca­<lb/>uitates la torum interiorem partem minimè <expan abbr="re&longs;piciãt">re&longs;piciant</expan>. Sed h&ecedil;c <lb/>po&longs;&longs;unt e&longs;&longs;e, & non e&longs;&longs;e, vt in figura M, cuius centrum extra <lb/>e&longs;&longs;e pote&longs;t in N. quamuis (vt an tea diximus) centrum graui-
<pb pagenum="37"/>tatis in tra figuram &longs;emper exi&longs;tere aliquo modo intelligi po­<lb/>te&longs;t. </s></p>
<p type="margin">
<s><margin.target id="marg22"></margin.target><emph type="italics"/>per def. <lb/><expan abbr="c&etilde;t">cent</expan>. grau.<emph.end type="italics"/></s></p>
<figure id="fig16"></figure>
<figure id="fig17"></figure>
<p type="main">
<s>Refort Eutocius hocloco, Geminum rectè dicere, dum a&longs;&longs;e­<lb/>rit Archimedem dignitates peritiones apellare. æqualia enim <lb/>grauia ex di&longs;tantijs æqualibus æqueponderare, dignitas eft; & <lb/>quæ deinceps. <expan abbr="Verũ">Verum</expan> &longs;i hæc principra ab Archimede tradita re <lb/>ctèperpendamus, omnia dignitates e&longs;&longs;e minimè reperiemus. <lb/>nam &longs;eptimum po&longs;tulatum e&longs;t definitio, non dignitas. veluti <lb/>alia forta&longs;&longs;e nonnulla non &longs;unt dignitates, vt &longs;ecundum; quod <lb/>aliquo modo probari pote&longs;t, vt diximus. &longs;extum quoque po­<lb/>tiùs e&longs;t &longs;uppo&longs;ito, quàm dignitas. Quoniam autem vt clarè <lb/>con&longs;picitur Archimedes &longs;ub vno tantùm titulo pauca hæc <lb/>principia complecti voluit; quippe quod in&longs;titutum quàm plu <lb/>rimis mathematicis &longs;olemne fuit, qui principia vnico tantum <lb/>nomine nuncuparunt, modò vno, modò altero; nimirum, <lb/>vel petitionis, vel dignitatis, vt refert Proclus &longs;ecundo libro, & <lb/>tertio &longs;uorum commentariorum in primum elementorum. Eu <lb/>clidis; qui de Archimede peculiariter mentionem faciens, in­<lb/>quit illum in his libris ptincipia vnico tantùm nomine (peti­<lb/>tionis &longs;cilicet) nuncupa&longs;&longs;e. Hæc tamen potiùs petitionum, <lb/>quàm definitionum, vel dignitatum nomine nuncupare vo­<lb/>luit; nam &longs;i dignitares appella&longs;&longs;et; ea principia, quæ non &longs;unt <lb/>dignitates, inter dignitates malè collo ca&longs;&longs;ct. nulla quippè defi­<lb/>nitio dignitas dici debet; quandoquidem definitio terminos <lb/>declarat, atque con&longs;tituit. dignitas verò notos terminos copu­<lb/>lat. Pariquè ratione &longs;i de&longs;initionis nomine hæc principia nun <lb/>cupa&longs;&longs;et. dignitates malè &longs;ub hoc nomine complexus fui&longs;&longs;et, <lb/>quæ nullo modo rem definiunt, &longs;ed cùm &longs;int communes no <lb/>tiones, &longs;tatim cùm eas in tellectus apprehendit, quie&longs;cit. Qua­<lb/>reomnia &longs;ub petitionum nomine recte collocauit, non e&longs;t. n. <lb/>ab&longs;urdum dignitates, definitione&longs;què po&longs;&longs;e apellari petitio­<lb/>nes. etenim petimus, quæ &longs;unt concedenda, atque dignitates <lb/>&longs;unt concedend&ecedil;, ergo eas petere quoque po&longs;&longs;umus. Definitio <lb/>nibus verò rectè quoque hoc nomen conuenire pote&longs;t. Nam <lb/>dùm definitio terminos con&longs;tituat, atque declaret, cur non pe <lb/>tere po&longs;&longs;umus, terminos &longs;ic &longs;e habere, vel &longs;ice&longs;&longs;e rectè definitos? <lb/>vt exempli gratia, petit Archimedes puncta in figuris fimilitel
<pb pagenum="38"/>po&longs;ita, ita &longs;ehabere, vt &longs;untab ip&longs;o definita, vel rectè e&longs;&longs;e defi­<lb/>nita puncta, quæ &longs;unt in figuris &longs;imilibus po&longs;ita. Quapropter <lb/>hæc principia, quoniam pauca &longs;unt, &longs;ub petitionum nomine <lb/>Archimedes rectè collocauit. quòd &longs;i multa extiti&longs;&longs;ent, ea for <lb/>ta&longs;&longs;e di&longs;tinxi&longs;&longs;et. </s></p>
<p type="main">
<s><emph type="italics"/>His &longs;uppo&longs;itis.<emph.end type="italics"/> <expan abbr="po&longs;tquã">po&longs;tquam</expan> Archimedes <expan abbr="prĩcipia">principia</expan> po&longs;uit, ad theore <lb/>mata &longs;e conuertit, & inquit, <emph type="italics"/>his &longs;uppo&longs;itis<emph.end type="italics"/>, qua&longs;i dicat, ea, quæ <lb/>po&longs;uimus, &longs;ufficiuntad o&longs;ten denda theoremata, veluti. </s></p>
<p type="head">
<s>PROPOSITIO. I.</s></p>
<p type="main">
<s>Grauia, quæ ex æqualibus di&longs;tantijs æquepon­<lb/>derant, æqualia &longs;unt. </s></p>
<p type="main">
<s>Sint AD, & B grauia, <lb/>
<arrow.to.target n="fig18"></arrow.to.target><lb/>quæ ex æqualibus di&longs;tantijs <lb/>CA CB æqueponderent. di <lb/>co grauia AD, & B inter­<lb/>&longs;e&longs;e æqualia e&longs;&longs;e. <emph type="italics"/>&longs;i enim<emph.end type="italics"/> (&longs;i &longs;ie <lb/>ri pote&longs;t) <emph type="italics"/>e&longs;&longs;ent inæqualia<emph.end type="italics"/>; vt &longs;i <lb/>AD e&longs;&longs;et grauius, quàm B, <lb/>&longs;it D exce&longs;&longs;us, quo AD grauius e&longs;t, quàm B. <emph type="italics"/>ablato<emph.end type="italics"/> itaque <lb/><emph type="italics"/>exce&longs;&longs;u<emph.end type="italics"/> D <emph type="italics"/>àmaiori<emph.end type="italics"/> AD, <emph type="italics"/>reliqua<emph.end type="italics"/> grauia, quæ relinquuntur AB, <lb/>
<arrow.to.target n="marg23"></arrow.to.target> eruntinter&longs;e &ecedil;qualia; qu&ecedil; ex &ecedil;qualibus di&longs;tantijs CA CB æ­<lb/>queponderare deberent; tamen <emph type="italics"/>non æqueponderabunt. cùm<emph.end type="italics"/> enim <lb/>po&longs;itum &longs;it AD B &ecedil;queponderare, & <emph type="italics"/>ab altero aqueponderan-<emph.end type="italics"/><lb/>
<arrow.to.target n="marg24"></arrow.to.target> <emph type="italics"/>tium<emph.end type="italics"/> AD <emph type="italics"/>aliquod &longs;it ablatum<emph.end type="italics"/> D; reliqua grauia AB ex &ecedil;qua <lb/>libus di&longs;tantijs CA CB non &ecedil;queponderabunt quod fieri <lb/>non pote&longs;t; &longs;iquidem AB inter &longs;e &longs;unt &ecedil;qualia. <emph type="italics"/>Grauia igitur, <lb/>quæ ex æqualibus <expan abbr="distãtijs">distantijs</expan> æqueponderant, æqualia &longs;unt.<emph.end type="italics"/> quod de­<lb/>mon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg23"></margin.target>4. <emph type="italics"/>po&longs;tula­<lb/>tum huius<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg24"></margin.target><emph type="italics"/>contrapri­<lb/>mum post <lb/>huius.<emph.end type="italics"/></s></p>
<figure id="fig18"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Cùm &longs;it &longs;copus Archimedis (vt diximus) in primis octo <lb/>theorematibus, fun damentum tradere in hac &longs;cientia præci-
<pb pagenum="39"/>puum, nempè magnitudinum grauitates inter&longs;e ita &longs;e habe­<lb/>re, vt di&longs;tantiæ permutatim ex quibus &longs;u&longs;penduntur &longs;e <expan abbr="hab&etilde;t">habent</expan>. <lb/>primùm incipit o&longs;tendere, quomodo &longs;e habeant grauia in di <lb/>&longs;tantijs &ecedil;qua ibuspo&longs;ita; primùmquè in hac prima propo&longs;itio <lb/>ne o&longs;tendit, &longs;i grauia &ecedil;queponderant ex di&longs;tantijs &ecedil;qualibus, <lb/>&ecedil;qualia e&longs;&longs;e. in &longs;equenti verò, &longs;i grauia &longs;untin&ecedil;qualia, ex di­<lb/>&longs;tantijs &ecedil;qualibus nullo modo æqueponderare o&longs;tendet; &longs;ed <lb/>præponderare ad maius. </s></p>
<p type="head">
<s>PROPOSITIO. II.</s></p>
<p type="main">
<s>Inæqualia grauia ex æqualibus di&longs;tantijs non <lb/>æqueponderabunt, &longs;ed præponderabit ad maius. </s></p>
<figure></figure>
<p type="main">
<s>Sint gra­<lb/>uia in&ecedil;qua­<lb/>lia AB C in <lb/>di&longs;tantijs &ecedil;­<lb/>qualib^{9} DA <lb/>DC. &longs;itquè <lb/>grauius AB, <lb/>quàm C. di <lb/>co grauia AB C non &ecedil;queponderare, &longs;ed maius AB <expan abbr="deorsũ">deorsum</expan> <lb/>ferri. &longs;it B exce&longs;&longs;us, quo AB &longs;uperat C. <emph type="italics"/>ablato<emph.end type="italics"/> itaque à ma <lb/>iori AB <emph type="italics"/>exce&longs;&longs;u<emph.end type="italics"/> B, reliqua grauia AC &ecedil;qualia ex di&longs;tantijs <lb/>DA DC <emph type="italics"/>æqueponderabunt. cùm æqualia grauia ex distantiis æquali-<emph.end type="italics"/>
<arrow.to.target n="marg25"></arrow.to.target><lb/><emph type="italics"/>bus æqueponderent.<emph.end type="italics"/> &longs;i itaque grauia AC &ecedil;queponderant, <emph type="italics"/>adiecto <lb/>igitur<emph.end type="italics"/> ip&longs;i A <emph type="italics"/>ablato<emph.end type="italics"/> B, <emph type="italics"/>præponderabit ad maius<emph.end type="italics"/>, hoc e&longs;t ab deor
<arrow.to.target n="marg26"></arrow.to.target><lb/>&longs;um tendet. <emph type="italics"/>quoniam æqueponderantium altero<emph.end type="italics"/> nempè A <emph type="italics"/>adiectum <lb/>fuit<emph.end type="italics"/> B. Grauius igitur præponderatleuiori, ambobus in <expan abbr="di&longs;tã">di&longs;tam</expan> <lb/>tijs &ecedil;qualibus po&longs;itis. quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg25"></margin.target>1 <emph type="italics"/>po&longs;t hu­<lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg26"></margin.target>3 <emph type="italics"/>post hu­<lb/>ius.<emph.end type="italics"/></s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Hæc duo theoremata in gr&ecedil;co exemplari impre&longs;&longs;o &longs;equun <lb/>tur <expan abbr="quid&etilde;">quidem</expan> po&longs;tulata, & reliquis theorematibus &longs;unt pr&ecedil;po&longs;ita.
<pb pagenum="40"/>quia verò inter principia collocari non po&longs;&longs;unt; cùm &longs;uas ha­<lb/>beant propo&longs;itiones, &longs;uafquè &longs;eor&longs;um habeant demon&longs;tratio­<lb/>nes, ideo inter propo&longs;itiones ip&longs;a collocare nobis vi&longs;um e&longs;t. <lb/>cùm pr&ecedil;&longs;ertim nonnulla ex &longs;equentibus theorematibus, po­<lb/>ti&longs;&longs;i mùm verò proximum eiu&longs;dem cum his duobus ordinis, <lb/>& naturæ &longs;int. Neque enim propterea peruertitur ordo; non <lb/>enim h&ecedil; propo&longs;itiones in alium transcerun tur locum. &longs;ed <expan abbr="tã-ù">tan­<lb/>ù</expan> n inter alias numeris adnotantur. exi&longs;tim andum enim e&longs;t, <lb/>Archimedem propo&longs;itiones in &longs;erie propo&longs;itionum colloca&longs;­<lb/>&longs;e. hanc verò exiguam muta tionem accidi&longs;&longs;e <expan abbr="oblongitudin&etilde;">oblongitudinem</expan> <lb/>temporis; cuius proprium e&longs;t, res potiùs de&longs;truere, quàm ac­<lb/>comodare. Hocautem nobis hanc præbebit commoditatem, <lb/>vt, quando libuerit, has propo&longs;itiones numeris nominare <lb/>po&longs;&longs;imus. idip&longs;umquè numeri po&longs;tulata di&longs;tinguentes præ­<lb/>&longs;tant, quamuis in Gr&ecedil;co codice po&longs;tulata (Gr&ecedil;corum more) <lb/>numeris adnotata non &longs;int. </s></p>
<p type="head">
<s>PROPOSITIO. III.</s></p>
<p type="main">
<s>Inæqualia grauia ex di&longs;tantijs inæqualibus æ­<lb/>
<arrow.to.target n="marg27"></arrow.to.target> queponderabunt, maius quidem ex minori. </s></p>
<p type="margin">
<s><margin.target id="marg27"></margin.target>A</s></p>
<figure></figure>
<p type="main">
<s><emph type="italics"/>Sint in æqualia grauia AD, B<emph.end type="italics"/>; <lb/>
<arrow.to.target n="marg28"></arrow.to.target> <emph type="italics"/>&longs;it què maius AD<emph.end type="italics"/>, exce&longs;&longs;us ve <lb/>rò, quo AD &longs;uperat B, &longs;it <lb/>D. <emph type="italics"/><expan abbr="æquepõderentquè">æqueponderentquè</expan><emph.end type="italics"/> AD B <emph type="italics"/>ex <lb/>di&longs;tantiis AC C B. o&longs;tendendum <lb/>e&longs;t, minorem e&longs;&longs;e<emph.end type="italics"/> <expan abbr="diftantiã">diftantiam</expan> <emph type="italics"/>AC <lb/>ip&longs;a CB. Non &longs;it quidem, &longs;i fie­<lb/>ripotest<emph.end type="italics"/>, AC minor, quàm CB; erit nimirum, vel &ecedil;qualis, <lb/>vel maior. Quòd &longs;i AC fuerit &ecedil;qualis ip&longs;i CB, <emph type="italics"/>ablato &ecedil;nim <lb/>exce&longs;&longs;u<emph.end type="italics"/> D, <emph type="italics"/>quo AD &longs;uperat B. cùm ab aqueponderantium altero ab<emph.end type="italics"/><lb/>
<arrow.to.target n="marg29"></arrow.to.target> <emph type="italics"/>latum &longs;it aliquid<emph.end type="italics"/>, grauia AB non æqueponderabunt; &longs;ed <emph type="italics"/>præ-<emph.end type="italics"/><lb/>
<arrow.to.target n="marg30"></arrow.to.target> <emph type="italics"/>ponderabit ad B. non præponderabit autem; exi&longs;tente enim AC aqua <lb/>li CB<emph.end type="italics"/>, cùm ab in&ecedil;qualibus grauibus AD B ablatus &longs;<gap/> ex­<lb/>ce&longs;&longs;us D, <emph type="italics"/>grauia<emph.end type="italics"/>, quæ relinquuntur AB, eruntinter&longs;e <emph type="italics"/>æqualia<emph.end type="italics"/>;
<pb pagenum="41"/>quæ <emph type="italics"/>ex di&longs;tantiis æqualibus<emph.end type="italics"/> AC CB <emph type="italics"/>æqueponderarent.<emph.end type="italics"/> at non &ecedil;que <lb/>ponderant, quod e&longs;t ab&longs;urdum. di&longs;tantia igitur AC ip&longs;i CB <lb/>æqualis e&longs;&longs;e non pote&longs;t. <emph type="italics"/>&longs;i uerò AC maior fuerit CB<emph.end type="italics"/>; ab ato &longs;i­<lb/>militer exce&longs;&longs;u D, nihilominus &ecedil;qualia grauia AB non &ecedil;que <lb/>ponderabunt, &longs;ed <emph type="italics"/>inclinabitur ad A. æqualia enim grauia<emph.end type="italics"/> AB <emph type="italics"/>ex<emph.end type="italics"/>
<arrow.to.target n="marg31"></arrow.to.target><lb/><emph type="italics"/>distantiis inæqualibus non aqueponderant, &longs;ed inclinatur admaiorem <lb/>distantiam<emph.end type="italics"/> AC. ergo totum AD multò magis præpond&ecedil;rabit, <lb/><expan abbr="quãm">quamm</expan> B. quod fieri non pote&longs;t. po&longs;ita enim &longs;unt æqueponde <lb/>rare. Quare AC maior e&longs;&longs;e non pote&longs;t, quàm CB. &longs;ed o&longs;ten&longs;a <lb/>e&longs;t, neque ip&longs;i CB æqualis e&longs;&longs;e: <emph type="italics"/>ac propterea minor e&longs;t AC, quàm <lb/>CB. Mani&longs;estum e&longs;t itaque grauia ex distantiis inæqualibus æquepon­<lb/>derantia, inæqualia e&longs;&longs;e; maiu&longs;què in minori<emph.end type="italics"/> di&longs;tantia <emph type="italics"/>existere.<emph.end type="italics"/> quod <lb/>oportebat demon&longs;trare. </s></p>
<p type="margin">
<s><margin.target id="marg28"></margin.target>B</s></p>
<p type="margin">
<s><margin.target id="marg29"></margin.target>4 <emph type="italics"/>post hu­<lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg30"></margin.target>1 <emph type="italics"/>po&longs;t hu­<lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg31"></margin.target>2 <emph type="italics"/>post hu­<lb/>ius.<emph.end type="italics"/></s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>In propo&longs;itione verba illa, <emph type="italics"/>maius quidem ex minori<emph.end type="italics"/>, non <expan abbr="hab&etilde;">habem</expan>
<arrow.to.target n="marg32"></arrow.to.target><lb/>tur integra in codice græco, qui &longs;ic habet, <foreign lang="greek"><gap/> to/ aw_o\to_u ela/ssonos</foreign><lb/>vbi de&longs;iderari viderur <foreign lang="greek">me/izon</foreign>, vt integrèita legatur, <foreign lang="greek">kai\ to/ me/izon <lb/>a)w_o\ tou_ e)la/ssonos.</foreign></s></p>
<p type="margin">
<s><margin.target id="marg32"></margin.target>A</s></p>
<p type="main">
<s><emph type="italics"/>Sitquè maius A.<emph.end type="italics"/> Græcus codex, <foreign lang="greek">kai\ e)/sw to\ a</foreign>, vbi &longs;imiliter
<arrow.to.target n="marg33"></arrow.to.target> &longs;up­<lb/>plendum e&longs;t, <foreign lang="greek">kai\ e)/sw me/izon to\ a</foreign> Hæc verò ita &longs;untomnino re&longs;ti <lb/>tuenda, quia in vltima demon&longs;trationis conclu&longs;ione inquit <lb/>Archimedes, <emph type="italics"/>Manife&longs;tum est itaque grauia ex di&longs;tantiis inæqualibus <lb/>æqueponderantia inæqualia e&longs;&longs;e; maiu&longs;què in minori existere.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg33"></margin.target>B</s></p>
<p type="main">
<s><expan abbr="Po&longs;tquã">Po&longs;tquam</expan> Archimedes duab^{9} primis ppo&longs;itionib^{9} <expan abbr="o&longs;t&etilde;dit">o&longs;tendit</expan>, <expan abbr="qũo">quno</expan> <lb/>&longs;e <expan abbr="h&etilde;ant">henant</expan> grauia ex <expan abbr="di&longs;tãtijs">di&longs;tantijs</expan> &ecedil;qualib^{9}; in hac tertia <expan abbr="cõuertit&longs;e">conuertit&longs;e</expan> ad <lb/><expan abbr="o&longs;t&etilde;d&etilde;dũ">o&longs;tendendum</expan>, <expan abbr="qũo">quno</expan> &longs;e <expan abbr="h&etilde;nt">hennt</expan> ex <expan abbr="di&longs;tãtijs">di&longs;tantijs</expan> in&ecedil;qualib^{9}. & <expan abbr="q&mtilde;">qmm</expan> in <expan abbr="&longs;ecũdo">&longs;ecundo</expan> <lb/>po&longs;tulato <expan abbr="a&longs;sũp&longs;it">a&longs;sump&longs;it</expan>, <expan abbr="qũo">quno</expan> &longs;e <expan abbr="h&etilde;nt">hennt</expan> grauia &ecedil;qualia in <expan abbr="di&longs;tãtijs">di&longs;tantijs</expan> in &ecedil;­<lb/>qualibus <expan abbr="cõ&longs;tituta">con&longs;tituta</expan>; <expan abbr="nimirũ">nimirum</expan> qd e&longs;t in <expan abbr="lõgiori">longiori</expan> <expan abbr="di&longs;tãtia">di&longs;tantia</expan>, <expan abbr="pr&ecedil;põde-rat">pr&ecedil;ponde­<lb/>rat</expan> ei, qd e&longs;t in breuiori. <expan abbr="nũc">nunc</expan> <expan abbr="o&longs;t&etilde;dit">o&longs;tendit</expan>, <expan abbr="qũo">quno</expan> in&ecedil;qualia grauia &longs;e <lb/><expan abbr="h&etilde;nt">hennt</expan>, ita vt <expan abbr="&ecedil;quepõder&etilde;t">&ecedil;queponderent</expan>, in <expan abbr="di&longs;tãtijs">di&longs;tantijs</expan> in &ecedil;qualibus po&longs;ita. <expan abbr="demõ">demom</expan> <lb/>&longs;tratquè graue maius in breuiori <expan abbr="di&longs;tãtia">di&longs;tantia</expan> <expan abbr="e&etilde;">eem</expan> oportere, min^{9} ve­<lb/>rò graue in <expan abbr="lõgiori">longiori</expan>. & ecce quomodo Archimedes <expan abbr="paulatĩ">paulatim</expan> de <lb/>ducit nos in <expan abbr="cognition&etilde;">cognitionem</expan> principalis <expan abbr="fundam&etilde;ti">fundamenti</expan>, qd &longs;cilicetgra <lb/>ue ad graue e&longs;t, vt <expan abbr="di&longs;tãtia">di&longs;tantia</expan> ad <expan abbr="di&longs;tãtiã">di&longs;tantiam</expan> pmutatim. Ex hoc. n. pri <lb/>mùm cogno&longs;cimus grauius in minori, leuius <expan abbr="aut&etilde;">autem</expan> in maiori <lb/>di&longs;tantia e&longs;&longs;e debere, &longs;i &ecedil;queponderare debent. </s></p>
<pb pagenum="42"/>
<p type="head">
<s>PROPOSITIO. IIII.</s></p>
<p type="main">
<s>Si due magnitudines æquales non idem <expan abbr="centrũ">centrum</expan> <lb/>grauitatis habuerint, magnitudinis ex vtri&longs;que <lb/>magnitudinibus compo&longs;itæ centrum grauitatis <lb/>er it medium rectæ lineæ grauitatis centra magni <lb/>tudinum coniungentis. </s></p>
<p type="main">
<s><emph type="italics"/>Sit <expan abbr="quid&etilde;">quidem</expan> A<emph.end type="italics"/><lb/>
<arrow.to.target n="fig19"></arrow.to.target><lb/><emph type="italics"/><expan abbr="centrũ">centrum</expan> grauita <lb/>tis magnitudi­<lb/>nis A. B uerò<emph.end type="italics"/><lb/>&longs;it <expan abbr="c&etilde;trũ">centrum</expan> gra­<lb/>uitatis <emph type="italics"/>magni­<lb/>tudinis B iun­<lb/>staquè AB bifariam diuidatur in C. dico magnitudinis ex utri&longs;què ma­<lb/>gnitudinibus compo&longs;itæ centrum<emph.end type="italics"/> grauitatis <emph type="italics"/>e&longs;&longs;e punctum C. &longs;i. n. non; &longs;it <lb/>utrarumquè magnitudinum AB centrum grauitatis D, &longs;i <expan abbr="fieripõt">fieripont</expan>. Quòd <lb/>autem &longs;it in linea AB, præo&longs;ten&longs;um est. Quoniam igitur punstum D <expan abbr="c&etilde;">cem</expan><emph.end type="italics"/><lb/>
<arrow.to.target n="marg34"></arrow.to.target> <emph type="italics"/><expan abbr="trũ">trum</expan> e&longs;t grauitatis magnitudinisex AB <expan abbr="cõpo&longs;itæ">compo&longs;itæ</expan>, <expan abbr="&longs;u&longs;p&etilde;&longs;o">&longs;u&longs;pen&longs;o</expan> <expan abbr="pũcto">puncto</expan> D<emph.end type="italics"/>, magni <lb/>tudines AB <emph type="italics"/>æqueponderabunt. magnitudines igitur AB<emph.end type="italics"/> &ecedil;quales <emph type="italics"/>æque <lb/>ponderant ex di&longs;tantiis AD DB<emph.end type="italics"/> in &ecedil;qualibus exi&longs;tentibus; <emph type="italics"/>quod fie<emph.end type="italics"/><lb/>
<arrow.to.target n="marg35"></arrow.to.target> <emph type="italics"/>ri non pote&longs;t. æqualia. n.<emph.end type="italics"/> grauia <emph type="italics"/>ex di&longs;tantiis in a qualibus non <expan abbr="æquepõde-rãt">æqueponde­<lb/>rant</expan>.<emph.end type="italics"/> <expan abbr="Nõ">non</expan> e&longs;t igitur D <expan abbr="ip&longs;arũ">ip&longs;arum</expan> <expan abbr="magnitudinũ">magnitudinum</expan> <expan abbr="c&etilde;trũ">centrum</expan> grauitatis.. <emph type="italics"/>Qua <lb/>re manifestum est punstum C <expan abbr="centrũ">centrum</expan> e&longs;&longs;e grauitatis magnitudinis ex AB <lb/>compo&longs;itæ.<emph.end type="italics"/> quod demonftrare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg34"></margin.target><emph type="italics"/>def. centri <lb/>grauit. <lb/>contra 2. <lb/>post huins<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg35"></margin.target>2 <emph type="italics"/>post bu­<lb/>ius.<emph.end type="italics"/></s></p>
<figure id="fig19"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<figure></figure>
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<s>Po&longs;&longs;unt magnitudines &ecedil;quales <expan abbr="id&etilde;">idem</expan> <expan abbr="centrũ">centrum</expan> <lb/>grauitatis habere, vt duo <expan abbr="parallelogrãma">parallelogramma</expan> æ­<lb/>qualia ad rectos &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> angulos exi&longs;ten <lb/>tia: <expan abbr="triãgulũ">triangulum</expan> quoque & <expan abbr="parallelogrãmũ">parallelogrammum</expan> in­<lb/>ter&longs;e æqualia. <expan abbr="&ptilde;terea">pnterea</expan> cubos, piramides, cylin <lb/>dros, & nuiu&longs;modi alias magnitudines &ecedil;qua <lb/>les <expan abbr="id&etilde;">idem</expan> grauitatis <expan abbr="c&etilde;trũ">centrum</expan> <expan abbr="h&etilde;re">herre</expan> in telligere po&longs;&longs;u <lb/>mus. propterea in propo&longs;itione cùm inquit Archimedes <lb/><emph type="italics"/>&longs;i duæ magnitudines æquales non idem centrum grauitatis<emph.end type="italics"/>
<pb pagenum="43"/><emph type="italics"/>babuerint.<emph.end type="italics"/> intelligendum e&longs;t his verbis Archimedem &longs;uppo­<lb/>nere magnitudines ita e&longs;&longs;e con&longs;titutas, vt à centro ad centrum <lb/>duci po&longs;&longs;it recta linea. quod idem ob&longs;eruandum e&longs;t in prima <lb/>propo&longs;itione &longs;ecundi libri huius. </s></p>
<p type="main">
<s>Súmoperè <expan abbr="aũtanimaduert&etilde;da">auntanimaduertenda</expan> &longs;unt nonulla, quibus vtitur <lb/>Archimedes in hac propo&longs;itione, cùm &longs;int communi&longs;&longs;ima, <lb/>& maximè vtilia in hac &longs;cientia. ac primùm quidem con&longs;ide <lb/>randum occurrit, quid &longs;ibi vult Archimedes per magnitudi <lb/>nem ex vtri&longs;que magnitudinibus AB compo&longs;itam. Nam ma­<lb/>gnitudines AB &longs;unt inuicem &longs;eparat&ecedil;, & &longs;unt du&ecedil;, ip&longs;e autem <lb/>vtram quovnam tantùm con&longs;iderat. quod quidem ita in <expan abbr="telli-gendũ">telli­<lb/>gendum</expan> e&longs;t. <expan abbr="quoniã&longs;cilicet">quonian&longs;cilicet</expan> recta linea AB eas coniungit; ideo <lb/>Archimedes con&longs;iderat vnam tantùm e&longs;&longs;e <expan abbr="magnitudin&etilde;">magnitudinem</expan>; qu&ecedil; <lb/>con&longs;tat ex ip&longs;is AB, & efficitur vna magnitudo àlinea AB. <lb/>cuius munus e&longs;t non &longs;olùm connectere magnitudines AB, <lb/>ita vtneque ad &longs;e ampliùs accedere, neque recedere inuicem <lb/>po&longs;&longs;int; &longs;intquè ab hac linea qua&longs;i compul&longs;&ecedil; eundem &longs;emper <lb/>in ter&longs;e &longs;eruare &longs;i tum: verum etiam &longs;i &longs;u&longs;pendantur ex C, in­<lb/>tellig endum e&longs;t linea AB in rectitudin em iacere, in&longs;uperquè <lb/>&longs;u&longs;tinere magnitudines AB. Neque magis vna e&longs;t magnitudo <lb/>quadrilaterum, <expan abbr="p&etilde;tagonum">pentagonum</expan>, cubus, & huiu&longs;modi aliæ, quàm <lb/>&longs;it magnitudo, quæ componitur ex magnitudinibus AB v­<lb/>nà cum linea AB. quòd &longs;i e&longs;t vna tantùm magnitudo, ergo <lb/>vnum habet <expan abbr="c&etilde;trum">centrum</expan> grauitatis. Archimed esigitur qu&ecedil;rit cen <lb/>trum grauitatis huiu&longs;ce magnitudinis; demon&longs;tratquè cen <lb/>trum e&longs;&longs;e in puncto C. quod e&longs;t medium lineæ AB. notan <lb/>dum e&longs;t autem Archimedem non con&longs;iderare grauitatem li­<lb/>ne&ecedil; AB. vt potè, qu&ecedil; longitudo tantùm exi&longs;tat. Quòd &longs;i quis <lb/>etiam mente concipere vellet lineam AB grauitate <expan abbr="pr&ecedil;ditã">pr&ecedil;ditam</expan> <lb/>e&longs;&longs;e; nihilominus centrum grauitatis line&ecedil; AB &longs;imiliter e&longs;&longs;et <lb/>in eius medio C. nam longitudo AC longitudini CB e&longs;t <lb/>æqualis; ac propterea h&ecedil; quidem longitudines e&longs;&longs;ent inter&longs;e&longs;o <lb/>&ecedil;queponderantes. Quare, &longs;iue <expan abbr="cõ&longs;iderata">con&longs;iderata</expan> grauitate line&ecedil; AB, <lb/>&longs;iue minùs, centrum grauitatis magnitudinis ex AB compo <lb/>&longs;it&ecedil; e&longs;t mediu rect&ecedil; line&ecedil;, quæ centra grauitatis <expan abbr="magnitudinũ">magnitudinum</expan> <lb/>coniungit. Et hoc modo &longs;i plures etiam e&longs;&longs;ent magnitudines <lb/>à recta linea coniunct&ecedil;, eodem modo eas pro vna tan tùm ma
<pb pagenum="44"/>gnitudine ex plurib^{9} magnitudinibus compo&longs;ita accipere po <lb/>terimus, veluti Archimedes in &longs;equenti bus accipiet. </s></p>
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<s>Argumen tandi modus in e&longs;t in hac demon&longs;tratione maxi­<lb/>ma con&longs;ideratione dignus, & huius &longs;cientiæ maximè pro­<lb/>prius. cùm enim dixi&longs;&longs;et Archimedes po&longs;ito centro grauitatis <lb/>magnitudinis ex AB compo&longs;itæ in puncto D, &longs;tatim infert. <lb/><emph type="italics"/>Quoniam igitur punctum D centrum e&longs;t grauitatis magnitudinis ex <lb/>AB compo&longs;ita, &longs;u&longs;pen&longs;o puncto D, magnitudines AB æquepondera­<lb/>bunt.<emph.end type="italics"/> hoc e&longs;t &longs;i magnitudo ex AB compo&longs;ita &longs;u&longs;pendatur ex <lb/>D, manebit, vt reperitur; nec amplius in alteram partem in cli <lb/>nabit. quod euenit ob naturam centri grauitatis, quod talis <lb/>e&longs;t naturæ (&longs;icuti initio explicauimus) ut &longs;i graue in eius cen­<lb/>tro grauitatis &longs;u&longs;tineatur, eo modo manet, quo reperitur, <expan abbr="dũ">dum</expan> <lb/>&longs;u&longs;penditur; parte&longs;què undiquè æqueponderant. & ob id &longs;i <lb/>magnitudo ex AB compo&longs;ita &longs;u&longs;pendatur in eius centro gra <lb/>uitatis, manet; parte&longs;què AB æqueponderant. ac propterea <lb/>quando in &longs;equentibus quærit Archimedes, quoniam grauia <lb/>æqueponderare debent, tunc tan tùm quærit ip&longs;orum <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis, utin &longs;exta, &longs;eptimaquè propo&longs;itione in quit Archi­<lb/>medes magnitudines &ecedil;queponderare ex di&longs;tantijs, quç permu <lb/>tatim proportionem habent, utip&longs;arum grauitates, in <expan abbr="demõ">demom</expan> <lb/>&longs;tratione tamen quærit, vbi nam e&longs;t <expan abbr="c&etilde;trum">centrum</expan> grauitatis magni <lb/>tudinis ex vtrisquè compo&longs;it&ecedil;. quo inuento, &longs;tarim nece&longs;&longs;ariò <lb/>&longs;equitur, magnitudines, &longs;i ex ip&longs;o centro &longs;u&longs;pendantur, æque <lb/>ponderare. </s></p>
<p type="main">
<s>Hinc colligere po&longs;&longs;umus alterum argumentandi modum, <lb/>conuer&longs;o nempè modo, veluti in eadem &longs;igura, &longs;i dicamus <lb/>grauia AB &longs;u&longs;pen&longs;a ex C æqueponderant, &longs;tatim inferre <lb/>po&longs;&longs;umus, punctum C ip&longs;orum &longs;imul grauium, hoc e&longs;t ma <lb/>gnitudinis ex ip&longs;is AB compo&longs;it&ecedil; centrum e&longs;&longs;e grauitatis. <lb/>Quare ad &longs;e inuicem conuertuntur, hoc punctum e&longs;t horum <lb/>grauium cen trum grauitatis; ergo h&ecedil;c grauia ex hoc puncto <lb/>æqùeponderant; & è conuer&longs;o, nempè hæc grauia ex hoc pun <lb/>cto æqueponderant, ergo idem punctum e&longs;t ip&longs;orum <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis. &longs;ed ad uertendum hanc &longs;equi <expan abbr="conuertibilitat&etilde;">conuertibilitatem</expan>, <expan abbr="quã-do">quan­<lb/>do</expan> præfatum punctum e&longs;t in recta linea, quæ centra grauita­<lb/>tum ponderum coniungit; deinde quando h&ecedil;c linea non e&longs;t
<pb pagenum="45"/>horizonti perpendicularis. &longs;ecus aurem minimè. Nam &longs;i pon <lb/>dera AB &longs;intin libra ADB, qu&ecedil; &longs;itarcuata, vel angulum <expan abbr="cō-&longs;tituat">con­<lb/>&longs;tituat</expan>, &longs;iue intelligatur libra recta linea AB, cui affixa &longs;it <lb/>perpendicularis CD. vt in tractatu de libra no&longs;trorum Me­<lb/>chanicorum diximus. &longs;u&longs;pendantur autem pondera AB ex <lb/>
<arrow.to.target n="fig20"></arrow.to.target><lb/>D, & æqueponderent; <expan abbr="nõ">non</expan> <lb/>&longs;equitur tamen, ergo D <lb/><expan abbr="c&etilde;trum">centrum</expan> e&longs;t grauitatis ma­<lb/>gnitudinis ex AB com­<lb/>po&longs;it&ecedil;. centrum enim gra <lb/>uita tis in linea exi&longs;tit AB <lb/>quæ centra grauitatis ma <lb/>gnitudinum AB coniun <lb/>git, nempein C. Verùm coniungat recta linea AB centra <lb/>
<arrow.to.target n="fig21"></arrow.to.target><lb/>grauita tis æqualium ponderum AB, lineaquè <lb/>AB, cuius medium &longs;it C, in centrum mundi <expan abbr="t&etilde;-dat">ten­<lb/>dat</expan>, magnitudoquè ex ip&longs;is AB compo&longs;ita vbi­<lb/>cunque &longs;u&longs;pendatur in linea AB, vtin E; ma <lb/>nebuntvtique pondera AB ex E &longs;u&longs;pen&longs;a, vt in <lb/>prima propo&longs;itione de libra no&longs;trorum Mecha­<lb/>nicorum o&longs;ten dimus. cùm C &longs;it ip&longs;orum <expan abbr="centrū">centrum</expan> <lb/>grauita tis, & EC &longs;it horizonti erecta. Et quam­<lb/>uis magnitudo ex ip&longs;is AB compo&longs;ita ex E &longs;u <lb/>&longs;pen&longs;a maneat; non propterea &longs;equitur ergo E <lb/>centrum e&longs;t grauitatis magnitudinis ex ip&longs;is AB <lb/>compo&longs;it&ecedil;. ni&longs;i fortè accidat &longs;u&longs;pen&longs;io ex puncto <lb/>C. Præterea verò aduertendum e&longs;t in hoc ca&longs;u <expan abbr="põ">pom</expan> <lb/>dera AB, dici quidem po&longs;&longs;e, manere, non autem <lb/>æqueponderare. omnia nimirum, qu&ecedil; æqueponderant, ma­<lb/>nent; &longs;ed non è conuer&longs;o, quæ manent, æqueponderant. Nam <lb/>&longs;i pondus A maius fuerit pondere B; &longs;iue B maius, quàm <lb/>A, vbicunque fiat &longs;u&longs;pen&longs;io in linea AB, &longs;emper ob <expan abbr="eãdem">eandem</expan> <lb/>cau&longs;am, quomodocun que &longs;int pondera, manebunt; non ta­<lb/>men æqueponderabunt. Vt enim pondera æqueponderent, <lb/>requiritur, vt pars parti, virtu&longs;què vnius virtuti alterius hinc <lb/>inde re&longs;i&longs;tere, & æquipollere po&longs;&longs;it; vt propriè dici po&longs;&longs;int <expan abbr="põ">pom</expan> <lb/>dera æqueponderare. & vt hoc euenire po&longs;&longs;it, oportet, vt par­
<pb pagenum="46"/>tes ex determinatis di&longs;tantijs determinatas quoque habeant <lb/>grauita tes; &longs;i ex dato puncto æqueponderare debent. Quòd <lb/>&longs;i in hoc ca&longs;u datum fuerit punctum C, ex quo pondera AB <lb/>ex æqualibus di&longs;tantijs CA CB &ecedil;quepo nderare debeant: o­<lb/>porteret, vt pondera AB (ex demon&longs;tratis) &longs;emper e&longs;&longs;ent æ­<lb/>qualia. <expan abbr="Quoniã">Quoniam</expan> <expan abbr="aut&etilde;">autem</expan> <expan abbr="quomodocũque">quomodocunque</expan> &longs;int pondera, hoc e&longs;t; &longs;i <lb/>ue pondus A maius, &longs;iue minus fuerit, quàm B, manent, &longs;i <lb/>igitur dixerimus, ergo pondus A ponderi B &ecedil;queponderat; <lb/>e&longs;&longs;et o mnino inconueniens. cùm ex ijsdem di&longs;tantijs <expan abbr="eid&etilde;">eidem</expan> <expan abbr="põ">pom</expan> <lb/>deri pondus quandoquè maius, quandoquè minus &ecedil;quepon­<lb/>derare non po&longs;&longs;it; vt in hoc ca&longs;u accidere pote&longs;t. Quocirca <lb/>nec propriè dici po&longs;&longs;unt pondera, &longs;iue in libra AB, &longs;iue ex <lb/>di&longs;tantijs CA CB con&longs;tituta e&longs;&longs;e. Vndè neque Archimedis <lb/>propo&longs;itiones in hoc ca&longs;u &longs;unt in telligend&ecedil; quandoquidem <lb/>in his propriè quærit ponderum, magnitudinumquè æque­<lb/>ponderationes. neque enim in hac quarra demon&longs;tratione in <lb/>hoc ca&longs;u potui&longs;&longs;et Archimedes ab&longs;urdum o&longs;tendere, &longs;i C <expan abbr="nõ">non</expan> <lb/>e&longs;t grauitatis centrum magnitudinis ex AB compo&longs;itæ, &longs;it <lb/>E. facta igitur ex E &longs;u&longs;pen&longs;ione, magnitudines æquales AB <lb/>ex in æquali bus di&longs;tantijs EA EB &ecedil;queponderabunt. quod <lb/>&longs;ieri non pote&longs;t. non enim hoc e&longs;t ab&longs;urdum; cùm pondera <lb/>ex E &longs;u&longs;pen&longs;a <expan abbr="maneãt">maneant</expan> idcirco quando linea AB e&longs;t <expan abbr="horizõ">horizom</expan> <lb/>ti erecta; propriè ad rem no&longs;tram minimè pertinet. Ex dictis <lb/>igitur &longs;emper valet con&longs;equentia, hoc punctum horum pon <lb/>derum centrum e&longs;t grauitatis, ergo &longs;i ex hoc &longs;u&longs;pendantur, <expan abbr="põ">pom</expan> <lb/>dera &ecedil;queponderant. non autem è conuer&longs;o. ni&longs;i quando ar­<lb/>gumentatio &longs;umitur &longs;emper ex recta linea, quæ centra graui <lb/>tatis magnitudinum coniungit, & quando h&ecedil;c linea non e&longs;t <lb/>
<arrow.to.target n="fig22"></arrow.to.target><lb/>horizonti erecta. hac enim <lb/>ratione quocunque modo <lb/>recta linea &longs;e habeat, &longs;em­<lb/>per &longs;equitur idem. Vt &longs;i li­<lb/>nea AB fuerit, &longs;iue <expan abbr="nõ">non</expan> fue­<lb/>rit horizonti æquidi&longs;tans, <lb/>ip&longs;ius medium C centrum <lb/>erit grauitatis magnitudi­<lb/>nis ex magnitudinibus AB æqualibus compo&longs;it&ecedil;. vnde &longs;equi
<pb pagenum="47"/>tur, &longs;i appendantur pondera AB ex C, æqueponderare. & <lb/>è conuer&longs;o, &longs;i AB pondera ex C æqueponderant, ergo C <lb/>centrum grauitatis exi&longs;tit. ex quibus &longs;equitur lineam AB, <expan abbr="põ">pom</expan> <lb/>deraquè manere eo modo, quo reperiuntur. vt in no&longs;tro me­<lb/>chanicorum libro in codem tractatu de libra demon&longs;traui­<lb/>mus, & aduer&longs;us illos, qui aliter &longs;entiunt, abundè &longs;atis
<arrow.to.target n="marg36"></arrow.to.target> di&longs;pu­<lb/>tauimus. </s></p>
<p type="margin">
<s><margin.target id="marg36"></margin.target><emph type="italics"/>po&longs;t quar­<lb/>tam propo <lb/>&longs;itionem.<emph.end type="italics"/><lb/>*</s></p>
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<figure id="fig22"></figure>
<p type="main">
<s>In demon&longs;tratione autem huius quartæ propo&longs;itionis in­<lb/>quit Archimedes. <emph type="italics"/>Quòd autem &longs;it in linea AB, præosten&longs;um e&longs;t.<emph.end type="italics"/> qua <lb/>&longs;i dicat Archimedes, &longs;e priùs o&longs;ten di&longs;&longs;e centrum grauitatis ma <lb/>gnitudinis ex AB compo&longs;itæ e&longs;&longs;ein linea AB; quod tamen <lb/>in ijs, quæ dicta &longs;unt, non videtur expre&longs;&longs;um. virtute tamen &longs;i <lb/>con&longs;ideremus ea, qu&ecedil; in prima, tertiaquè propo&longs;itione dicta <lb/>&longs;unt, facilè ex his concludi pote&longs;t, centrum grauitatis magni­<lb/>tudinis ex duabus magnitudinibus compo&longs;itæ e&longs;&longs;e in recta li <lb/>nea, quæ ip&longs;arum centra grauitatis coniungit. Quare memi­<lb/>ni&longs;&longs;e oportet eorum, qu&ecedil; a nobis in expo&longs;itione primi po&longs;tu <lb/>lati huius dicta fuere, nempè Archimedem &longs;upponere, di&longs;tan­<lb/>tias e&longs;&longs;e in vna, eademquè recta linea con&longs;titutas. ideoquè in <lb/>prima propo&longs;itionec inquit, Grauia, qu&ecedil; ex <expan abbr="di&longs;tãtijs">di&longs;tantijs</expan> &ecedil;quali <lb/>bus <expan abbr="æquepõderãt">æqueponderant</expan>, æqualia e&longs;&longs;e inter&longs;e; Archimedes què <expan abbr="demõ">demom</expan> <lb/>&longs;trat, quòd quando æqueponderant, &longs;unt æqualia: ex dictis <lb/>&longs;equitur, &longs;i æqueponderant, ergo centrum grauitatis magni­<lb/>tudinis ex ip&longs;is compo&longs;it&ecedil; erit in eo puncto, vbi æqueponde­<lb/>rant; hoc e&longs;t in medio di&longs;tantiarum, line&ecedil; &longs;cilicet, qu&ecedil; <expan abbr="grauiũ">grauium</expan> <lb/>centra grauitatis coniungit. quod idem e&longs;t, ac &longs;i Archimedes <lb/>dixi&longs;&longs;et. Grauia, qu&ecedil; habent centrum grauitatis in medio li­<lb/>ne&ecedil;, qu&ecedil; magnitudinum centra grauitatis coniungit, &ecedil;qua­<lb/>lia &longs;unt inter&longs;e. cuius quidem h&ecedil;c quarta propo&longs;itio videtur <lb/>e&longs;&longs;e conuer&longs;a. quamuis Archimedes loco grauium nominet <lb/>magnitudines. Pr&ecedil;terea in tertia propo&longs;itione, quoniam <expan abbr="o&longs;t&etilde;-dit">o&longs;ten­<lb/>dit</expan> Archimedes, in&ecedil;qualia grauia &ecedil;queponderare ex <expan abbr="di&longs;tãtijs">di&longs;tantijs</expan> <lb/>in&ecedil;qualibus, ita vt grauius &longs;it in minori di&longs;tantia, &longs;equitur er <lb/>go centrum grauitatis e&longs;t in eo puncto, vbi æqueponderant; <lb/>& idem e&longs;t, ac &longs;i dixi&longs;&longs;et, in æqualium grauium centrum gra­<lb/>uitatis e&longs;t in recta linea, quæ ip&longs;orum centra grauitatis con­<lb/>iungit; ita vt &longs;it propinquius grauiori, remotius uerò leuiori.
<pb pagenum="48"/>vnde &longs;equitur centrum grauitatis ip&longs;orum grauium ubicum <lb/>que e&longs;&longs;e po&longs;&longs;e in recta linea, qu&ecedil; ipiorum centra grauitatis <expan abbr="cõ">com</expan> <lb/>iungit. Ex quibus concludi potelt, <expan abbr="c&etilde;trum">centrum</expan> grauitatis magni­<lb/>tudinis ex duabus magnitudinibus compo&longs;it&ecedil; e&longs;&longs;e in recta li <lb/>nea, quæ ip&longs;orum centra grauitatis connectit. </s></p>
<p type="main">
<s>Po&longs;tremò notandum e&longs;t, Archimedem ea, quæ in &longs;uperio <lb/>ribus propo&longs;itionibus nuncupauit grauia, in hac quarta pro <lb/>po&longs;itione, veluti etiam in &longs;equentibus, non ampliùs grauia, <lb/>&longs;ed (vti diximus) magnitudines nominare. quod quidem his <lb/>de cau&longs;is id ab ip&longs;o factum exi&longs;timo. primùm enim, quia in <lb/>his expre&longs;se quærit centrum grauitatis; quod quidem <expan abbr="c&etilde;trum">centrum</expan>, <lb/>quamuis &longs;it centrum grauitatis, potiùs re&longs;picit <expan abbr="magnitudin&etilde;">magnitudinem</expan>, <lb/>quàm graue aliquod. Nam cùm dicim us centrum grauitatis, <lb/>&longs;tatim innuim us &longs;i tum, &longs;itum inquàm determinatum &longs;igu­<lb/>ræ, in qua e&longs;t; &longs;iquidem centrum grauitatis e&longs;t punctum, & <lb/>(vtita dicam) punctum grauitatis eius, in quo e&longs;t. & ideo, <lb/>quoniam magnitudo formam habet dete mina tam, <expan abbr="centrũ">centrum</expan> <lb/>grauitatis rectè pote&longs;t re&longs;picere &longs;itum re&longs;pectu magnitudinis, <lb/>in qua e&longs;t; quod tamen efficere non pote&longs;t re&longs;pectu grauis. <lb/>etenim graue, ut graue e&longs;t, non habet formam determina <expan abbr="tã">tam</expan>; <lb/>cùm eadem grauitas e&longs;&longs;e po&longs;&longs;itin cubo, in piramide, alii&longs;què <lb/>corporibus quibu&longs;cunque, modò minoribus, modò maiori­<lb/>bus, prout &longs;unt diuer&longs;arum &longs;pecierum. quare centrum grauita <lb/>tis non pote&longs;t re&longs;picere &longs;itum in grauibus, quatenus grauia <expan abbr="cõ">com</expan> <lb/>&longs;iderantur; &longs;ed quatenus magnitudines exi&longs;tunt. Præterea Ar­<lb/>chimedes loco grauium magnitudines nominat, quia eas di­<lb/>ui&longs;ibiles con&longs;iderat, quod e&longs;t proprium magnitudinis; vt in &longs;e <lb/>xta, &longs;eptima, & octaua propo&longs;itione. & quamuis, dum <expan abbr="diuidũ">diuidum</expan> <lb/>tur magnitudines, grauia quoque diui&longs;a proueniant; non ta­<lb/>men propterea grauia diuiduntur, ut grauia. <expan abbr="nõ">non</expan>.n. hoc ip&longs;is <lb/>competit, vt grauibus; &longs;ed vt magnitudinibus, quæ &longs;unt por <lb/>&longs;e diui&longs;ibiles. Archimedes igitur his de cau&longs;is nomen <expan abbr="grauiũ">grauium</expan> <lb/>in magnitudines mutauit. in &longs;uperioribus enim theoremati­<lb/>bus pertractauit, quomodo res æqueponderant ex di&longs;tantijs <lb/>modò æqualibus, modò in æqualibus. & quoniam res <expan abbr="&ecedil;quepõ">&ecedil;quepom</expan> <lb/>derant, prout &longs;unt magis grauia, & minùs grauia; non ut <expan abbr="sũt">sunt</expan> <lb/>maiores, vel minores magnitudines, &longs;iquidem talis naturæ
<pb pagenum="49"/>e&longs;&longs;e pote&longs;t minor magnitudo, qu&ecedil; maiore magnitudine alte <lb/>rius nature grauior exi&longs;tat; proindé Archimedesin &longs;uperiori­<lb/>busrectè grauia nuncupauit; optimèquè in his magnitudines <lb/>vocat. Atverò aduertendum e&longs;t, quòd quamuis Archimedes <lb/>in his magnitudines nominet, non propterea exi&longs;tim andum <lb/>e&longs;t, eum intelligere magnitudines tantùm; &longs;ed magnitudines <lb/>grauitate prçditas, ita utin ip&longs;is omnino grauitatem re&longs;piciat. <lb/>Etenim pluribus modis in telligere po&longs;&longs;umus magnitudines, <lb/>vel enim ut &longs;int inter&longs;e eiu&longs;dem &longs;peciei, vel diuer&longs;æ; nec <expan abbr="nõ">non</expan> <lb/>in&longs;uper homogeneæ, vel heterogeneæ. vt in hac propo&longs;itione <lb/><expan abbr="quãdo">quando</expan> Archimedes pponit duas magnitudines &ecedil;quales, tuc <lb/>intelligere po&longs;&longs;umus eas e&longs;&longs;e eiu&longs;dem &longs;peciei, & homogeneas; <lb/>quæ, cùm &longs;int æquales, erit & grauitas vnius grauita ti alterius <lb/>æqualis. &longs;i verò con&longs;ideremus eas e&longs;&longs;e diuer&longs;æ &longs;peciei, & e­<lb/>tiam heterogeneas; tunc quando Archimedes proponit has <lb/>magnitudines æ quales; intelligendum e&longs;t, eas e&longs;&longs;e æ quales in <lb/>grauita te; quæ quidem efficit, vt demon&longs;tratio, quod propo­<lb/>&longs;itum e&longs;t, concludat. vtex eius demon&longs;tratione patet. Et his <lb/>quoque modis intelligere po&longs;&longs;umus magnitudines in &longs;equen <lb/>tibus v&longs;que ad nonam propo&longs;itionem in quibus &longs;cilicet intel <lb/>ligere po&longs;&longs;umus magnitudines e&longs;&longs;e non &longs;olùm eiu&longs;dem &longs;pe­<lb/>ciei, vel diuer&longs;æ, verùm etiam & homogeneas. & heteroge­<lb/>neas. ut po&longs;t &longs;eptimam clariùs o&longs;tendemus. Verùm de­<lb/>mon&longs;trationes clariores red duntur, &longs;i intelligamus magnitu­<lb/>dines e&longs;&longs;e eiu&longs;dem &longs;peciei, & homogeneas, in quibus graui­<lb/>tas magnitudini re&longs;pondet, vt &longs;i ip&longs;arum altera fuerit alte­<lb/>rius dupla, & grauitas vnius grauitatis alterius dupla exi&longs;tat. <lb/>Quòd &longs;i magnitudo fuerit alterius tripla, vel quadrupla, &c. <lb/>erit & grauitas grauitatis tripla, vel quadrupla, & &longs;ic dein­<lb/>ceps. deinde &longs;i magnitudo bifariam diui&longs;a fuerit, & ip&longs;ius gra <lb/>uitas in duas &ecedil;quas partes &longs;it quoque diui&longs;a. quòd &longs;i magnitu­<lb/>do in plures diuidatur partes, & grauitas quoque in totidem <lb/>eiu&longs;dem proportionis diui&longs;a proueniat. </s></p>
<pb pagenum="50"/>
<p type="head">
<s>PROPOSITIO. V.</s></p>
<p type="main">
<s>Si trium magnitudinum centra grauitatis in re <lb/>cta linea fuerint po&longs;ita, & magnitudines æ qualem <lb/>habuerint grauitatem, acrectæ lineæ inter centra <lb/>fuerint æ quales, magnitudinis ex omnibus magni <lb/>tudinibus compo&longs;itæ centrum grauitatis erit <expan abbr="pũ">pum</expan> <lb/>ctum, quod & ip&longs;arum mediæ centrum grauitatis <lb/>exi&longs;tit. </s></p>
<figure></figure>
<p type="main">
<s><emph type="italics"/>Sint tres magnitudines ACB. ip&longs;arum autem centra grauitatis &longs;int <lb/>puncta ACB in resta linea<emph.end type="italics"/> ACB <emph type="italics"/>po&longs;ita. &longs;int verò magnitudines ACB <lb/>æquales; rectæquè lineæ AC CB<emph.end type="italics"/> inter centra ip&longs;arum <emph type="italics"/>aquales. Di <lb/>co magnitudims ex omnibus<emph.end type="italics"/> ACB <emph type="italics"/>magnitudinibus compo&longs;it æ <expan abbr="centrũgra">centrungra</expan> <lb/>uitatis e&longs;&longs;e punetum C.<emph.end type="italics"/> quod e&longs;t centrum grauitatis mediæ ma­<lb/>gnitudinis. <emph type="italics"/>Quoniam enim magnitudines AB æqualem habent graui<emph.end type="italics"/><lb/>
<arrow.to.target n="marg37"></arrow.to.target> <emph type="italics"/>tatem<emph.end type="italics"/>; magnitudinis ex vtri&longs;que AB compo&longs;itæ <emph type="italics"/>centrum graui <lb/>tatis erit punctum C: cùm &longs;int AC CB æquales.<emph.end type="italics"/> &longs;itquè propterea <lb/>punctum C medium rectæ line&ecedil; AB. <emph type="italics"/>Sed & magnitudinis C <expan abbr="cē">cem</expan> <lb/>trum grauitatis est<emph.end type="italics"/> idem <emph type="italics"/>punctum C.<emph.end type="italics"/> punctum ergo C <expan abbr="triũ">trium</expan> ma­<lb/>gnitudinum ABC centrum quoque grauitatis erit. <emph type="italics"/>Quare pa <lb/>tet magnitudinis ex omnibus magnitudinibus<emph.end type="italics"/> ACB <emph type="italics"/>compo&longs;itæ centrum <lb/>grauitatis e&longs;&longs;e punctum, quod &<emph.end type="italics"/> magnitudinis <emph type="italics"/>mediæ centrum graui­<lb/>tatis existit.<emph.end type="italics"/> quod demon&longs;trare oportebat. </s></p>
<pb pagenum="51"/>
<p type="margin">
<s><margin.target id="marg37"></margin.target>4 <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="head">
<s>COROLLARIVM. I.</s></p>
<p type="main">
<s>Ex hoc autem manife&longs;tum e&longs;t, &longs;i quotcunquè <lb/>magnitudinum, & numero imparium, centra
<arrow.to.target n="marg38"></arrow.to.target> gra­<lb/>uitatis in re cta linea con&longs;tituta fuerint; & magni­<lb/>tudines æ qualem habuerint grauitatem; rectæquè <lb/>lineæ inter ip&longs;arum centra fuerint æ quales, ma­<lb/>gnitudinis ex omnibus magnitudinibus compo&longs;i <lb/>tæ centrum grauitatis e&longs;&longs;e punctum, quod & ip&longs;a­<lb/>rum mediæ centrum grauitatis exi&longs;tit. </s></p>
<p type="margin">
<s><margin.target id="marg38"></margin.target>*</s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<figure></figure>
<p type="main">
<s>Ex demon&longs;tratione colligit Archimedes &longs;i plures fuerint <lb/>magnitudines, <expan abbr="quã">quam</expan> tres; dummodo &longs;int numero impares, vt <lb/>ABCDE; quarum centra grauitatis ABCDE reperiantur in li <lb/>nea recta AE. fuerint autem h&ecedil; magnitudines æ quales in gra <lb/>uitate. in&longs;uper rect&ecedil; line&ecedil; AB BC CD DE, qu&ecedil; &longs;unt in ter <expan abbr="c&etilde;-tra">cen­<lb/>tra</expan> grauita tis, fuerint æ quales: magnitudinis ex omnibus ma <lb/>gnitudinibus ABCDE compo&longs;itæ centrum grauita tis e&longs;&longs;e <lb/>punctum C. quod e&longs;t centrum grauitatis magnitudinis <lb/>mediæ. </s></p>
<p type="main">
<s>Eodem enim modo, ac primùm quidem ex demon&longs;tratio <lb/>ne patet <expan abbr="punctũ">punctum</expan> C centrum e&longs;&longs;e grauita tis <expan abbr="triũ">trium</expan> <expan abbr="magnitudinũ">magnitudinum</expan> <lb/>BCD, & quoniam AB BC &longs;unt æquales ip&longs;is CD DE,
<pb pagenum="52"/>erit AC ip&longs;i CE &ecedil;qualis. cùm què &longs;it grauitas magnitudinis <lb/>
<arrow.to.target n="marg39"></arrow.to.target> A &ecedil;qualis grauitati ip&longs;ius E, erititidem punctum C magni <lb/>tudinum AE centrum grauitatis. ergo punctum C magni <lb/>tudinis ex omnibus magnitudinibus ABCDE compo&longs;itæ <lb/>centrum grauitatis exi&longs;tit. </s></p>
<p type="margin">
<s><margin.target id="marg39"></margin.target>4 <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="main">
<s>Quòd &longs;i fuerint ad huc plures magnitudines, impares verò <lb/>extiterint; quæ ita &longs;e habeant, vt expo&longs;itum e&longs;t; &longs;imiliter <expan abbr="o&longs;t&etilde;">o&longs;tem</expan> <lb/>detur, centrum grauitatis mediæ magnitudinis centrum e&longs;&longs;e <lb/>grauitatis magnitudinis ex omnibus magnitudinibus com­<lb/>po&longs;itæ. </s></p>
<p type="main">
<s>
<arrow.to.target n="marg40"></arrow.to.target> In hoc corollario, verba illa, <emph type="italics"/>& magnitudines æqualem habue­<lb/>rint grauitatem<emph.end type="italics"/> in greco codiceita habentur. <foreign lang="greek">e(ika tate i)/son a)w_e\xon­<lb/>ta a)w_o\ tou= me\sou mege\qeos i)\sonba/<10>os e)/xwnt<gap/></foreign> quorum multa &longs;uperuaca­<lb/>nea nobis vi&longs;a &longs;unt; loco quorum (vt arbitror) rectè <expan abbr="congru&etilde;t">congruent</expan> <lb/><foreign lang="greek">kai) ta\ mege\qea i)/son ba<10>os e)/xwnti</foreign>, vt vertimus. Nam &longs;i ordinis atque <lb/><expan abbr="cõditionum">conditionum</expan> propo&longs;it&ecedil; propo&longs;itionis ratio habenda e&longs;t, opor <lb/>tet vt magnitudines &ecedil;qualem habeant grauitatem; Nam & <lb/>Archimed es in &longs;equentibus demon&longs;trationibus ijs vtitur, ut <lb/>&longs;unt æquegraues. Adhuc tamen veritatem habebit &longs;i cæteris <lb/>conditionibus illud quoque addere voluerimus, nempe &longs;i <emph type="italics"/>ma <lb/>gnitudines à media magnitudine æqualiter di&longs;tantes æqualem habuerint <lb/>grauitatem<emph.end type="italics"/> eodem modo punctum C centrum erit grauitatis <lb/>
<arrow.to.target n="fig23"></arrow.to.target><lb/>magnitudinis ex omnibus ABCDE compo&longs;it&ecedil;, Nam &longs;i ma­<lb/>gnitudines à media magnitudine &longs;unt &ecedil;quegraues; &ecedil;qualem <lb/>quoque habebunt grauitatem magnitudines AE; veluti ma­<lb/>gnitudines BD, quæ æ qualiter à media magnitudine C di­<lb/>&longs;tant. & quam uis non &longs;int omnes æ quegraues, &longs;ufficit, vt AE <lb/>quæ &ecedil;qualiter à media magnitudine di&longs;tant, &longs;int &ecedil;quegraues. <lb/>&longs;imiliter BD &ecedil;quegraues. Eadem enim ratione, quoniam <lb/>BD &longs;untæ quegraues, & di&longs;tantiæ BC CD &ecedil;quales; erit C ip&longs;a-
<pb pagenum="53"/>rum BD ccntrum grauitatis. pari què ratione C erit centrum <lb/>grauitatis magnitudinum AE &ecedil;quegrauium. cum &longs;int AC <lb/>CE &ecedil;quales, & idem C e&longs;t grauitatis centrum magnitudinis <lb/>C. ergo punctum C magnitudinis ex omnibus magnitudini­<lb/>bus ABCDE compo&longs;it&ecedil; centrum grauitatis exi&longs;tit. </s></p>
<p type="margin">
<s><margin.target id="marg40"></margin.target>*</s></p>
<figure id="fig23"></figure>
<p type="head">
<s>COROLL ARIVM. II.</s></p>
<p type="main">
<s>Si verò magnitudines fuerint numero pares; <lb/>& ip&longs;arum centra grauitatis in recta linea extite­<lb/>rint, magnitudine&longs;què æ qualem habuerint graui
<arrow.to.target n="marg41"></arrow.to.target><lb/>tatem, rectæ què lineæ inter centra fuerint æ qua <lb/>les: magnltudinis ex omnibus magnitudinibus <expan abbr="cõ">com</expan> <lb/>po&longs;itæ centrum grauitatis erit medium rectæ li­<lb/>neæ, quæ magnitudinum centra grauitatis <expan abbr="coniũ-git">coniun­<lb/>git</expan>. vt in &longs;ubiecta figura. </s></p>
<p type="margin">
<s><margin.target id="marg41"></margin.target>*</s></p>
<figure></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Colligit præterea Archimedes &longs;i magnitudines ABCDEF <lb/>fuerint numero pares, quarum centra grauitatis ABCDEF in <lb/>recta linea AF &longs;int con&longs;tituta; magnitudine&longs;què &longs;int æquales <lb/>in grauitate; &longs;intquè inter centra line&ecedil; AB BC CD DE EF <lb/>æ quales. diuidatur autem AF bifariam in G. erit punctum <lb/>G centrum grauita tis magnitudinis ex omnibus compo&longs;i­<lb/>tæ quod quidem, figura tantùm in&longs;pecta, per&longs;picuum e&longs;t. <lb/>Cùm enim magnitudines AF &longs;int æquegraues, & AG GF
<pb pagenum="54"/>
<arrow.to.target n="marg42"></arrow.to.target> &longs;int æ quales, erit G centrum grauitatis magnitudinis ex AF <lb/>compo&longs;itæ. quia verò AB e&longs;t ip&longs;i EF æqualis, reliqua BG <lb/>ip&longs;i GE æqualis exi&longs;ter. & &longs;unt magnitudines BE çquegra­<lb/>ues, eritidem G centrum grauitatis <expan abbr="magnitudinũ">magnitudinum</expan> BE. &longs;imili­<lb/>ter cùm &longs;it BC æ qualis DE, relin quetur CG ip&longs;i GD &ecedil;qua­<lb/>lis; magnitudinesquè CD &longs;unt &ecedil;quegraues. ergo <expan abbr="pũctum">punctum</expan> G <expan abbr="c&etilde;">cem</expan> <lb/>trum e&longs;t quoque magnitu dinum CD. Vnde &longs;equitur, <expan abbr="punctũ">punctum</expan> <lb/>G magnitudinis ex omnibus magnitudinibus ABCDEF <expan abbr="cõ-po&longs;itæ">con­<lb/>po&longs;itæ</expan> centrum grauitatis exi&longs;tere. </s></p>
<p type="margin">
<s><margin.target id="marg42"></margin.target>4 <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="main">
<s>
<arrow.to.target n="marg43"></arrow.to.target> Hoc quoque loco verba illa <emph type="italics"/>magnitudine&longs;què æqualem habuerint <lb/>grauitatem.<emph.end type="italics"/> Græ cus cod ex ita mendosè legit. <foreign lang="greek">kai\ ta/| me\sa auths i)\son <lb/>ba/<10>os e)/xwnti</foreign>, quæ quidem verba hoc modo re&longs;titui po&longs;&longs;unt. <lb/><foreign lang="greek">kai\ ta mege\qea i)/son ba/<10>os e)/xwnti. </foreign></s></p>
<p type="margin">
<s><margin.target id="marg43"></margin.target>*</s></p>
<p type="main">
<s>In præcedenti propo&longs;itione o&longs;tendir Archimedes, quomo <lb/>do &longs;e habet centrum gra uitatis magnitudinis ex duabus ma­<lb/>gnitudinibus equalibus compo&longs;itæ. In hac autem <expan abbr="demõ&longs;trat">demon&longs;trat</expan>, <lb/>vbi &longs;imiliter grauitatis cen trum reperitur inter plures magni­<lb/>tudines æquegraues, & inter &longs;e &ecedil;qualiter di&longs;tantes. ex quibus <lb/>randem colliget fun damentum &longs;æpiùs dictum. nempè &longs;i ma­<lb/>gnitudines &ecedil;queponderare debent; ita &longs;e habebit magnitudi­<lb/>num grauitas ad grauitatem, ut &longs;e habent di&longs;tantiæ permuta <lb/>tim, ex quibus &longs;u&longs;pen duntur. & hoc demon&longs;trat Archimedes <lb/>in duabus &longs;equen tibus propo&longs;itionibus. nam magnitudines, <lb/>vel &longs;unt commen&longs;urabiles in ter&longs;e&longs;e, vel incommen&longs;urabiles. <lb/>de commen&longs;urabilibusaget in &longs;equenti: de incommen&longs;urabi <lb/>libusverò in &longs;eptima propo&longs;itione. & Archimedes duas <expan abbr="&longs;equ&etilde;-tes">&longs;equen­<lb/>tes</expan> propo&longs;itiones ueluti coniunctas proponit. Nam in &longs;exta <lb/>inquit <emph type="italics"/>Magnitudines commen&longs;urabiles,<emph.end type="italics"/> &c. in &longs;<gap/>ptima uerò in­<lb/>quit, <emph type="italics"/>Si autem magnitudines &longs;uerint incommen&longs;urabiles,<emph.end type="italics"/> qua&longs;i vna <expan abbr="tã">tam</expan> <lb/>tùm &longs;it propo&longs;itio in duas partes diui&longs;a. ita ut neque numeris <lb/>e&longs;&longs;ent di&longs;ting uende, &longs;ed pro vna tantùm propo&longs;iuone &longs;um n<gap/><expan abbr="&etilde;">em</expan> <lb/>dæ, ob&longs;equen tis autem demon&longs;trationis faciliorem <expan abbr="intellig&etilde;">intelligem</expan> <lb/>tiam hecpriùs præmittimus. </s></p>
<p type="head">
<s>LEMMA.</s></p>
<p type="main">
<s>Si du&ecedil; fuerint magnitudines in æquales, quarum maior &longs;it <lb/>alterius dupla, tertia verò qu&ecedil;dam magnitudo minorem me-
<pb pagenum="55"/>tiatur. maiorem quoque in partes numero pares metietur. </s></p>
<p type="main">
<s>Sint du&ecedil; in &ecedil;quales magni <lb/>
<arrow.to.target n="fig24"></arrow.to.target><lb/>tudines AB, &longs;itquè A ip&longs;ius <lb/>B duplex. magnitudo <expan abbr="aut&etilde;">autem</expan> <lb/>C <expan abbr="magnitudin&etilde;">magnitudinem</expan> B metia­<lb/>tur. Dico C <expan abbr="magnitudin&etilde;">magnitudinem</expan> <lb/>A metiri, men&longs;urationesquè numero pares e&longs;&longs;e. Quoniam <lb/>enim C metitur B, eodem numero C metietur medietates <lb/>ip&longs;ius A, quæ &longs;untip&longs;i B æquales. ergo duplo plures erunt nu <lb/>mero men&longs;urationes ip&longs;ius A, quàm ip&longs;ius B. quare men&longs;u­<lb/>rationes ip&longs;ius A &longs;unt numero pares. duplum enim &longs;emper <lb/>paritatem &longs;ecum affert. quod demon&longs;trare oportebat. </s></p>
<figure id="fig24"></figure>
<p type="main">
<s>Porrò maxima in his duabus &longs;equentibus propo&longs;itionibus <lb/>adhibenda e&longs;t diligentia; quibus tota rerum Mechanicarum <lb/>ratio in nititur. Quocirca vt harum propo&longs;itionum demon­<lb/>&longs;trationes perfectè intelligere po&longs;&longs;imus; præter eos argumen­<lb/>tandi modos, quorum ante quintam huius propo&longs;itionem <lb/>meminimus; alterum quoque modum, quo Archimedes in <lb/>
<arrow.to.target n="fig25"></arrow.to.target><lb/>hac&longs;exta propo&longs;itione vtitur, noui&longs;&longs;e oportet. vt &longs;cilicet, &longs;i ma <lb/>gnitudo A æqueponderatip&longs;is BC facta &longs;u&longs;pen&longs;ione ex <expan abbr="pũ-cto">pun­<lb/>cto</expan> D; ita &longs;cilicet, vt D &longs;it centrum grauitatis magnitudinis <lb/>ex omnibus ABC magnitudinibus compo&longs;itæ; ip&longs;arum verò
<pb pagenum="56"/>magnitudinum BC, hoc e&longs;t magnitudinis ex BC compo&longs;i­<lb/>tæ centrum grauitatis &longs;it punctum E; auferantur verò BC <lb/>à linea EA, & ip&longs;arum loco ponatur in E magnitudo; <lb/>quæ &longs;it vtri&longs;que &longs;imul BC &ecedil;qualis, vtin &longs;ecunda figura. Dico <lb/>eodem modo pondera ABC &ecedil;queponderare in prima figu­<lb/>ra, veluti grauia AE in &longs;ecunda. </s></p>
<figure id="fig25"></figure>
<p type="main">
<s>Primum autem, vthoc recte per <lb/>
<arrow.to.target n="fig26"></arrow.to.target><lb/>pendamus, intelligantur pondera <lb/>BC (vt in tertia figura) &longs;eor&longs;um <lb/>à linea CA, & penes di&longs;tantias EC <lb/>EB con&longs;tituta. quorum quidem <expan abbr="põ-derum">pon­<lb/>derum</expan> &longs;it centrum grauitatis E. &longs;i igitur intelligatur poten <lb/>
<arrow.to.target n="marg44"></arrow.to.target> tia in E &longs;u&longs;tinere pondera BC, hoc e&longs;t pondus exip&longs;is BC <lb/>compo&longs;itum: pondera utique manebunt. quòd &longs;i ambo pe­<lb/>penderint, vt quinquaginta, potentia in E tantùm quinqua <lb/>ginta &longs;u&longs;tinebit. quoniam totum &longs;u&longs;tinebit pondus ex ip&longs;is <lb/>compo&longs;itum, auferantur verò pondera BC à &longs;itu BC, intelli <lb/>ganturquè pondera e&longs;&longs;e in E con&longs;tituta; hoc e&longs;t vnum &longs;it <lb/>pondus ex ip&longs;is &longs;imul iun ctis compo&longs;itum, cuius <expan abbr="c&etilde;trum">centrum</expan> gra­<lb/>uitatis &longs;itin E con&longs;titutum; tunc eadem potentia in E eo­<lb/>dem modo hoc pondus &longs;u&longs;tinebit; propterea quod <expan abbr="eod&etilde;">eodem</expan> mo­<lb/>do quinquagin ta tantùm &longs;u&longs;tinebit. Quare pondera BC <expan abbr="tã">tam</expan> <lb/>ex di&longs;tan tijs EC EB grauitant, quàm &longs;i vtraque in E con <lb/>&longs;tituta fuerint; vel quod idem e&longs;t, quàm pondus ip&longs;is BC &longs;i­<lb/>mul æ quale in E po&longs;itum. Ex quo patetid, quod initio pr&ecedil;­<lb/>fati &longs;um us, nempe, vnumquodquè graue in eius centro gra­<lb/>uitatis propriè grauitare. Quocum que enim modo <expan abbr="ead&etilde;">eadem</expan> gra <lb/>uia &longs;e&longs;e habent, eodem &longs;emper modo in eius grauitatis <expan abbr="c&etilde;tro">centro</expan> <lb/>grauitant. </s></p>
<p type="margin">
<s><margin.target id="marg44"></margin.target><emph type="italics"/>per def. <lb/>cent. grau.<emph.end type="italics"/></s></p>
<figure id="fig26"></figure>
<p type="main">
<s>Quibus cognitis, intelligantur nunc grauia BC in linea <lb/>CA po&longs;ita e&longs;&longs;e; ut in &longs;uperiori figura: & ut quod propo&longs;itum <lb/>fuit, o&longs;tendatur; hoc modo argumentari licebit. Quoniam <lb/>enim magnitudines BC &longs;uam habent grauitatem in E, &longs;iqui <lb/>dem pro vna tantùm in telliguntur magnitudine ex BC com <lb/>po&longs;ita, cuius punctum E centrum grauitatis exi&longs;tit. in <expan abbr="&longs;ecũ">&longs;ecum</expan> <lb/>da verò figura magnitudo E &longs;imiliter &longs;uam habet <expan abbr="grauitat&etilde;">grauitatem</expan> <lb/>in puncto E; quod e&longs;t eius <expan abbr="centrũ">centrum</expan> grauitatis. atque magnitu
<pb pagenum="57"/>do E e&longs;tip&longs;is BC &longs;imul &longs;umptis &ecedil;qualis. di&longs;tanti&ecedil; verò AD <lb/>DE &longs;unt æquales, cum &longs;int &ecedil;edem; erit vtique punctum D in <lb/>&longs;ecunda figura centrum grauitatis magnitudinis ex AE com­<lb/>po&longs;itæ, veluti D in prima figura ip&longs;arum ABC centrum gra <lb/>uitatis exi&longs;tit. ac propterea in vtraque figura pondera æque­<lb/>ponderabunt: </s></p>
<p type="main">
<s>Cæterum hoc quoque o&longs;tendemus hoc pacto. </s></p>
<figure></figure>
<p type="main">
<s>Ii&longs;dem namque po&longs;itis; æqueponderarent &longs;cilicet grauia <lb/>ABC facta ex D &longs;u&longs;pen&longs;ione. &longs;itquè punctum E <lb/>centrum grauitatis ponderum CB. quæ quidem pondera <lb/>CB grauitatis centrum habeantin linea CB. Dico pondus <lb/>A ponderi ip&longs;is CB &longs;imul &longs;umptis æquali in E con&longs;ti­<lb/>tuto æqueponderare. Mente concipiamus di&longs;tantias EC <lb/>EB, manente centro E, circa ip&longs;um circumuerti po&longs;&longs;e; <lb/>vt modò &longs;intin FEG, modòin HEK. &longs;imiliter in­<lb/>telligantur pondera CB, modò in FG, modò in HK <lb/>exi&longs;tere. Quoniam igitur punctum E. centrum e&longs;t <lb/>grauitatis ponderum CB; erit idem E (cùm &longs;itum <lb/>nonmutet) centrum grauitatis ponderum in &longs;itu FG, ac <lb/>ponderum in HK exi&longs;tentium. Quiaverò vnumquod­<lb/>que pondus (ex dictis) propiè in eius centro grauitatis graui <lb/>tat; pondera &longs;imul CB &longs;iue &longs;intin FG, &longs;iuein HK, proprie <lb/>in puncto E grauitabunt. At verò quoniam idem
<pb pagenum="58"/>pondus vnam & eandem &longs;emper habet grauitatem; erit <expan abbr="põdus">pondus</expan> <lb/>ex CB compo&longs;itum æquegraue, tam in &longs;itu CB, quàm in <lb/>FG, & in &longs;itu HK. con&longs;iderando nempe pondera CB (ut <lb/>revera &longs;unt) nilaliud e&longs;&longs;e ni&longs;i vnum tantùm pondus ex CB <lb/>compo&longs;itum. Ex quibus per&longs;picuum e&longs;t, punctum E eodem <lb/>&longs;emper modo grauitare. Quare quoniam pondera CB in &longs;i­<lb/>tu CB ip&longs;i A &ecedil;queponderant, &longs;uamquè habent grauitatem <lb/>in puncto E; eadem pondera CB &longs;iue &longs;int in FG, &longs;iue in <lb/>HK, eidem ponderi A æqueponderabunt. &longs;iquidem propriè <lb/>&longs;emper grauitantin E, & eandem &longs;emper habent <expan abbr="grauita-t&etilde;">grauita­<lb/>tem</expan> Intelligatur denique HEK in centrum mundi tendere; e­<lb/>runtvtique vtraque pondera HK, tanquam in puncto E <expan abbr="cõ">com</expan> <lb/>&longs;tituta, vt ex prima propo&longs;itione no&longs;trorum Mechanicorum <lb/>elici pote&longs;t, quamuis per&longs;e notum &longs;it. &longs;iquidem &longs;eor&longs;um pon <lb/>dus H &longs;ecundùm eius centrum grauitatis propriè grauitat &longs;u <lb/>per puncto E; pondus verò K e&longs;t, tanquam ex E appen&longs;um; <lb/>vndè & in eodem puncto E quoque grauitat. Itaque <expan abbr="quoniã">quoniam</expan> <lb/>ambo propriè grauitant in E, erunt pondera HK perinde, <lb/>ac&longs;i vnum e&longs;&longs;et pondusip&longs;is HK, hoc e&longs;tip&longs;is CB æquale, cu <lb/>ius centrum grauitatis &longs;itin E con&longs;titutum. atverò pondus <lb/>A ip&longs;is CB in &longs;itu HK exi&longs;tentibus æqueponderat. ergo <expan abbr="id&etilde;">idem</expan> <lb/>pondus A ip&longs;is CB in E con&longs;titutis, hoc e&longs;t ponderi ip&longs;is CB <lb/>&longs;imul &longs;umptis &ecedil;quali in E po&longs;ito æqueponderabit. quod de­<lb/>mon&longs;trare oportebat. </s></p>
<p type="main">
<s>Quod idem quoque, &longs;i plura e&longs;&longs;ent pondera, &longs;imiliter o­<lb/>&longs;tendetur. </s></p>
<p type="main">
<s>Valetitaque con&longs;equentia, punctum D centrum e&longs;tgra­<lb/>uitatis magnitudinis ex ponderibus ABC compo&longs;it&ecedil;; ergoi­<lb/>dem punctum D centrum e&longs;t grauitatis ponderis in A, & <expan abbr="põ">pom</expan> <lb/>derisip&longs;is BC &longs;imul &ecedil;qualis in E con&longs;tituti. ex quo con&longs;equi­<lb/>tur, quòd &longs;i magnitudines ABC ex D æqueponderant, ergo <lb/>ex eodem D magnitudo ip&longs;is BC &longs;imul æqualis in E po&longs;ita, <lb/>& magnitudo A æqueponderabunt. quòd &longs;i rectè perpenda­<lb/>mus, nil aliud &longs;unt pondera in BC, ni&longs;i magnitudo in E con­<lb/>&longs;tituta. &longs;iquidem punctum E ip&longs;ius centrum grauitatis <lb/>exi&longs;tit </s></p>
<p type="main">
<s>In no&longs;tro autem Mechanicorum libro in quinta propo&longs;i-
<pb pagenum="59"/>tione tractatus de libra duas attulimus demon &longs;trationes <expan abbr="o&longs;t&etilde;-tes">o&longs;ten­<lb/>tes</expan> duo pondera vt CB tam in punctis CB ponderare, quàm &longs;i <lb/>vtraque ex puncto E &longs;u&longs;pendantur. At verò quo niam demon <lb/>&longs;trationes ibi allatæ ijs indigent, qu&ecedil; Archimedes in &longs;equen­<lb/>ti &longs;exta propo&longs;itione demon&longs;trauit, idcirco demon&longs;trationes <lb/>illæ huic loco non &longs;unt oportunæ; vt ex ip&longs;is&longs;umi po&longs;&longs;it tan­<lb/>quam demon&longs;tratum pondera CB, tam in punctis CB pon­<lb/>derare, quàm &longs;i vtraque ex E &longs;u&longs;pendantur. Quare hoc loco h&ecedil; <lb/>tantùm &longs;ufficiant rationes, quæ dictæ &longs;unt. Ex quibus pote&longs;t <lb/>Archime des di&longs;tam con&longs;equentiam colligere; nempè magni­<lb/>tudines ABC ex D æqueponderant, auferantur autem BC, <lb/>& loco ip&longs;arum vtri&longs;que &longs;imul &ecedil;quegrauis ponatur magnitu­<lb/>do in E; &longs;imiliter h&ecedil;c magnitudo ip&longs;i A æqueponderabit. Po­<lb/>&longs;tea verò ex ijs, quæ Archimedes demon&longs;trauit, fieri pote&longs;t re <lb/>gre&longs;&longs;us; v<gap/>apertiùs, manife&longs;tiù&longs; què cogno&longs;cere valeamus, pon <lb/>dera BC ita ponderare, ac &longs;i vtraque ex puncto E &longs;u&longs;pen­<lb/>dantur. </s></p>
<figure></figure>
<p type="main">
<s>C&ecedil;terum hoc loco Archimedes non &longs;olùm de duobus, <expan abbr="verũ">verum</expan> <lb/>etiam de pluribus ponderibus idip&longs;um <expan abbr="intelligendũ">intelligendum</expan> admittit. <lb/>vt &longs;i magnitudines STVXZM æqueponderent facta <expan abbr="&longs;u&longs;p&etilde;&longs;io">&longs;u&longs;pen&longs;io</expan> <lb/>ne ex puncto C. &longs;itquè magnitudinum MZ <expan abbr="centrũ">centrum</expan> grauitatis <lb/>D; ip&longs;arum verò STVX &longs;it centrum grauitatis E. &longs;i itaque ma <lb/>gnitudines STVX, & ZM ex C æqueponderant; auferantur <lb/>STVX, quarum loco ponatur in E magnitudo ip&longs;is STVX &longs;i <lb/>mul &longs;umptis &ecedil;qualis: auferanturquè ZM, atque <expan abbr="ip&longs;arũ">ip&longs;arum</expan> loco po <lb/>natur in D magnitudo ip&longs;is ZM &longs;imul &ecedil;qualis; tunclicetinfer <lb/>re, ergo hæ magnitudines in ED po&longs;itæ &ecedil;quepondera­<lb/>bunt. Quod quidem ijsdem pror&longs;us modis o&longs;tendentur. <lb/>præ&longs;ertim &longs;i mente concipiamus di&longs;tantias ES EX,
<pb pagenum="60"/>nec non magnitudines STVX in &longs;uis di&longs;tantijs circa <expan abbr="centrũ">centrum</expan> <lb/>grauitatis E circumuerti po&longs;&longs;e; veluti di&longs;tantias DZ DM, ma <lb/>gnitudine&longs;què ZM circacentrum D. moueantur autem <lb/>SEX, & ZDM, donec in centrum mundi vergant. &longs;imiliter <lb/>o&longs;tendetur magnitudines STVX e&longs;&longs;e, ac &longs;i in E e&longs;&longs;ent appen <lb/>&longs;&ecedil;, &longs;iue con&longs;titut&ecedil;; magnitudines verò ZM ac &longs;i in D po&longs;i­<lb/>tæ fuerint. &c. Ex quibus &longs;equitur, &longs;i punctum C centrum <lb/>e&longs;t grauitatis magnitudinum STVXZM. ponatur magnitu­<lb/>do ip&longs;is STVX &longs;imul &longs;umptis &ecedil;qualis in E; magnitudo au <lb/>tem ip&longs;is ZM &longs;imul æqualis in D; punctum C &longs;imiliter <lb/>ip&longs;arum quoque centrum grauitatis exi&longs;tet. vnde vtroque mo <lb/>do æqueponderabunt. & ita in alijs, &longs;i plures fuerint magni­<lb/>tudines. </s></p>
<p type="head">
<s>PROPOSITIO. VI.</s></p>
<p type="main">
<s>Magnitudines commen&longs;urabiles ex di&longs;tantijs <lb/>eandem permutatim proportionem habentibus, <lb/>vt grauitates, æqueponderant. </s></p>
<p type="main">
<s><emph type="italics"/>Commen&longs;urabiles &longs;int magnitudines AB quarum centra<emph.end type="italics"/> grauita­<lb/>tis <emph type="italics"/>AB, & quædam &longs;it di&longs;tantia E D, & vt<emph.end type="italics"/> &longs;e habet grauitas ma­<lb/>gnitudinis <emph type="italics"/>A ad<emph.end type="italics"/> grauitatem magnitudinis <emph type="italics"/>B, ua &longs;it <expan abbr="di&longs;tãtia">di&longs;tantia</expan> <lb/>DC ad distantiam CE. <expan abbr="ostend&etilde;dũ">ostendendum</expan> e&longs;i<emph.end type="italics"/>, &longs;i centra grauitatis AB fue <lb/>rint in punctis ED con&longs;tituta, hoc e&longs;t A in E, & B in D; <lb/><emph type="italics"/>magnitudinis ex vtri&longs;què<emph.end type="italics"/> magnitudinibus <emph type="italics"/>AB compo&longs;itæ centrum <lb/>grauitatis e&longs;&longs;e punctum C. Quoniam enim ita est<emph.end type="italics"/> magnitudo <emph type="italics"/>A ad<emph.end type="italics"/><lb/>magnitudinem <emph type="italics"/>B, vt DC ad CE. e&longs;t autem<emph.end type="italics"/> magnitudo <emph type="italics"/>A ip&longs;i <lb/>
<arrow.to.target n="marg45"></arrow.to.target> B commen&longs;urabilis; erit & CD ip&longs;i CE commen&longs;urabilis; hoc e&longs;t <lb/>recta linea rectæ lineæ<emph.end type="italics"/> commen&longs;urabilis exi&longs;tet. <emph type="italics"/>Quare ip&longs;arum EC <lb/>CD communis reperitur men&longs;ura. quæ quidem &longs;it N. deinde ponatur <lb/>ip&longs;i EC æqualis vtraque DG DK; ip&longs;i verò DC æqualis EL. & <lb/>quoniam æqualis est DG ip&longs;i CE<emph.end type="italics"/>, communi addita CG, <emph type="italics"/>erit DC <lb/>ip&longs;i EG æqualis<emph.end type="italics"/>; &longs;ed DC e&longs;t ip&longs;i EL &ecedil;qualis: <emph type="italics"/>erit igitur LE æqua­<lb/>lis ip&longs;i EG.<emph.end type="italics"/> quare vtraque LE EG &ecedil;qualis e&longs;t ip&longs;i DC. <emph type="italics"/>ac propte<emph.end type="italics"/>
<pb pagenum="61"/><emph type="italics"/>rea dupla est LG ip&longs;ius DC.<emph.end type="italics"/> quia verò vtraque DG DK æqualis <lb/>facta e&longs;t ip&longs;i CE, erit <emph type="italics"/>& ip&longs;a quoque GK ip&longs;ius CE<emph.end type="italics"/> dupla. <emph type="italics"/>Quare <lb/>N <expan abbr="vtrãque">vtranque</expan> LG Gk metitur, cùm & ip&longs;arum medietates<emph.end type="italics"/> DC CE <lb/>
<arrow.to.target n="fig27"></arrow.to.target><lb/>metiatur. <emph type="italics"/>Et quoniam<emph.end type="italics"/> magnitudo <emph type="italics"/>A ita e&longs;t ad<emph.end type="italics"/> magnitudinem <lb/><emph type="italics"/>B, vt DC ad CE, ut autem DC ad CE, ita e&longs;t LG ad G<emph.end type="italics"/>K, <emph type="italics"/>utraque <lb/>enim vtriu&longs;que duplex exi&longs;tit<emph.end type="italics"/> (&longs;iquidem LG dupla e&longs;t ip&longs;ius DC, <lb/>& GK itidem ip&longs;ius CE duplex) <emph type="italics"/>erit<emph.end type="italics"/> magnitudo <emph type="italics"/>A ad<emph.end type="italics"/>
<arrow.to.target n="marg46"></arrow.to.target> magni­<lb/>tudinem <emph type="italics"/>B, ut LG ad G<emph.end type="italics"/>k; & conuertendo magnitudo B ad <lb/>magnitudinem A, vt KG ad GL. <emph type="italics"/>Quotuplex autem est LG ip&longs;ius <lb/>N, totuplex &longs;it<emph.end type="italics"/> magnitudo <emph type="italics"/>A ip&longs;ius F, erit vtique LG ad N, vt<emph.end type="italics"/><lb/>magnitudo <emph type="italics"/>A ad F, atqui est KG ad LG, vt<emph.end type="italics"/> magnitudo <emph type="italics"/>B ad<emph.end type="italics"/><lb/>magnitudinem <emph type="italics"/>A:<emph.end type="italics"/> LG verò ad N e&longs;t, vt magnitudo A ad
<arrow.to.target n="marg47"></arrow.to.target> <expan abbr="i-psã">i­<lb/>psam</expan> F, <emph type="italics"/>ex æquali igitur erit KG ad N, vt<emph.end type="italics"/> magnitudo <emph type="italics"/>B ad F quare æ­<lb/>quemultiplex e&longs;t<emph.end type="italics"/> kG <emph type="italics"/>ip&longs;ius N, veluti<emph.end type="italics"/> magnitudo <emph type="italics"/>B ip&longs;ius F. demon <lb/><expan abbr="&longs;tratũ">&longs;tratum</expan> <expan abbr="aũt">aunt</expan> e&longs;t<emph.end type="italics"/> <expan abbr="magnitudin&etilde;">magnitudinem</expan> <emph type="italics"/>A ip&longs;ius F multiplicem e&longs;&longs;e<emph.end type="italics"/>, &longs;iquidem e&longs;t <lb/>magnitudo A ad ip&longs;am F, vt LG ad N, quæ quidem LG mul <lb/>tiplex e&longs;t ip&longs;ius N. <emph type="italics"/>qua propter F ip&longs;arum AB communis existit men <lb/>&longs;ura. Jtaque diui&longs;a LG in partes<emph.end type="italics"/> LH, HE, EC, CG, <emph type="italics"/>ip&longs;i N aquales<emph.end type="italics"/>, <lb/>cadent vtique diui&longs;iones in punctis EC, quoniam <expan abbr="Nipsã">Nipsam</expan> EC
<arrow.to.target n="marg48"></arrow.to.target><lb/>metitur, nec non ip&longs;am quoque LE metitur; cùm &longs;it LE ip&longs;i <lb/>CD æqualis. eruntquè diui&longs;iones LH, HE, EC, CG, numero <lb/>pares; cùm N dimidiam ip&longs;ius LG, hoc e&longs;t CD metiatur.
<pb pagenum="62"/><emph type="italics"/>Averò<emph.end type="italics"/> &longs;imiliter diui&longs;a <emph type="italics"/>in partes<emph.end type="italics"/> OP QR <emph type="italics"/>ip&longs;i F æquales; &longs;ectio­<lb/>nes<emph.end type="italics"/> LH, HE, EC, CG <emph type="italics"/>in LG existentes magnitudini N æqua­<lb/>les, erunt numero æquales &longs;ectionibus<emph.end type="italics"/> OPQR <emph type="italics"/>in<emph.end type="italics"/> magnitudine <emph type="italics"/>A <lb/>existentibus ip&longs;i F æqualibus.<emph.end type="italics"/> Diuidantur &longs;ectiones LH, HE, EC, <lb/>
<arrow.to.target n="fig28"></arrow.to.target><lb/>CG bifariam in punctis STVX. <emph type="italics"/>&longs;i it aque in vnaquaque &longs;estio <lb/>ne ip&longs;ius LG apponatur magnitudo æqualis ip&longs;i F, quæ centrum gra­<lb/>uitatis babeat in medio &longs;ectionis<emph.end type="italics"/>; vt &longs;i in LH ponatur magnitudo <lb/>S, in HE magnitudo T, in EC magnitudo V, & in <lb/>CG magnitudo X; ip&longs;arum què vna quæque STVX &longs;it ip&longs;i <lb/>F æqualis: habeat verò magnitudo S &longs;uum grauitatis <expan abbr="centrũ">centrum</expan>, <lb/>quod &longs;it punctum S, in medio &longs;ectionis LH, nempè in <expan abbr="pũ-cto">pun­<lb/>cto</expan> S; &longs;imiliter cæteræ magnitudines TVX habeant <expan abbr="c&etilde;rra">cerrra</expan> <lb/>grauitatis; quæ &longs;int puncta TVX, in medio &longs;ectionum HE, <lb/>EC, CG, in punctis nempè TVX, erunt centra grauitatisma <lb/>gnitudinum STVX in recta linea con&longs;tituta, & quoma<gap/>o <lb/>SH dimidia e&longs;t ip&longs;ius LH, veluti HT ip&longs;ius HE, erit ST, <lb/>ip&longs;ius LE dimidia, vnaquæque verò LH HE dimidia <lb/>quoque e&longs;t ip&longs;ius LE, &longs;iquidem LH, HE inter&longs;e &longs;unt &ecedil;qua <lb/>les; eritigitur ST vnicuique LH, & HE æqualis. eodem què <lb/>pror&longs;us modo o&longs;tendeturi TV &ecedil;qualem e&longs;&longs;e vnicuique HE <lb/>EC. & VX æqualem EC. & CG. & quoniam omnes
<pb pagenum="63"/>LH, HE, EC, CG, inter&longs;e &longs;unt æquales; erunt ST TV VX in <lb/>ter&longs;e æquales. quare lineæ inter centra grauitatis magnitudi­<lb/>num STVX exi&longs;tentes &longs;untinter&longs;e &ecedil;quales. <emph type="italics"/>omnes verò magni <lb/>tudines<emph.end type="italics"/> STVX &longs;imul <emph type="italics"/>&longs;unt æquales ip&longs;i A<emph.end type="italics"/>, quandoquidem ip&longs;is <lb/>OPQR, & numero, & magnitudine &longs;unt &ecedil;quales; ergo <emph type="italics"/>magni­<lb/>tudinis ex omnibus<emph.end type="italics"/> magnitudinibus STVX <emph type="italics"/>compo&longs;itæ centrumgra <lb/>uitatis erit punstum E. cùm omnes<emph.end type="italics"/> magnitudines STVX <emph type="italics"/>&longs;int nu­<lb/>mero pares.<emph.end type="italics"/> quippe cùm &longs;int in &longs;ectionibus LH HE EC CG nu <lb/>mero paribus. & <emph type="italics"/>LE ip&longs;i EG æqualis exi&longs;tat.<emph.end type="italics"/> quòd &longs;i LE e&longs;tip&longs;i <lb/>EG æqualis, demptis æqualibus LS GX æqualibus, &longs;iquidem <lb/>&longs;unt dimidiæ &longs;ectionum LH CG æqualium: erunt SE EX
<arrow.to.target n="marg49"></arrow.to.target> in­<lb/>ter&longs;e æquales, vnde ex præcedenti colligitur, punctum E cen­<lb/>trum e&longs;&longs;e grauitatis magnitudinum STVX. <emph type="italics"/>&longs;imiliter autem <expan abbr="o&longs;t&etilde;">o&longs;tem</expan> <lb/>detur, quòd &longs;i<emph.end type="italics"/> diuidatur GK in partes GD DK ip&longs;i N æquales; <lb/>cadetvtique diui&longs;ionum aliqua in <expan abbr="pũcto">puncto</expan> D; &longs;iquidem Nip&longs;as <lb/>GD DK metitur; cùm vtraque &longs;it æqualisip&longs;i EC. diui&longs;ione&longs;­<lb/>què GD DK numero pares erunt; cùm N dimidiam ip&longs;ius
<arrow.to.target n="marg50"></arrow.to.target><lb/>GK, ip&longs;am &longs;cilicet EC metiatur. &longs;i itaque diuidatur GD DK <lb/>bifariam in punctis ZM. deinde diuidatur magnitudo B <lb/>in partes ip&longs;i F æquales; &longs;ectiones GD DH in GK exi&longs;tentes <lb/>ip&longs;i N æquales, erunt numero æquales &longs;ectionibus in ma <lb/>gnitudine B exi&longs;tentibus ip&longs;i F æqualibus. quare <emph type="italics"/>vnicuique <lb/>partium ip&longs;ius GK apponatur magnitudo æqualis ip&longs;i F; centrum gra­<lb/>uitatis habens in medio &longs;ectionis<emph.end type="italics"/>; vt <expan abbr="ponãtur">ponantur</expan> magnitudines ZM in <lb/>&longs;ectionibus GD DK, ita vt magnitudinum centra grauita­<lb/>tis, quæ &longs;int ZM, in medio &longs;ectionum GD DK, in punctis <lb/>nempè ZM &longs;int con&longs;tituta, <emph type="italics"/>omnes autem magnitudines<emph.end type="italics"/> ZM &longs;i <lb/>mul <emph type="italics"/>&longs;unt æquales ip&longs;i B. magnitudinis ex omnibus<emph.end type="italics"/> magnitudinibus <lb/>ZM <emph type="italics"/>compo&longs;itæ centrum grauitatis erit punctum D.<emph.end type="italics"/> cùm &longs;it ZD <lb/>&ecedil;qualis DM. <emph type="italics"/>&longs;ed<emph.end type="italics"/> magnitudines STVX &longs;unt magnitudini A <lb/>æquales, & ZM ip&longs;i B ergo <emph type="italics"/>magnitudo A e&longs;t<emph.end type="italics"/> tanquam <emph type="italics"/>impo&longs;ita <lb/>ad E, ip&longs;a verò B ad D.<emph.end type="italics"/> eodem &longs;cilicet modo &longs;e habebit ma­<lb/>gnitudo A impo&longs;ita ad E, vt &longs;e habent magnitudines STVX; <lb/>ip&longs;a verò B &longs;e habebit ad D, vt magnitudines ZM. <emph type="italics"/>&longs;unt au <lb/>tem magnitudines<emph.end type="italics"/> STVXZM <emph type="italics"/>inter&longs;e æquales<emph.end type="italics"/>, cùm vnaquæ que &longs;it <lb/>ip&longs;i F &ecedil;qualis: &longs;untquè omnes, (hoc e&longs;t ip&longs;arum centra graui <lb/>tatis) <emph type="italics"/>inrecta linea po&longs;itæ; quarum centragrauitatis po&longs;ita &longs;unt inter&longs;e<emph.end type="italics"/>
<pb pagenum="64"/><emph type="italics"/>æqualiter di&longs;tantia;<emph.end type="italics"/> &longs;iquidem o&longs;ten&longs;um e&longs;t ST TV VX inter­<lb/>&longs;e æquales e&longs;&longs;e. Eodemquè modo o&longs;tendetur XZ ZM cæteris <lb/>æquales e&longs;&longs;e. <emph type="italics"/>& &longs;unt<emph.end type="italics"/> magnitudines STVXZM <emph type="italics"/>numero pares,<emph.end type="italics"/><lb/>cùm &longs;ectiones totius LK, (in quibus in&longs;unt) ip&longs;i N æquales <lb/>&longs;intinter&longs;e &ecedil;quales, & numero pares. cùm o&longs;ten&longs;um &longs;it &longs;ectio <lb/>
<arrow.to.target n="marg51"></arrow.to.target> nes in LG, & in Gk exi&longs;tentes numero pares e&longs;&longs;e. <emph type="italics"/>con&longs;tat magni­<lb/>tudinis ex omnibus<emph.end type="italics"/> STVXZM magnitudinibus <emph type="italics"/>compo&longs;itæ centrum<emph.end type="italics"/><lb/>
<arrow.to.target n="marg52"></arrow.to.target> <emph type="italics"/>grauitatis e&longs;&longs;e medietatem restæ lineæ, in qua centra grauitatis magnitu <lb/>dinum habentur. Itaque cùm LE &longs;it æqualis C D, EC verò ip&longs;i D<emph.end type="italics"/>k, <lb/><emph type="italics"/>tota LC æqualis erit CK.<emph.end type="italics"/> cùm autem &longs;int LHDK æquales; &longs;i­<lb/>qui dem &longs;unt eidem N æquales, & harum medietates, hoc e&longs;t <lb/>LS ip&longs;i MK &ecedil;qualis erit. & ob id SC ip&longs;i CM e&longs;t æqualis. <lb/>at verò linea SM magnitudinum centra grauitatis <expan abbr="coniũgit">coniungit</expan>, <lb/><emph type="italics"/>ergo magnitudinis ex omnibus<emph.end type="italics"/> STVXZM magnitudinibus <emph type="italics"/>compo&longs;i <lb/>tæcentrum grauitatis est punctum C. Quare<emph.end type="italics"/> loco magnitudinum <lb/>STVX, <emph type="italics"/>po&longs;ito<emph.end type="italics"/> centro grauitatis <emph type="italics"/>A ad E, B verò<emph.end type="italics"/> loco ip&longs;arum <lb/>ZM po&longs;ito <emph type="italics"/>ad D,<emph.end type="italics"/> erit punctum C grauitatis centrum ma­<lb/>gnitudinis ex vtri&longs;que magnitudinibus AB compo&longs;itæ. ac <lb/>prop terea <emph type="italics"/>ex puncto C æqueponderabunt.<emph.end type="italics"/> ergo magnitudines AB <lb/>ex di&longs;tantijs DC CE, qu&ecedil; permutatim eandem habent pro. <lb/>portionem, vt grauitates, &ecedil;queponderant. quod demon&longs;trare <lb/>oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg45"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 3 <emph type="italics"/>de­<lb/>cimi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg46"></margin.target>11 <emph type="italics"/>quinti. <lb/>cor.<emph.end type="italics"/> 4. <emph type="italics"/>quin <lb/>ti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg47"></margin.target>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg48"></margin.target><emph type="italics"/>iemme.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg49"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 2. <emph type="italics"/>cor.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg50"></margin.target><emph type="italics"/>lemma.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg51"></margin.target>2.<emph type="italics"/>cor. quin <lb/>tæ huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg52"></margin.target>*</s></p>
<figure id="fig27"></figure>
<figure id="fig28"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>
<arrow.to.target n="marg53"></arrow.to.target> Circa finem Gr&ecedil;cus codex habet, <foreign lang="greek">ta ke/nt<10>a tw=n me/swn megeqw=n</foreign>, <lb/>qua&longs;i dicat, centrum grauitatis magnitudinis ex omnibus <lb/>magnitudinibus STVXZM compo&longs;it&ecedil; medietatem e&longs;&longs;e rect&ecedil; <lb/>line&ecedil; VX, qu&ecedil; centra mediarum magnitudinum VX coniun <lb/>git; quòd cùm &longs;int omnes magnitudines numero pares; <expan abbr="itid&etilde;">itidem</expan> <lb/>e&longs;&longs;et punctum C, & quamuis hoc &longs;it verum, non tamen ad hoc <lb/>re&longs;pexit Archimedes duabus de cau&longs;is. <expan abbr="Nãin">Nanin</expan> &longs;ecudo corollario <lb/>pr&ecedil;cedentis o&longs;tendit centrum grauitatis omnium magnitu­<lb/>dinum e&longs;&longs;e medietatem rect&ecedil; line&ecedil;, qu&ecedil; grauitatis centra om­<lb/>nia coniungit. Dein de concludere volens punctum C <expan abbr="centrũ">centrum</expan> <lb/>e&longs;&longs;e grauito tis omnium magnitudinum, &longs;tatim inquit hoc &longs;e <lb/>qui, quia LC e&longs;t ip&longs;i CK &ecedil;qualis, qu&ecedil; &longs;unt medietates totius
<pb pagenum="65"/>rectæline&ecedil; LK. Et non dixit, quia VC &longs;itip&longs;i CX &ecedil;qualis. <lb/>Quare codicem græcum ita re&longs;tituendum cen&longs;eo. <foreign lang="greek">ta\ke/nt<10>k tw=n <lb/>tou= ba\<10>eos megeqw=n</foreign>, vt vertimus. </s></p>
<p type="margin">
<s><margin.target id="marg53"></margin.target>*</s></p>
<p type="main">
<s>Ob &longs;equentis verò demon&longs;trationis cognitionem, hoc pro <lb/>blema priùs o&longs;tendemus. </s></p>
<p type="head">
<s>PROBLEMA.</s></p>
<p type="main">
<s>Duarum expo&longs;itarum magnitudinum incommen&longs;urabi­<lb/>lium altera vtcumque &longs;ecetur; magnitudinem tota &longs;ecta ma­<lb/>gnitudine minorem, & altero &longs;egmentomaiorem, alteri ve­<lb/>rò expo&longs;itæ magnitudini commen&longs;urabilem inuenire. </s></p>
<p type="main">
<s>Sint duæ magnitudi­<lb/>nes incommen&longs;urabiles <lb/>
<arrow.to.target n="fig29"></arrow.to.target><lb/>AE BC. &longs;eceturquè ip&longs;a­<lb/>rum altera, putà BC, vt­<lb/>cumque in D. oportet <lb/>magnitudinem inuenire <lb/>minorem quidem BC, <lb/>maiorem verò BD, quæ &longs;itip&longs;i AE commen&longs;urabilis. Au­<lb/>feratur ab AE pars dimidia, rur&longs;us dimidiæ partis ip&longs;ius AE <lb/>dimidia auferatur; & eius, quæ remanet, adhuc dimidia; idquè <lb/>&longs;emper fiat, donec relinquatur magnitudo minor, quàm DE. <lb/>quod quidem per&longs;picuum e&longs;t po&longs;&longs;e fieri ex prima decimi Eu­<lb/>clidis propo&longs;itione. &longs;ititaque AF, quæ minor exi&longs;tat, quàm <lb/>DC. quippe qu&ecedil; AF, cùm &longs;it abla ta ex AE &longs;emper per dimi <lb/>diam partem, metietur vtique AF ip&longs;am AE. Deinde mul­<lb/>tiplicetur AF &longs;uper BD, tum demum multiplicatio vltima, <lb/>vel in puncto D cadet, vel minus. &longs;i cadet; &longs;eceturex DE <lb/>magnitudo DG &ecedil;qualis AF. quod quidem fiet, <expan abbr="quoniã">quoniam</expan> AF <lb/>minor e&longs;t DC. Quoniam igitur AF metitur BD, & DG; <lb/>metietur AF totam BG. Sed & ip&longs;am AE metitur; etgo <lb/>AF ip&longs;arum BG AE communis exi&longs;tit men&longs;ura, ac propte­<lb/>rea BG ip&longs;i AE commen&longs;urabilis exi&longs;tir; quæ quidem BG <lb/>minor e&longs;t BC, maior verò BD. Si verò vltima
<arrow.to.target n="marg54"></arrow.to.target> multi­<lb/>plicatio ip&longs;ius AF &longs;uper BD non cadet in D. &longs;ed in H, <lb/>erit vtique HD minor AF. nam &longs;i HD ip&longs;i AF e&longs;&longs;et &ecedil;qualis,
<pb pagenum="66"/>vltima multiplicatio caderet in D. &longs;i verò maior e&longs;&longs;et HD, <lb/>quàm AF tunc non e&longs;&longs;et vltima multiplicatio. quare cùm &longs;it <lb/>DC maior AF; erit & HC ip&longs;a FA maior. &longs;i itaque fiat HK <lb/>æqualis AF; erit punctum K inter puncta DC. BK igitur <lb/>minor erit, quàm BC, & maior BD; eodemquè modo o­<lb/>&longs;tendetur AF ip&longs;arum Bk AE communem e&longs;&longs;e men&longs;u­<lb/>ram. & obid BK ip&longs;i AF commen&longs;urabilem exi&longs;tere. quod <lb/>facere oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg54"></margin.target>1.<emph type="italics"/>def.deci­<lb/>mi.<emph.end type="italics"/></s></p>
<figure id="fig29"></figure>
<p type="main">
<s>Cùm autem verba &longs;equentis demon&longs;trationis aliquantu­<lb/>lum &longs;int ob&longs;cura, vt vim demon&longs;trationis rectè petcipiamus, <lb/>hoc quoque theorema ex ijs, quæ ab Archimede hactenus de­<lb/>mon&longs;trata &longs;unt, o&longs;tendemus. ad quod demon&longs;trandum com <lb/>muni notione indigemus, quam nos in no&longs;tro Mechanico­<lb/>rum libro po&longs;uimus. Nempè. </s></p>
<p type="main">
<s>Quæ eidem æquepondeiant, inter&longs;e æquè &longs;unt grauia. </s></p>
<p type="head">
<s>PROPOSITIO.</s></p>
<p type="main">
<s>Si commen&longs;urabiles magnitudines minorem habuerint <lb/>proportionem, quàm di&longs;tanti&ecedil; permutatim habent; vt &ecedil;que­<lb/>ponderent, maiori opus erit magnitudine, quàm &longs;it ea, qu&ecedil; <lb/>ad alteram magnitudinem minorem proportionem habet. </s></p>
<figure></figure>
<p type="main">
<s>Sint magnitudines AC commen&longs;urabiles, di&longs;tanti&ecedil; ve­<lb/>rò &longs;int ED EF. minorem autem habeat pro-
<pb pagenum="67"/>portionem A ad C, quàm ED ad EF. Dico, vt magnitu­<lb/>dines ex di&longs;tantijs ED EF æqueponderent, maiori o­<lb/>pus e&longs;&longs;e magnitudine in F, quàm &longs;it magnitudo A; <lb/>ita vt ip&longs;i C in D æqueponderare po&longs;&longs;it. fiat ED <lb/>ad EG, vt magnitudo A ad magnitudinem C. <lb/>Deindefiat EK æqualis EG. exponaturquè altera ma­<lb/>gnitudo L ip&longs;i A &ecedil;qualis. Quoniam igitur minorem <lb/>habet proportionem A ad C, quàm ED ad EF, & <lb/>vt A ad C, ita ED ad EG; habebit ED ad <lb/>EG minorem proportionem, quàm ad EF. ac propterea
<arrow.to.target n="marg55"></arrow.to.target><lb/>EF minor e&longs;t, quàm EG. quoniam ausem A ad C <lb/>e&longs;t, vt ED ad EG, commen&longs;urabiles magnitudines <lb/>AC ex di&longs;tantijs ED EG æqueponderabunt. Cùm
<arrow.to.target n="marg56"></arrow.to.target><lb/>verò EK &longs;it æqualis EG, magnitudines AL æ­<lb/>quales ex di&longs;tantis æqualibus EK EG &longs;imiliter æque­<lb/>ponderabunt. At verò quoniam C in D æque­<lb/>ponderat ip&longs;i A in G, &longs;imiliter L in K eidem A in <lb/>G &ecedil;queponderat; &ecedil;qualem habebit grauitatem C in D, vt
<arrow.to.target n="marg57"></arrow.to.target><lb/>L in K. Itaque quoniam di&longs;tantia EG æqualis e&longs;t di&longs;tan <lb/>tiæ Ek, longitudo EK maior erit longitudine EF. ergo <lb/>magnitudines AL &ecedil;quales ex inæqualibus di&longs;tantijs EK
<arrow.to.target n="marg58"></arrow.to.target><lb/>EF non &ecedil;queponderabunt. &longs;ed magnitudo L deor&longs;um ver­<lb/>get. &longs;i igitur in F collocanda &longs;it magnitudo, quæ æquepon <lb/>deret ip&longs;i L in K, proculdubiò h&ecedil;c magnitudine A ma­<lb/>ior exi&longs;tet. Inæqualia enim grauia, nempè L, & magnitu
<arrow.to.target n="marg59"></arrow.to.target><lb/>do maior, quàm A, exinæqualibus di&longs;tantijs EK EF æ­<lb/>queponderant, dummodo maius, hoc e&longs;t magnitudo maior, <lb/>quàm A, &longs;it in di&longs;tantia minori EF. minusverò, hoc e&longs;t ma­<lb/>gnitudo L, &longs;it in minori EK. Quoniam itaque magnitudo <lb/>C in D e&longs;t &ecedil;quegrauis, vt L in K, magnitudo, quæ in F <lb/>ip&longs;i L in K æqueponderat, eadem quoque in F ip&longs;i C in D <lb/>æqueponderabit maior verò magnitudo, quàm &longs;it A, in F ip&longs;i <lb/>L in K æqueponderat, ergo maior magnitudo, quàm A in <lb/>F, ip&longs;i C in D æqueponderabit. quod demon&longs;trare opor­<lb/>tebat. </s></p>
<p type="margin">
<s><margin.target id="marg55"></margin.target>10. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg56"></margin.target>6. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg57"></margin.target><emph type="italics"/><expan abbr="cõm">comm</expan>. not.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg58"></margin.target>2. <emph type="italics"/>po&longs;t bu­<lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg59"></margin.target>3. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="main">
<s>His cognitis po&longs;&longs;umus ad Archimedis demon&longs;trationem <lb/>accedere. </s></p>
<pb pagenum="68"/>
<p type="head">
<s>PROPOSITIO. VII.</s></p>
<p type="main">
<s>Si autem magnitudines fuerint incommen&longs;ura <lb/>biles, &longs;imiliter æqueponderabunt ex di&longs;tantijs per <lb/>mutatim eandem, atque magnitudines, propor­<lb/>tionem habentibus. </s></p>
<figure></figure>
<p type="main">
<s><emph type="italics"/>Sint incommen&longs;urabiles magnitudines AB C. Distantiæ verò <lb/>DE EF. Habeat autem AB ad C proportionem eandem, quam di <lb/>stantia ED ad ip&longs;am EF. Dico,<emph.end type="italics"/> &longs;i ponatur AB ad F, C ve­<lb/>rò ad D, <emph type="italics"/>magnitudinis ex vtri&longs;que AB C compo&longs;itæ centrum gra <lb/>uitatis e&longs;&longs;e punctum E. &longs;i enim non æqueponderabit<emph.end type="italics"/> (&longs;i fieri pote&longs;t) <lb/><emph type="italics"/>AB po&longs;ita ad F ip&longs;i C po&longs;itæ ad D; velmaior est AB, quàm C, ita <lb/>vt<emph.end type="italics"/> AB ad F <emph type="italics"/>æqueponderet ip&longs;i C<emph.end type="italics"/> ad D; <emph type="italics"/>vel non. Sit maior<emph.end type="italics"/>; &longs;itquè <lb/>exce&longs;&longs;us HL; ita vt KH ad F, & C ad D &ecedil;queponderent. <lb/>
<arrow.to.target n="marg60"></arrow.to.target> <emph type="italics"/>auferaturquè ab ip&longs;a AB<emph.end type="italics"/> magnitudo NL, quæ &longs;it <emph type="italics"/>minor exce&longs;&longs;u<emph.end type="italics"/><lb/>HL, <emph type="italics"/>quo maior est<emph.end type="italics"/> tota <emph type="italics"/>AB, quàm C, ita vt æqueponderent<emph.end type="italics"/>; vt <expan abbr="dictũ">dictum</expan> <lb/>e&longs;t. <emph type="italics"/>& &longs;it quidem re&longs;iduum A,<emph.end type="italics"/> hoc e&longs;t KN, <emph type="italics"/>commen&longs;urabile ip&longs;i C.<emph.end type="italics"/><lb/>Et quoniam minor e&longs;t kN quàm KM, minorem quoque
<pb pagenum="69"/>habebit proportionem kN ad C, quàm kM ad eandem <lb/>C. tota verò KM ad C e&longs;t, vt DE ad EF; ergo KN ad <lb/>C minorem habet proportionem; quàm DE ad EF. <emph type="italics"/>Quo <lb/>niam igitur magnitudines AC,<emph.end type="italics"/> hoc e&longs;t KN C, <emph type="italics"/>&longs;unt commen&longs;urabi­<lb/>les, & minorem habet proportionem A,<emph.end type="italics"/> hoc e&longs;t kN <emph type="italics"/>ad C, quàm DE <lb/>ad EF; non æqueponderabunt A C,<emph.end type="italics"/> hoc e&longs;t KN C, <emph type="italics"/>ex distantiis<emph.end type="italics"/>
<arrow.to.target n="marg61"></arrow.to.target><lb/><emph type="italics"/>DE EF, po&longs;ito quidem A,<emph.end type="italics"/> hoc e&longs;t KN <emph type="italics"/>ad F, C verò ad D.<emph.end type="italics"/> & <lb/>vt æqueponderent, oporter, vt in F maior &longs;it magnitudo, <lb/>quàm KN; ita vt ip&longs;i C in D æqueponderate po&longs;&longs;it. Ac <lb/>propterea cùm &longs;it kH adhuc minor, quàm KN, &longs;i igitur <lb/>KH ponatur ad F, & C ad D, nullo modo æqueponde­<lb/>rabunt. quod tamen fieri non pote&longs;t. &longs;upponebatur enim eas <lb/>æqueponderare. Non igitur magnitudo minor, quàm tota <lb/>KM in F magnitudini C in D æqueponderat. <emph type="italics"/>Eadem au­<lb/>tem ratione, neque &longs;i C maior fuerit, quàm vt æqueponderet ip&longs;i A<emph.end type="italics"/>B, <lb/>hoc e&longs;t ip&longs;i KM. etenim grauiore <expan abbr="exi&longs;t&etilde;te">exi&longs;tente</expan> C ad D, quàm KM <lb/>ad F. primùm auferatur ex C exce&longs;&longs;us, quo C grauior e&longs;t, <lb/>quàm KM, ita vt æqueponderet ip&longs;i KM. Deinde rur&longs;us <lb/>auferatur quædam magnitudo minor exce&longs;&longs;u, quo grauior <lb/>e&longs;t C, quàm kM, ita vt æqueponderent; re&longs;iduum verò &longs;it <lb/>ip&longs;i KM commen&longs;urabile, & c. &longs;imiliter o&longs;tendetur <expan abbr="nullã">nullam</expan> <lb/>magnitudinem ip&longs;a C minorem po&longs;itam ad D vllo modo <lb/>æqueponderare ip&longs;i KM ad F po&longs;itæ. Quare magnitudo <lb/>C ad D, kM verò ad F &ecedil;queponderant. Vnde &longs;equitur ma <lb/>gnitudinis ex vtri&longs;que magnitudinibus compo&longs;itæ centrum <lb/>grauitatis e&longs;&longs;e punctum E. ac propterea incommen&longs;urabiles <lb/>magnitudines AB C ex di&longs;tantiijs ED EF, quæ permutatim <lb/>eandem habent proportionem, vt magnitudines, æquepon­<lb/>derare. quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg60"></margin.target><emph type="italics"/>ex proxi­<lb/>mo proble­<lb/>mate.<emph.end type="italics"/><lb/>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg61"></margin.target><emph type="italics"/>ex præce­<lb/>denti. <lb/>ex prima <lb/>propo&longs;itio­<lb/>ne.<emph.end type="italics"/></s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>In demon&longs;tratione occurrit ob&longs;eruandum, quòd &longs;i exce&longs;­<lb/>&longs;us HL ita diuideret magnitudinem KM, vt re&longs;iduum KH <lb/>fuerit commen&longs;urabile ip&longs;i C; tunc ab&longs;que alia con&longs;tructio­<lb/>ne, magnitudines commen&longs;urabiles KH C ex di&longs;tantijs DE <lb/>EF æqueponderarent; quod fieri non pote&longs;t. cùm minorem
<pb pagenum="70"/>habeat proportionem KH ad C, quàm ED ad EF. <expan abbr="&longs;iquid&etilde;">&longs;iquidem</expan> <lb/>&longs;upponitur KM ad C ita e&longs;&longs;e, vt ED ad EF. Archimed es ve <lb/>iò, vt demon&longs;tratio ab&longs;que di&longs;tinctione &longs;it vniuer&longs;alis, pr&ecedil;­<lb/>cipit (exi&longs;tente KH ip&longs;i C commen&longs;urabili, &longs;iu e incommen <lb/>&longs;urabili) vt auferatur pars aliqua minor exce&longs;&longs;u HL, ut AL, <lb/>ita tamen, vt reliqua KN &longs;it commen&longs;urabilis ip&longs;i C. quod qui <lb/>dem fieri po&longs;&longs;e o&longs;ten&longs;um e&longs;t in proximo problemate. ex tota <lb/>enim magnitudine KM partem ab&longs;cindere po&longs;&longs;umus, vt KN <lb/>minorem quidem tota KM, maiorem verò KH, quæ ip&longs;i <lb/>C commen&longs;urabilis exi&longs;tat. </s></p>
<p type="main">
<s>Cognita Archimedis demon&longs;tratione de incommen&longs;ura­<lb/>bilibus magnitudinibus, idem alio quoque modo o&longs;tendere <lb/>po&longs;&longs;umus, applicando nempè diui&longs;ibilitatem, & commen&longs;ura <lb/>bilitatem non magnitudinibus, verùm di&longs;tantijs. hac autem <lb/>priùs demon&longs;trata propo&longs;itione. </s></p>
<p type="head">
<s>PROPOSITIO.</s></p>
<p type="main">
<s>Si commen&longs;urabiles di&longs;tanti&ecedil; maiorem habuerint pro­<lb/>portionem, quàm magnitudines permutatim habent; vt <lb/>&ecedil;queponderent, maiori opus erit longitudine, quàm &longs;it <lb/>ea, ad quam altera longitudo maiorem habet proportio­<lb/>nem. </s></p>
<figure></figure>
<p type="main">
<s>Sint di&longs;tantiæ DE EH commen&longs;urabiles, magnitudines <lb/>verò &longs;int A C. habeatquè ED ad EH maiorem proportio­<lb/>nem, quàm A ad C. Dico vt AC &ecedil;queponderent, maiori opus
<pb pagenum="71"/>e&longs;&longs;e longitudine, quàm &longs;it EH. exponatur altera magnitu­<lb/>do G, quæ ad C eandem habeat proportionem, quàm habet <lb/>DE ad EH. erunt vtique magnitudines GC inter &longs;e <expan abbr="comm&etilde;">commen</expan> <lb/>&longs;urabiles. Deinde fiat EK æqualis EH, exponaturquè ma­<lb/>gnitudo L ip&longs;i G æqualis. Quoniam igitur G ad C e&longs;t, <lb/>vt DE ad EH, ob commen&longs;urabilitatem æquepondera bunt
<arrow.to.target n="marg62"></arrow.to.target><lb/>G in H, & C in D. &longs;imiliter æquepondera bunt magnitudi­<lb/>nes æquales GL ex æqualibus di&longs;tantijs EK EH. Cùm igitur <lb/>C in D ip&longs;i G in H æqueponderet; L verò in K ip&longs;i quo­<lb/>que G in H æqueponderet; eandem habebit grauitatem C
<arrow.to.target n="marg63"></arrow.to.target><lb/>in D, ut L in K. Quoniam autem maiorem habet propor­<lb/>tionem DE ad EH, quàm A ad C, & vt DE ad EH, ita e&longs;t <lb/>G ad C; maiorem habebit proportionem G ad C, quàm A <lb/>ad C. ergo maior e&longs;t G, quàm A. ac propterea magnitudo A
<arrow.to.target n="marg64"></arrow.to.target><lb/>minor e&longs;t magnitudine L. po&longs;ita igitur magnitudine L in K, <lb/>& A in H, non æquepondera bunt; & vt &ecedil;queponderent, o­<lb/>portet, vt A in longiori &longs;it di&longs;tantia, quàm &longs;it EH: In&ecedil;qualia <lb/>enim grauia LA ex in&ecedil;qualibus di&longs;tantijs &ecedil;queponderant,
<arrow.to.target n="marg65"></arrow.to.target><lb/>maius quidem L in minori di&longs;tantia EK, minus verò graue <lb/>A in maiori, quàm &longs;it EK, hoc e&longs;t in maiori, quàm &longs;it EH. <lb/>Itaque cùm &longs;it C in D æquegrauis, vt L in k; longitudo, <lb/>quæ efficit, vt A æqueponderetip&longs;i L in K; eadem pror&longs;us <lb/>efficiet, vt A ip&longs;i C in D &ecedil;queponderare po&longs;&longs;it. A verò in <lb/>maiori di&longs;tantia, quàm EH, ip&longs;i L in K &ecedil;queponderat; ergo <lb/>in maiori di&longs;tantia, quàm EH, magnitudo A ip&longs;i C in D <lb/>&ecedil;queponderabit. quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg62"></margin.target>6. <emph type="italics"/>buius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg63"></margin.target><emph type="italics"/><expan abbr="cõmunis">communis</expan> no <lb/>tio &longs;upradi <lb/>cta.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg64"></margin.target>10. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg65"></margin.target>3. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="main">
<s>Hoc demon&longs;trato Archimedis propo&longs;itionem de incom­<lb/>men&longs;urabilibus magnitudinibus aliter o&longs;tendemus hoc <lb/>pacto. </s></p>
<p type="head">
<s>ALITER.</s></p>
<p type="main">
<s>Incommen&longs;urabiles magnitudines ex di&longs;tantijs permuta­<lb/>tim eandem, atque magnitudines, proportionem habenti­<lb/>bus; &ecedil;queponderant. </s></p>
<pb pagenum="72"/>
<p type="main">
<s>Sint incom­<lb/>
<arrow.to.target n="fig30"></arrow.to.target><lb/><expan abbr="m&etilde;&longs;urabiles">men&longs;urabiles</expan> ma <lb/>gnitudines AC, <lb/>di&longs;tantiæ verò <lb/>DE EF. &longs;itquè vt <lb/>A ad C, ita DE <lb/>ad EF. Dico A <lb/>in F, C verò in <lb/>D æqueponde­<lb/>rare. Si autem (&longs;i fieri pote&longs;t) non æquepondera bunt; <expan abbr="di&longs;tã">di&longs;tam</expan> <lb/>tiæ DE EF aliter &longs;e&longs;e habere debebunt, vt magnitudines AC <lb/>&ecedil;queponderent. Quocirca vel longior e&longs;t EF, quàm opus <lb/>&longs;it, vel longior e&longs;t ED. &longs;it EF longior. &longs;itquè exce&longs;&longs;us GF, ita <lb/>vt po&longs;ita magnitudine A in G ip&longs;i C in D æqueponde­<lb/>
<arrow.to.target n="marg66"></arrow.to.target> ret. Fiat EH maior EG, minor verò EF. &longs;it autem EH <lb/>ip&longs;i ED commen&longs;urabilis. Quoniam igitur DE ad EH <lb/>maiorem habet proportionem, quàm ad EF; & vt DE ad <lb/>EF, ita e&longs;t A ad C; maiorem habebit proportionem DE <lb/>ad EH, quàm A ad C. &longs;untquè longitudines ED EH in­<lb/>ter&longs;e commen&longs;urabiles; ergo magnitudo A in H ip&longs;i C in <lb/>
<arrow.to.target n="marg67"></arrow.to.target> D non æqueponderabit, &longs;ed vt &ecedil;queponderet, maiori opus <lb/>e&longs;t longitudine, quàm &longs;it EH; ita vt A ip&longs;i C in D æque <lb/>ponderare po&longs;&longs;it. atque adeò cùm adhuc minor &longs;it EG, quàm <lb/>EH; magnitudo A in G magnitudini C in D nullo modo <lb/>æqueponderabit. quod fieri non pote&longs;t. &longs;upponebatur enim <lb/>A in G, & C in D &ecedil;queponderare. eademquè pror&longs;us ra­<lb/>tione, &longs;i ED longior fuerit, quàm opus &longs;it, ita vt magnitu­<lb/>dines æqueponderent, o&longs;tendetur <expan abbr="magnitudin&etilde;">magnitudinem</expan> C nullo pa­<lb/>cto æqueponderare po&longs;&longs;e ip&longs;i A in F in minori di&longs;tantia, <lb/>quàm DE. Quare magnitudines in commen&longs;urabiles AC ex <lb/>di&longs;tantijs ED EF, quæ eandem permutatim habent propor­<lb/>tionem, vt magnitudines, æqueponderant. quod demon&longs;tra­<lb/>re oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg66"></margin.target><emph type="italics"/>problema <lb/>ante<emph.end type="italics"/> 7. <emph type="italics"/>bu­<lb/>ius<emph.end type="italics"/> 8. <emph type="italics"/>quinti<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg67"></margin.target><emph type="italics"/>ex pxima <lb/>ppo&longs;itione<emph.end type="italics"/></s></p>
<figure id="fig30"></figure>
<p type="main">
<s>In prioribus &longs;ermonibus ante quintam propo&longs;itionem ha­<lb/>bitis, diximus propo&longs;itionum præcedentium demon&longs;tratio­<lb/>nes planiores euadere, &longs;i intelligamus magnitudines eiu&longs;dem <lb/>e&longs;&longs;e &longs;peciei, & homogeneas. Quòd quidem &longs;i Archimedem
<pb pagenum="73"/>his, vel de rectilineis tan tùm demon&longs;trationes attuli&longs;&longs;e (vt <expan abbr="nõ-nulli">non­<lb/>nulli</expan> forta&longs;&longs;e falsò exi&longs;timarunt) intelligeremus; ita vt ex Ar­<lb/>chimedis demon&longs;trationibus non &longs;it adhuc vniuer&longs;aliter de­<lb/>mon&longs;tratum hoc pr&ecedil;cipuum fun damentum; nempè magni­<lb/>tudines ex di&longs;tantijs permutatim <expan abbr="proportion&etilde;">proportionem</expan> habentibus, vt <lb/>ip&longs;arum grauitates, &ecedil;queponderare; in hoc certè rationes ab <lb/>Archimede allatas, ip&longs;arum què demon&longs;trationum vim mini­<lb/>mè percipiemus. Quapropter ea, quæ demon&longs;trauit, omni­<lb/>bus magnitudinibus vniuer&longs;aliter competere ip&longs;um volui&longs;&longs;e <lb/>nullatenus e&longs;t dubitandum. Neque enim, vt perfectè, & vni­<lb/>uer&longs;aliter&longs;ciamus, magnitudines çqueponderare ex di&longs;tantijs <lb/>permutatim proportionem habentibus, vt ip&longs;arum grauita­<lb/>tes, alijs, quàm pr&ecedil;cedentibus propo&longs;itionibus indigemus. <lb/>In hoc enim fundamento demon&longs;trando minimè diminu­<lb/>tus extitit Archimede. Nam &longs;i ad propo&longs;itiones ab ip&longs;o alla­<lb/>tas, pr&ecedil;cipuèquè ad vim demon&longs;trationum re&longs;piciamus, &longs;iuè <lb/>magnitudines intelligantur eiuldem &longs;peciei, &longs;iue diuer&longs;&ecedil;, &longs;i­<lb/>ue homogene&ecedil;, &longs;iue heterogene&ecedil;, &longs;iue plan&ecedil;, &longs;iue &longs;olid&ecedil;, & <lb/>h&ecedil; quidem, &longs;iue rectiline&ecedil;, &longs;iue quom odocunque mixt&ecedil;; ni­<lb/>hilominus demon&longs;trationes idem pror&longs;us concludent, ita vt <lb/>Archimedes non de aliquibus magnitudimbus tantùm de­<lb/>mon&longs;trationes attulerit; &longs;ed de omnibus pror&longs;us demon&longs;tra­<lb/>uerit. In his enim Archimedes non ad magnitudines tantùm, <lb/>verùm ad magnitudinum grauitates poti&longs;&longs;imùm re&longs;pexit. <lb/>quandoquidem loco grauium magnitudines nominat; vt <lb/>po&longs;t quartam huius propo&longs;itionem adnotauimus. quod qui­<lb/>dem facilè ex verbis ip&longs;ius rectè in tellectis apparere pote&longs;t. <expan abbr="Nã">Nam</expan> <lb/>in quærta propo&longs;itione cùm inquit, <emph type="italics"/>&longs;i duæ fuerint magnitudines <lb/>æquales<emph.end type="italics"/>, vt antea diximus, intelligendum e&longs;t eas &ecedil;quales <lb/>e&longs;&longs;e grauitate. quod non &longs;olùm ex eius demon&longs;trationeli­<lb/>quet, verùm etiam ex modo loquendi, quo v&longs;us e&longs;t Archime­<lb/>des in alijs propo&longs;itionibus. In quinta enim propo&longs;itione, <lb/>qu&ecedil; eiu&longs;dem e&longs;t cum quarta ordinis, & natur&ecedil;, in quit; <lb/><emph type="italics"/>Sitrium magnitudinum centra grauitatis in recta linea fuerint po&longs;i­<lb/>ta, & magnitudines æqualem habuerint grauitatem.<emph.end type="italics"/> &longs;imlli­<lb/>ter po&longs;t quintam demon&longs;trationem bis quoquè eodem v­<lb/>titur loquendi modo, nempè cùm adhuc proponit
<pb pagenum="74"/>plures magnitudines, inquit, <emph type="italics"/>& magnitudines æqualem habuerint <lb/>grauitatem.<emph.end type="italics"/> ex quibus con&longs;tat Archimedem ad magnitudinum <lb/>grauitates omnino re&longs;pexi&longs;&longs;e. ita vt quando Archimedes in­<lb/>quit, <emph type="italics"/>& magnitudines æquales<emph.end type="italics"/>, idem e&longs;t, ac &longs;i dixi&longs;&longs;et, <emph type="italics"/>& magnitu­<lb/>dines æqualem habuerint grauitatem.<emph.end type="italics"/> Præterea in &longs;exta propo&longs;itio <lb/>ne inquit magnitudines &ecedil;queponderare ex di&longs;tantijs permu­<lb/>tàtim proportionem habentibus, vt grauitates. ita ut cau&longs;a <lb/>huius æqueponderationis &longs;it (vt reuera e&longs;t) magnitudinum <lb/>grauitas. & <expan abbr="quãquam">quanquam</expan> in hac &longs;eptima propo&longs;itione dicat, ma <lb/>gnitudines æqueponderare ex di&longs;tantijs permutatim propor­<lb/>tionem habentibus, vt magnitudines, & non dixit, vt grauita <lb/>tes; intelligendum tamen e&longs;t, ac &longs;i dixi&longs;&longs;et, eas &ecedil;quepondera­<lb/>re, vt magnitudinum grauitates. h&ecedil;c enim &longs;eptima propo&longs;i­<lb/>tio e&longs;t pars &longs;extæ propo&longs;itionis, vt iam pr&ecedil;fati fum^{9}; vnde &longs;i in <lb/>&longs;exta magnitudines &ecedil;queponderant ob earum grauitatem, ob <lb/>eandem quoque cau&longs;am & in hac &longs;eptima æqueponderare de <lb/>bent. Pr&ecedil;terea in &longs;equenti etiam propo&longs;itione dum proponit <lb/>o&longs;tendere quam proportionem habere debent &longs;ectiones line&ecedil; <lb/>intercentra grauitatum diui&longs;&ecedil; magnitudinis <expan abbr="exi&longs;t&etilde;tes">exi&longs;tentes</expan>, inquit, <lb/><emph type="italics"/>quam habet grauitas magnitudinis ablatæ ad grauitatem re&longs;iduæ<emph.end type="italics"/> hoc <lb/>autem deinceps exponens, <expan abbr="nõ">non</expan> inquit oportere &longs;ectiones lineæ <lb/>eam habere proportionem, quàm grauitas ad grauitatem ha­<lb/>bet; &longs;ed horum loco inquit, quàm magnitudo ad magnitudi <lb/>nem. ex quibus omnibus clarè per&longs;picitur, quòd quando Ar­<lb/>chimedes magnitudines nominat, omnino magnitudinum <lb/>grauitates vult intelligere. </s></p>
<p type="main">
<s>Ad eorum autem <expan abbr="intelligentiã">intelligentiam</expan>, qu&ecedil; dicta &longs;unt in &longs;exta, &longs;epti <lb/>maquè propo&longs;itione, <expan abbr="earũquè">earunquè</expan> <expan abbr="demõ&longs;trationibus">demon&longs;trationibus</expan>, <expan abbr="ob&longs;eruandũ">ob&longs;eruandum</expan> <lb/>e&longs;t, quòd in &longs;exta propo&longs;itione pro magnitudinibus commen <lb/>&longs;urabilibus intelligere oportet magnitudines grauitate com­<lb/>men&longs;urabiles; ita nempe, vt numeris exprimi po&longs;&longs;int; quam­<lb/>quam non &longs;int mole, & magnitudine commen&longs;urabiles, vt <lb/>in figura &longs;ext&ecedil; propo&longs;itionis magnitudo A ponderet exempli <lb/>gratia vt XVI. B verò vt VIII. <expan abbr="intelligatur&qacute;">intelligaturque</expan>; F <expan abbr="magnitudinũ">magnitudinum</expan>
<pb pagenum="75"/>AB <expan abbr="cõmunis">communis</expan> men&longs;ura in grauitate, ita vt &longs;it æquegrauis vni­<lb/>cuique parti OPQR, quæ quidem, & &longs;i non &longs;int magnitu­<lb/>dine inter&longs;e &ecedil;quales, &longs;ufficit, vt &longs;int æquegraues: veluti magni <lb/>
<arrow.to.target n="fig31"></arrow.to.target><lb/>tudines quoque STVX inter&longs;e, <expan abbr="ip&longs;is&qacute;">ip&longs;isque</expan>; OPQR tantùm &ecedil;que <lb/>graues; ita ut vnaquæque ponderet, vt IIII. veluti etiam par <lb/>tes ip&longs;ius B, & vnaquæque ZM. hi&longs;què ita po&longs;itis <expan abbr="demõ&longs;tra">demon&longs;tra</expan> <lb/>tio rectè concludet. </s></p>
<figure id="fig31"></figure>
<p type="main">
<s>In hacverò &longs;eptima Archimedis propo&longs;itione &longs;imiliter
<arrow.to.target n="marg68"></arrow.to.target> in­<lb/>telligantur magnitudines kMC incommen&longs;urabiles graui­<lb/>tate, vt in eius figura grauitas ip&longs;ius C ponderet, vt XII. gra <lb/>uitas verò ip&longs;ius KM maior &longs;it, quàm XX. ita vth&ecedil; graui­<lb/>tates &longs;intincommen&longs;urabiles. auferaturquè grauitas exce&longs;&longs;us <lb/>HL, quæ &longs;it vt IIII. ita vt quæ relinquiturgrauitas, ip&longs;ius <expan abbr="n&etilde;-pè">nen­<lb/>pè</expan> KH, qu&ecedil; quidem maior e&longs;t, quàm XVI, in F po&longs;ita, gra <lb/>uitati ip&longs;ius C, quæ e&longs;t XII, in D po&longs;itæ æqueponderet, <lb/>Auferatur deinde NL minor exce&longs;&longs;u HL; cuius quidem gra <lb/>uitas &longs;it maior, quàm II. ita vt grauitas re&longs;idui KN, quæ <lb/>nimirum &longs;it XVIII, &longs;it commen&longs;urabilis grauitati <lb/>XII. ip&longs;ius C. & <expan abbr="quãuis">quanuis</expan> magnitudines KM C, & KN C &longs;int, <lb/>vel <expan abbr="nõ">non</expan> &longs;int inter&longs;e magnitudine <expan abbr="cõmen&longs;urabiles">commen&longs;urabiles</expan>, vel incom­
<pb pagenum="76"/>men&longs;urabiles; eadem pror&longs;us demon&longs;tratio idem concludet. <lb/>quæ quidem omnia in &longs;equenti quoque propo&longs;itione <expan abbr="con&longs;i-derãda">con&longs;i­<lb/>deranda</expan> occurrunt. Vnde per&longs;picuum e&longs;t has Archime dis pro <lb/>po&longs;itiones, ac demon&longs;trationes vniuer&longs;ali&longs;&longs;imas e&longs;&longs;e, arque o­<lb/>mnibus, & quibu&longs;cunque magnitudinibus conuenientes. </s></p>
<p type="margin">
<s><margin.target id="marg68"></margin.target><emph type="italics"/>re&longs;pice <expan abbr="fi-gurã">fi­<lb/>guram</expan> &longs;epti­<lb/>mæ propo&longs;i <lb/>tionis Ar­<lb/>chimedis.<emph.end type="italics"/></s></p>
<p type="main">
<s>Iacto hoc pr&ecedil;cipuo, ac pr&ecedil;&longs;tanti&longs;&longs;imo mechanico funda­<lb/>mento; in &longs;equenti propo&longs;itione colligit ex hoc Archimedes, <lb/>quomodo &longs;e habent centra grauitatis magnitudinis diui&longs;æ. </s></p>
<p type="head">
<s>PROPOSITIO. VIII.</s></p>
<p type="main">
<s>Si ab aliqua magnitudine magnitudo aufera­<lb/>tur; quæ non habeat idem centrum cum tota; re­<lb/>liquæ magnitudinis centrum grauitatis e&longs;t in re­<lb/>cta linea, quæ coniungit centra grauitatum to tius <lb/>magnitudinis, & ablatæ, ad eam partem produ­<lb/>cta, vbi e&longs;t centrum to tius magnitudinis, ita vt a&longs;­<lb/>&longs;umpta aliqua ex producta, quæ coniungit <expan abbr="c&etilde;tra">centra</expan> <lb/>prædicta eandem habeat proportionem ad eam, <lb/>quæ e&longs;t inter centra, quam habet grauitas magni­<lb/>tudinis ablatæ ad grauitatem re&longs;iduæ, centrum e­<lb/>rit terminus a&longs;&longs;umptæ. </s></p>
<p type="main">
<s><emph type="italics"/>Sit alicuius magnitudinis AB centrum grauitatis C. auferatur­<lb/>què ex AB magnitudo AD; cuius centrum grauitatis &longs;it E. coniuncta <lb/>verò EC, &<emph.end type="italics"/> ex parte C <emph type="italics"/>producta, a&longs;&longs;umatur CF, quæ ad CE <expan abbr="eã">eam</expan> <lb/>dem habeat proportionem, quam habet magnitudo AD ad DG. osten­<lb/>dendum est, magnitudinis DG centrumgrauitatis e&longs;&longs;e punctum F. <expan abbr="Nõ">non</expan> <lb/>&longs;it autem; &longs;ed, &longs;i fieri potest, &longs;it punctum H. Quoniam igitur magnitudi­<lb/>nis AD centrum grauitatis est punctum E; magnitudinis verò DG <lb/>e&longs;t punctum H; magnitudinis ex vtri&longs;que magnitudinibus AD DG,<emph.end type="italics"/><lb/>
<arrow.to.target n="marg69"></arrow.to.target> <emph type="italics"/>compo&longs;itæ centrum grauitatis erit in linea EH, ita diui&longs;a, ut pirtes ip&longs;ius <lb/>permutatim eandem <expan abbr="habeãt">habeant</expan> proportionem, vt magnitudines. Quare non<emph.end type="italics"/>
<pb pagenum="77"/><emph type="italics"/>erit punctum C &longs;ecundùm diui&longs;ionem proportione re&longs;pondentem prædi­<lb/>etæ.<emph.end type="italics"/> vt &longs;cilicet &longs;it HC ad CE, vt AD ad DG. etenim ut AD <lb/>ad DG; ita <expan abbr="factũ">factum</expan> fuit FC ad CE. &longs;i igitur &longs;ecetur linea EH &longs;e <lb/>cundùm proportionem ip&longs;ius AD ad DG; non terminabit <lb/>
<arrow.to.target n="fig32"></arrow.to.target><lb/>diui&longs;io ad punctum C. cùm &longs;it impo&longs;&longs;ibile eandem habere <lb/>proportionem FC ad CE, quam. HC ad eandem CE. di­<lb/>ui&longs;io igitur ad aliud terminabitur punctum, vt K; ita vt HK
<arrow.to.target n="marg70"></arrow.to.target><lb/>ad KE &longs;it, vt AD ad DG. vnde &longs;equitur punctum K cen­<lb/>trum e&longs;&longs;e grauitatis magnitudinis ex AD DG compo&longs;itæ. <lb/><emph type="italics"/>Non e&longs;t igitur punctum C centrum magnitudinis ex AD DG compo <lb/>&longs;itæ; hoc est ip&longs;ius AB. e&longs;t autem; &longs;uppo&longs;itum e&longs;t enim<emph.end type="italics"/> ip&longs;um e&longs;&longs;e. <emph type="italics"/>er­<lb/>go neque punctum H centrum est grauitatis magnitudinis DG.<emph.end type="italics"/> e&longs;t <lb/>igitur punctum F; quod quidem e&longs;t terminus product&ecedil; line&ecedil; <lb/>CF; quæ eandam habet proportionem ad lineam CE inter <lb/>centra exi&longs;tentem; quam habet grauitas magnitudinis AD <lb/>ad grauitatem ip&longs;ius DG. quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg69"></margin.target><emph type="italics"/>ex præce­<lb/>dentibus.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg70"></margin.target><emph type="italics"/>ex præce­<lb/>dentibus.<emph.end type="italics"/></s></p>
<figure id="fig32"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>In hac demon&longs;tratione intelligendum e&longs;t etiam punctum <lb/>H e&longs;&longs;e po&longs;&longs;e extra lineam EF, ita vt EFH non &longs;itirecta linea. <lb/>quòd &longs;i H non e&longs;&longs;et in linea EF, idem &longs;equi ab&longs;urdum adeò <lb/>per&longs;picuum e&longs;t; vt nec demon&longs;tratione egeat. Quoniam &longs;i in <lb/>telligatur H extra lineam EF; iuncta EH, & ita diui&longs;a intel­<lb/>ligatur, vt ip&longs;ius partes permutatim grauitatibus magnitudi­<lb/>num AD DG re&longs;pondeant; e&longs;&longs;et vtique hoc punctum <expan abbr="inu&etilde;-tum">inuen­<lb/>tum</expan>, quod extra lineam EF reperiretur, centrum grauitatis to
<pb pagenum="78"/>tius AB quod fieri non pote&longs;t. &longs;iquidem e&longs;t punctum C, vt <lb/>&longs;uppo&longs;itum fuit. Vnde neque illud punctum H ip&longs;ius DG <expan abbr="c&etilde;">cem</expan> <lb/>trum grauitatis exi&longs;teret. </s></p>
<p type="main">
<s>Hic e&longs;t terminus prim&ecedil; partis principalis, in qua Archime <lb/>des (vt initio dixim^{9}) de magnitudinib^{9}, & degrauibus in <lb/>communi pertractauit; quandoquidem propo&longs;itiones, ac de­<lb/>mon&longs;trationes tam planis, quàm &longs;olidis quibu&longs;cunque &longs;unt <lb/>accomodatæ; vt manife&longs;tum fecimus. </s></p>
<p type="main">
<s>Nunc ita que &longs;e conuertit Archimedes ad <expan abbr="inue&longs;tigandũ">inue&longs;tigandum</expan> cen <lb/>tra grauitatis planorum. primùm què perquirit centrum gra­<lb/>uitatis parallelogrammorum; o&longs;ten detquè centrum grauitatis <lb/>cuiu&longs;libet parallelogrammi e&longs;&longs;e in recta linea, quæ coniungit <lb/>oppo&longs;ita latera bifariam diui&longs;a. ob cuius intelligentiam hæc <lb/>priùs lemmata in vnum collecta noui&longs;&longs;e erit valdè vtile. </s></p>
<p type="head">
<s>LEMMA.</s></p>
<p type="main">
<s>Sit parallelogrammum ABCD, cuius oppo&longs;ita latera AB <lb/>CD &longs;int bifariam diui&longs;a in EF. connectaturquè EF, quæ ni <lb/>mirum æquidi&longs;tans eritip&longs;is AC BD. Deinde diuidatur v­<lb/>
<arrow.to.target n="fig33"></arrow.to.target><lb/>naquæque AE EB in partes numero pares, & inuicem &ecedil;qua <lb/>les; vt in AG GE; & EH HB. <expan abbr="ducãturquè">ducanturquè</expan> GK HL ip&longs;i <lb/>EF &ecedil;quidi&longs;tantes. &longs;it verò centrum grauitatis ip&longs;ius AK pun <lb/>ctum M. ipfius verò GF punctum N, & ip&longs;ius EL pun­<lb/>ctum O deniquè ip&longs;ius HD punctum P. Dico primùm <expan abbr="pũ">pum</expan> <lb/>cta MNOP e&longs;&longs;e in linea recta. deinde lineas MN NO OP <lb/>inter centra exi&longs;tentes inter&longs;e æquales e&longs;&longs;e. Denique centrum <lb/>grauitatis parallelogrammi AD e&longs;&longs;e in linea NO, qu&ecedil; con <lb/>iungit centra grauitatis &longs;patiorum mediorum; parallelogram <lb/>morum &longs;cilicet GF EL.
<pb pagenum="79"/>Ducantur à punctis MN ip&longs;i AGE &ecedil;quidi&longs;tantes QMR <lb/>SNT. erunt vtique AQRG, & GSTE parallelogramma. <lb/>Quoniam igitur parallelogramma AK GF in æqualibus <lb/>&longs;untba&longs;ibus AG GE, & in ij&longs;dem parallelis; erunt AK GF
<arrow.to.target n="marg71"></arrow.to.target><lb/>inter&longs;e &ecedil;qualia. & quoniam AC GK EF &longs;unt <expan abbr="&ecedil;quidi&longs;tãtes">&ecedil;quidi&longs;tantes</expan>; <lb/>erit angulus CAG ip&longs;i KGE &ecedil;qualis, & KGA ip&longs;i FEG
<arrow.to.target n="marg72"></arrow.to.target><lb/>æqualis; & horum oppo&longs;iti inter&longs;e &longs;unt &ecedil;quales; ergo
<arrow.to.target n="marg73"></arrow.to.target> paralle­<lb/>logrammum GF ip&longs;i AK &ecedil;quale, & &longs;imile exi&longs;tit. Itaque <lb/>&longs;i GF collocetur&longs;uper AK, rectè congruet: eruntquè paral­<lb/>lelogramma inuicen coaptata. line&ecedil;què GE AG, GK AC, & <lb/>reliquæ coaptatæ erunt. quare eorum centra grauitatis
<arrow.to.target n="marg74"></arrow.to.target> inui­<lb/>cem coaptata erunt. hoc e&longs;t N erit in puncto M. Quoniam <lb/>autem à punctis MN (quod nunc intelligitur vnum tantum <lb/>e&longs;&longs;e punctum) ductæ fuerunt ST QR ip&longs;i AGE æquidi­<lb/>&longs;tantes, linea ST coaptabitur cum QR, quippe cùm ambæ <lb/>hæ lineæ ab vno puncto prodeuntes ip&longs;i AG &ecedil;quidi&longs;tantes <lb/>e&longs;&longs;e debeant. punctum igitur S in Q, & T in R coaptabi­<lb/>tur. eritquè QM ip&longs;i SN &ecedil;qualis, & MR ip&longs;i NT. ac pro <lb/>pterea linea GS parallelogrammi GT erit coaptata in <expan abbr="Aq;">Aque</expan> <lb/>& ET coaptata eritin GR parallelogrammi AR. Vnde e­<lb/>rit AQ &ecedil;qualis GS, cùm &longs;int coaptatæ; & GR ip&longs;i ET &ecedil;­<lb/>qualis; cùm &longs;int quoque coaptat&ecedil;. Quocirca quoniam
<arrow.to.target n="marg75"></arrow.to.target> pa­<lb/>rallelogramma AR GT &longs;unt inuicem coaptata, paral­<lb/>lelogrammorumquè oppo&longs;ita latera &longs;untinter&longs;e &ecedil;qualia, <expan abbr="erũt">erunt</expan> <lb/>AQ GS GR ET inter&longs;e &ecedil;qualia. Nunc autem <expan abbr="intelligãtur">intelligantur</expan> <lb/>parallelogramma AK GF non ampliùs coaptata. & <expan abbr="quoniã">quoniam</expan> <lb/>line&ecedil; QMR, & SNT &longs;untip&longs;i AGE parallel&ecedil;; & AQ GR, <lb/>GS ET, inter&longs;e &longs;untæquales, & &ecedil;quidi&longs;tantes; puncta RS in <lb/>vnum coincident punctum. eritquè QST linea recta. ex qui <lb/>bus patet, rectam <expan abbr="lineã">lineam</expan>, quæ coniungit centra grauitatis MN <lb/>ip&longs;i AGE æquidi&longs;tantem exi&longs;tere. eodemquè modo o&longs;tende­<lb/>tur rectas lineas, quæ coniungunt grauitatis centra NO, cen­<lb/>traquè OP, ip&longs;i AB <expan abbr="æquidi&longs;tãtes">æquidi&longs;tantes</expan> e&longs;&longs;e. Vnde &longs;equitur lineam <lb/>MNOP rectam e&longs;&longs;e. Quare primùm con&longs;tat grauitatis <expan abbr="c&etilde;tra">centra</expan> <lb/>in recta linea exi&longs;tere. </s></p>
<p type="margin">
<s><margin.target id="marg71"></margin.target>36. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg72"></margin.target>29. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg73"></margin.target>34. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg74"></margin.target>5. <emph type="italics"/>post, hu­<lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg75"></margin.target>34. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<figure id="fig33"></figure>
<p type="main">
<s>Quoniam autem o&longs;ten&longs;um e&longs;t QM æqualem e&longs;&longs;e ip&longs;i SN, <lb/>& MR ip&longs;i NT, eodem quoque modo o&longs;tendetur OT &ecedil;qua-
<pb pagenum="80"/>lem e&longs;&longs;e ip&longs;i SN. Quoniam igitur OT NS &longs;unt &ecedil;quales, iti­<lb/>demquè TN SM æquales, erit ON ip&longs;i NM æqualis. ea­<lb/>demquè ratione o&longs;tendetur OP &ecedil;qualem e&longs;&longs;e ip&longs;i ON. vn­<lb/>de colligitur lineas MN NO OP inter centra exi&longs;tentes in­<lb/>rer&longs;e &ecedil;quales e&longs;&longs;e. </s></p>
<p type="main">
<s>Po&longs;tremò quoniam parallelogramma AK GF EL HD <lb/>&longs;unt inuicem æqualia, & numero paria, centraquè grauitatis <lb/>&longs;unt in recta linea po&longs;ita. line&ecedil;què MN NO OP inter cen­<lb/>tra &longs;unt &ecedil;quales, magnitudinis ex omnibus AK GF EL HD <lb/>
<arrow.to.target n="marg76"></arrow.to.target> magnitudinibus compo&longs;itæ centrum grauitatis e&longs;t in linea <lb/>MP bifariam diui&longs;a. Et quoniam MN e&longs;t æqualis ip&longs;i OP, <lb/>punctum, quod bifariam diuidit MP cadet in linea NO. <lb/>centrum ergo grauitatis omnium magnitudinum AK GF <lb/>EL HD, hoc e&longs;t parallelogrammi AD e&longs;t in linea NO, qu&ecedil; <lb/>coniungit centra &longs;patiorum mediorum GF EL. qu&ecedil; <expan abbr="quid&etilde;">quidem</expan> <lb/>omnia o&longs;tendere oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg76"></margin.target>2.<emph type="italics"/>cor. quin <lb/>tæhuius.<emph.end type="italics"/></s></p>
<p type="main">
<s>Quoniam autem centrum grauitatis <expan abbr="parallelogrãmi">parallelogrammi</expan> AD <lb/>e&longs;t in linea NO, & in linea MP bifariam diui&longs;a; non repu­<lb/>gnare videtur, quin inferri po&longs;&longs;it, hoc centrum e&longs;&longs;e in puncto <lb/>T, in linea EF exi&longs;tente. Quòd tamen fal&longs;um e&longs;t. nam po&longs; <lb/>&longs;et quidem concludi centru e&longs;&longs;e in medio line&ecedil; NO (<expan abbr="&longs;iquid&etilde;">&longs;iquidem</expan> <lb/>e&longs;t in medio line&ecedil; MP, vt <expan abbr="dictũ">dictum</expan> e&longs;t) &longs;ed <expan abbr="nõ">non</expan> in <expan abbr="pũcto">puncto</expan> T; ex <expan abbr="demõ">demom</expan> <lb/>&longs;tratione enim o&longs;tenditur NS æqualem e&longs;&longs;e ip&longs;i TO. at verò <lb/>NT &ecedil;qualem e&longs;&longs;e ip&longs;i TO, nullo modo demon&longs;trari pote&longs;t; <lb/>ni&longs;i &longs;upponeremus centra grauitatis MNOP in parallelogra <lb/>mis ita &longs;e habere, vt MQ MR, & MR RN, & RN NT & <lb/>NT TO, &c. inter &longs;e &ecedil;quales e&longs;&longs;ent. quod nullo modo &longs;up­<lb/>poni pote&longs;t nam hoc modo centra grauitatis parallelogram­<lb/>morum AK GF &c. e&longs;&longs;entin lineis, qu&ecedil; bifariam &longs;ecant op <lb/>po&longs;ita latera. e&longs;&longs;ent quippè in lineis à punctis MN OP du­<lb/>ctisip&longs;is AC GK EF &c. æquidiftantibus, quæ oppo&longs;ita la <lb/>tera AG CK, GE KF, EH FL, &c. bifariam &longs;ecarent. quod <lb/>e&longs;t id, quod Archimedes demon&longs;trare in <expan abbr="&longs;equ&etilde;ti">&longs;equenti</expan> nititur. quod <lb/>quidem in cau&longs;a e&longs;t, vt demon&longs;tratione ad impo&longs;&longs;ibile id de­<lb/>ducat. &longs;uppo&longs;uimus autem (vt pare&longs;t) parallelogramma cen-
<pb pagenum="81"/>tra grauitatis habere; ac centra grauitatis MNOP intra pa­<lb/>rallelogramma exi&longs;tere, quoniam parallelogramma &longs;unt
<arrow.to.target n="marg77"></arrow.to.target> fi­<lb/>guræ ad ea&longs;dem partes concauæ. quod quidem eodem modo <lb/>ab Archimede in &longs;equenti &longs;upponitur. </s></p>
<p type="margin">
<s><margin.target id="marg77"></margin.target>9. <emph type="italics"/>po&longs;t hu­<lb/>ius.<emph.end type="italics"/></s></p>
<p type="head">
<s>PROPOSITIO. IX.</s></p>
<p type="main">
<s>Omnis parallelogrammi centrum grauitatis <lb/>e&longs;t in recta linea, quæ oppo&longs;ita latera parallelo­<lb/>grammi bifariam diui&longs;a coniungit. </s></p>
<figure></figure>
<p type="main">
<s><emph type="italics"/>Sit parallelogrammum ABCD, linea verò EF bifariam diuidat la <lb/>tera AB CD. Dico parallelogrammi ABCD centrum grauitatis e&longs;&longs;e<emph.end type="italics"/>
<arrow.to.target n="marg78"></arrow.to.target><lb/><emph type="italics"/>in linea EF. Non &longs;it quidem, &longs;ed, &longs;i fieri pote&longs;t, &longs;it H. &<emph.end type="italics"/> ab ip&longs;o <expan abbr="v&longs;q;">v&longs;que</expan> <lb/>ad lineam EF <emph type="italics"/>ducatur H<gap/> æquidistansip&longs;i AB. Diui&longs;a verò EB <lb/>&longs;emper bifariam<emph.end type="italics"/> in G. rur&longs;u&longs;què EG brfariam in K; idèquè <lb/>&longs;emper fiat, tandem <emph type="italics"/>quædam relinquetur linea,<emph.end type="italics"/> putà EK, <emph type="italics"/>minor <lb/>ip&longs;a HI. Diuidaturquè vtraque AE EB in partes<emph.end type="italics"/> AN NM ML
<arrow.to.target n="marg79"></arrow.to.target><lb/>LE GO OB <emph type="italics"/>ip&longs;i EK æquales.<emph.end type="italics"/> quod quidem fieri pote&longs;t, quia <lb/>diui&longs;a e&longs;t EB in partes &longs;emper &ecedil;quales. <emph type="italics"/>& ex<emph.end type="italics"/> his <emph type="italics"/>diui&longs;ionum pun <lb/>ctis ducantur<emph.end type="italics"/> NP MQ LR kS GT OV <emph type="italics"/>ip&longs;i EF æquidistantes. <lb/>diui&longs;um enim erit totum parallelogrammum in parallelogramma æqualia <lb/>& &longs;imiliaip&longs;i<emph.end type="italics"/> k<emph type="italics"/>F.<emph.end type="italics"/> cùm enim &longs;int parallelogrammorum ba&longs;es <lb/>EL LM MN NA KG GO OB ip&longs;i KE æquales,
<arrow.to.target n="marg80"></arrow.to.target> parallelo­<lb/>grammaquè in ij&longs;dem &longs;int parallelis AB CD con&longs;tituta; <lb/>erunt parallelogramma æqualia. &longs;imilia verò, quoniam <lb/>&longs;unt &ecedil;quiangula. <emph type="italics"/>Parallelogrammis igitur æqualibus, atque<emph.end type="italics"/>
<pb pagenum="82"/><emph type="italics"/>&longs;imilibus ip&longs;i KF inuicem coaptatis, & centra grauitatis inter&longs;e conue­<lb/>nient.<emph.end type="italics"/> quia verò in EB facta e&longs;t diui&longs;io &longs;emper in duas partes <lb/>&ecedil;quales erunt parallelogramma in ED numero paria. ac per <lb/>con&longs;equens & qu&ecedil; &longs;unt in EC numero paria. vnde & qu&ecedil; sut <lb/>in toto AD numero paria <expan abbr="erũt">erunt</expan>. <emph type="italics"/>Jtaque quædam erunt magnitudi­<lb/>nes æquidi&longs;tantium laterum æquales ip&longs;i KF numero pares,<emph.end type="italics"/> hoc e&longs;t o­<lb/>
<arrow.to.target n="marg81"></arrow.to.target> mnes, quæ &longs;untin AD, <emph type="italics"/>centraquè grauitatis ip&longs;arum in recta linea<emph.end type="italics"/><lb/>
<arrow.to.target n="marg82"></arrow.to.target> <emph type="italics"/>&longs;unt con&longs;tituta, & lineæ inter centra &longs;unt a quales magnitudinis ex ip&longs;is <lb/>omnibus compo&longs;itæ centrum grauitatis erit in recta linea, quæ coniungit <lb/>centra grauitatis mediorum &longs;patiorum,<emph.end type="italics"/> parallelogrammorum &longs;cili­<lb/>cet LF KF. <emph type="italics"/>Non est autem; punctum enim H,<emph.end type="italics"/> quod &longs;upponitur <lb/>e&longs;&longs;e centrum grauitatis omnium magnitudinum, hoc e&longs;t pa <lb/>rallelogrammi AD, <emph type="italics"/>extra media parallelogramma<emph.end type="italics"/> LF KF <emph type="italics"/>exi&longs;tit.<emph.end type="italics"/><lb/>etenim cùm &longs;it EK minor HI, linea KS ip&longs;i EF <expan abbr="&ecedil;quidi&longs;tãs">&ecedil;quidi&longs;tans</expan> <lb/>lineam HI ip&longs;i EK æquidi&longs;tantem &longs;ecabit, quippè quæ re­<lb/>linquet punctum H extra figuram KF, ac per con&longs;equens ex­<lb/>tra media parallelogramma LF KF. quare punctum H non <lb/>e&longs;t centrum grauitatis parallelogrammi AD, vt &longs;upponeba­<lb/>tur. <emph type="italics"/>ergo con&longs;tat, centrum grauitatis parallelogrammi ABCD e&longs;&longs;e in re <lb/>cta linea EF.<emph.end type="italics"/> quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg78"></margin.target>*</s></p>
<p type="margin">
<s><margin.target id="marg79"></margin.target><emph type="italics"/>ex prima <lb/>pr&ecedil;cedenti<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg80"></margin.target>36. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg81"></margin.target>*</s></p>
<p type="margin">
<s><margin.target id="marg82"></margin.target><emph type="italics"/>lemma.<emph.end type="italics"/></s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>
<arrow.to.target n="marg83"></arrow.to.target> Græcus codex po&longs;t verba, <emph type="italics"/>centraquè grauitatis ip&longs;arum in recta <lb/>linea &longs;unt constituta,<emph.end type="italics"/> habet, <foreign lang="greek">kai\ ta\ me\sa i)/sa, kai\ w_a\nta ta\ ef) eka/teza <lb/>tw=n me)swn auta/ te i)/sa e)nti/</foreign>, quæ quidem omnino &longs;uperflua nobis <lb/>ui<gap/>a &longs;unt, & <expan abbr="tanquã">tanquam</expan> ab aliquo addita. Nam &longs;i Archimedes di­<lb/>xit omnia parallelogramma e&longs;&longs;e inter&longs;e, & &ecedil;qualia, & &longs;imilia; <lb/>non opus e&longs;t addere, media LF ES e&longs;&longs;e inter&longs;e &ecedil;qualia, & <lb/>qu&ecedil; ab his &longs;unrad vtramque partem, vt MR KT, NQ GV, <lb/>AP OD, e&longs;&longs;e inter&longs;e æqualia; cum omnia (vt dictum e&longs;t) &longs;int <lb/>&ecedil;qualia. quare verba h&ecedil;c (meo quidem iudicio) delenda &longs;unt. <lb/>demon&longs;trationes enim mathematic&ecedil; nullum admittunt &longs;u­<lb/>perfluum. & Archim edes non tantùm &longs;uperfluus, quin potiùs <lb/>ob cius breuitatem diminutus ferè videatur. </s></p>
<pb pagenum="83"/>
<p type="margin">
<s><margin.target id="marg83"></margin.target>*</s></p>
<p type="main">
<s>Ex hac nona propo&longs;itione duo corolloria elicere po&longs;&longs;um^{9}; <lb/>quæ quidem tanquam valde nota fortaf&longs;e videtur omi&longs;i&longs;&longs;e Ar <lb/>chimedes. quamuis <expan abbr="primũ">primum</expan> in &longs;equenti <expan abbr="demõ&longs;tratione">demon&longs;tratione</expan> in&longs;eruit. </s></p>
<p type="head">
<s>COROLLARIVM. I.</s></p>
<p type="main">
<s>Ex hoc per&longs;picuum e&longs;t cuiu&longs;libet parallelogrammi <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis e&longs;&longs;e punctum, in quo coincidunt rectæ lineæ, quæ <lb/>oppo&longs;ita latera bifariam &longs;ecant. </s></p>
<p type="main">
<s>Nam (vt Archimedes etiam &longs;e <lb/>
<arrow.to.target n="fig34"></arrow.to.target><lb/>quenti demon&longs;tratione inquit) <lb/>&longs;i parallelogrammi ABCD line&ecedil; <lb/>EF GH bifariam diuident late­<lb/>ra oppo&longs;ita AB DC, & AD BC. <lb/>patetin EF centrum e&longs;&longs;e graui­<lb/>tatis parallelogrammi AC. &longs;imi <lb/>liter con&longs;tat idem centrum e&longs;&longs;e <lb/>in linea GH, quæ oppo&longs;ita latera AD BC bifariam &longs;ecat. e­<lb/>ritigitur in K, vbi EF GH &longs;einuicem &longs;ecant. </s></p>
<figure id="fig34"></figure>
<p type="head">
<s>COROLLARIVM. II.</s></p>
<p type="main">
<s>Ex hoc patet etiam, cuiu&longs;libet parallelogrammi <expan abbr="centrũ">centrum</expan> gra <lb/>uitatis e&longs;&longs;e in medio rectæ line&ecedil;, quæ bifariam oppo&longs;ita latera <lb/>di&longs;pe&longs;cit. </s></p>
<p type="main">
<s>Cùm enim o&longs;ten&longs;um &longs;it centrum grauitatis parallelogram <lb/>mi AC e&longs;&longs;e punctum K. & ob parallelogrammum EH e&longs;t <lb/>EK æqualis BH. propter parallelogrammum verò KC
<arrow.to.target n="marg84"></arrow.to.target><lb/>linea KF e&longs;t æqualis HC. &longs;untquè BH HC æqua­<lb/>les. erit EK ip&longs;i KF æqualis. punctum ergo K e&longs;tin medio <lb/>rectæ line&ecedil; EF, quæ oppo&longs;ita latera AB DC bifariam diui­<lb/>dit. <expan abbr="Eod&etilde;&qacute;">Eodenque</expan>; pror&longs;us modo <expan abbr="o&longs;t&etilde;detur">o&longs;tendetur</expan>, K <expan abbr="mediũ">medium</expan> e&longs;&longs;e rect&ecedil; line&ecedil; <lb/>GH, quæ bifariam &longs;ecat oppo&longs;ita latera AD BC. </s></p>
<p type="margin">
<s><margin.target id="marg84"></margin.target>34. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="main">
<s>In &longs;equenti Archimedes adhuc per&longs;i&longs;tit in inuentione cen­<lb/>tri grauitatis parallelogrammorum, alia tamen methodo. <lb/>nam hoc perip&longs;orum parallelogrammorum diametros duo­<lb/>bus modis a&longs;&longs;equitur. </s></p>
<pb pagenum="84"/>
<p type="head">
<s>PROPOSITIO. X.</s></p>
<p type="main">
<s>Omnis parallelogrammi centrum grauitatis <lb/>e&longs;t punctum, in quo diametri coincidunt. </s></p>
<p type="main">
<s><emph type="italics"/>Sit parallelogrammum <lb/>ABCD. & in ip&longs;o &longs;it li­<lb/>nea EF<emph.end type="italics"/> bifariam <emph type="italics"/><expan abbr="&longs;ecãs">&longs;ecans</expan><emph.end type="italics"/><lb/>
<arrow.to.target n="fig35"></arrow.to.target><lb/><emph type="italics"/>latera AB CD. itidem­<lb/>què &longs;it KL <expan abbr="&longs;ecãs">&longs;ecans</expan> AC BD<emph.end type="italics"/><lb/>bifariam. conueniant­<lb/>què EF kL in H. <emph type="italics"/>est <lb/>vtique parallelogrammi<emph.end type="italics"/><lb/>
<arrow.to.target n="marg85"></arrow.to.target> <emph type="italics"/>ABCD centrum grauita <lb/>tis in linea EF. hoc enim <lb/>o&longs;ten&longs;um e&longs;t. eadem verò de cau&longs;a<emph.end type="italics"/> centrum grauitatis ip&longs;ius AD <emph type="italics"/>est <lb/>etiam in linea<emph.end type="italics"/> K<emph type="italics"/>L. quare punctum H<emph.end type="italics"/> parallelogrammi AD <emph type="italics"/>cen­<lb/>trum grauitatis existit. Verùm in puncio H diametri parallelogram­<lb/>mi concurrunt.<emph.end type="italics"/> ductis enim lineis AH HB CH HD; quoniam <lb/>lineæ AE EB EF FD inter&longs;e &longs;unt &ecedil;quales. &longs;imiliter quoque <lb/>AK KC BL LD inter&longs;e &ecedil;quales; erit EH ip&longs;i HF &ecedil;qua <lb/>lis, cùm &longs;int ip&longs;is BL LD &ecedil;quales. duæ igitur AE EH dua <lb/>
<arrow.to.target n="marg86"></arrow.to.target> bus DF FH &longs;unt æquales, & angulus AEH angulo DFH <lb/>
<arrow.to.target n="marg87"></arrow.to.target> &ecedil;qualis; erit triangulum AEH triangulo DFH &ecedil;quale. ac <lb/>propterea angulus EHA angulo FHD æqualis. cùm igitur <lb/>&longs;it EHF recta linea, eruntangnli EHA FHD adverticem, <lb/>& obid AHD recta exi&longs;tit linea. ac per con&longs;equens diame­<lb/>ter parallelogrammi AD. pariquè ratione o&longs;tendetur BHC <lb/>rectam e&longs;&longs;e lineam. ex quibus patet in puncto H <expan abbr="vtrãque">vtranque</expan> dia <lb/>metrum conuenire. centrum igitur grauitatis parallelogram­<lb/>mi AD e&longs;t <expan abbr="pũctum">punctum</expan>, in quo diametri concurrunt. <emph type="italics"/>Quare demon <lb/>stratume&longs;t, quod propo&longs;itum fuit.<emph.end type="italics"/></s></p>
<pb pagenum="85"/>
<p type="margin">
<s><margin.target id="marg85"></margin.target>9 <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg86"></margin.target>29, <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg87"></margin.target>4. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<figure id="fig35"></figure>
<p type="main">
<s>ALITER. </s></p>
<p type="main">
<s><emph type="italics"/>Hoc autem aliter quo­<lb/>que o&longs;tendetur. &longs;it paralle<emph.end type="italics"/><lb/>
<arrow.to.target n="fig36"></arrow.to.target><lb/><emph type="italics"/>logrammum ABCD. <lb/>ip&longs;ius verò diameter &longs;it<emph.end type="italics"/>
<arrow.to.target n="marg88"></arrow.to.target><lb/><emph type="italics"/>B D. triangula<emph.end type="italics"/> vtique <lb/>ABD BDC <emph type="italics"/>erunt in­<lb/>ter&longs;e æqualia, & &longs;imilia. <lb/>quare triangulis inuicem <lb/>coaptatis; centra quoque <lb/>grauitatis ip&longs;orum inuicem coaptabuntur. Sit autem trianguli ABD cen<emph.end type="italics"/>
<arrow.to.target n="marg89"></arrow.to.target><lb/><emph type="italics"/>trum grauitatis punctum E; lineaquè BD bifariam &longs;ecetur in H. con <lb/>nectaturquè EH, & producatur. &longs;umaturquè FH æqualisip&longs;i HE. <lb/>Itaque coaptato triangulo ABD cumtriangulo B DC, po&longs;itoquè latere <lb/>AB in DC,<emph.end type="italics"/> hoc e&longs;t A in C, & B in D. <emph type="italics"/>AD autem<emph.end type="italics"/> po&longs;ito <emph type="italics"/>in <lb/>BC;<emph.end type="italics"/> A &longs;cilicet in C, & D in B. vnde & BD cum ip&longs;amet <lb/>DB coaptatur, B &longs;cilicet in D, & D in B. quia verò pun­<lb/>ctum H &longs;ibi ip&longs;i coaptatur, cùm fitmedium line&ecedil; BD. & an <lb/>guli EHD FHB ad verticem &longs;unt æquales; lineaquè EH e&longs;t <lb/>ip&longs;i HF &ecedil;qualis; <emph type="italics"/>congruet etiam recta HE cum recta FH, & <expan abbr="pũ-ctum">pun­<lb/>ctum</expan> E cum F conueniet, &longs;ed<emph.end type="italics"/> quoniam punctum E centrum <lb/>e&longs;t grauitatis trianguli ABD idem punctum E <emph type="italics"/>cum centro e­<lb/>tiam grauitatis trianguli B DC<emph.end type="italics"/> conueniet. ergo punctum F <expan abbr="c&etilde;-trum">cen­<lb/>trum</expan> e&longs;t grauitatis trianguli BDC. Nunc verò intelligantur <lb/>triangula non ampliùs coaptata. <emph type="italics"/>Quoniam igitur centrum graui­<lb/>tatis trianguli ABD e&longs;t punctum E, ip&longs;ius verò DBC est punctum F,<emph.end type="italics"/><lb/>triangulaquè ABD DBC &longs;unt &ecedil;qualia, <emph type="italics"/>patet magnitudinis ex v­<lb/>tri&longs;que triangulis compo&longs;it<gap/> centrum grauitatis e&longs;&longs;e medium rectæ lineæ<emph.end type="italics"/>
<arrow.to.target n="marg90"></arrow.to.target><lb/><emph type="italics"/>EF; quod e&longs;t punctum H,<emph.end type="italics"/> vt factum furt. Quoniam autem dia­<lb/>metri cuiu&longs;libet parallelogrammi &longs;e&longs;e bifariam di&longs;pe&longs;cunt, e­<lb/>rit punctum H, vbi diametri parallelogrammi ABCD con­<lb/>currunt. ergo punctum H, in quo diametri coincidunt; ip&longs;ius <lb/>ABCD centrum grauitatis exi&longs;tit. quod demon&longs;trare opor­<lb/>rebat. </s></p>
<pb pagenum="86"/>
<p type="margin">
<s><margin.target id="marg88"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 34.<emph type="italics"/>pri <lb/>mi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg89"></margin.target>5. <emph type="italics"/>post hu­<lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg90"></margin.target>4. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<figure id="fig36"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Cognito centro grauitatis cuiu&longs;libet parallelogrammi, <lb/>vult Archimedes o&longs;ten dere centrum grauitatis triangulorum. <lb/>& quoniam in hac po&longs;trema demon&longs;tratione a&longs;&longs;ump&longs;it cen­<lb/>trum grauitatis trianguli ABD e&longs;&longs;e punctum E, videtur or <lb/>dinem peruerti&longs;&longs;e, & per ignotiora doctrinam tradidi&longs;&longs;e; cùm <lb/>non &longs;it adhuc o&longs;ten&longs;um, in quo &longs;itu dictum centrum in <expan abbr="triã-gulis">trian­<lb/>gulis</expan> reperiatur. quod tamen &longs;i rectè perpendamus, non ita &longs;e <lb/>habet. Nam vis demon&longs;trationis e&longs;t in hoc con&longs;tituta, vt <lb/>&longs;upponatur triangulum habere centrum grauitatis, idquè tan <lb/>
<arrow.to.target n="marg91"></arrow.to.target> <gap/>ùm e&longs;&longs;e intra ipsum triangulum, quod quidem &longs;upponi po­<lb/>te&longs;t. cùm triangulum &longs;it figura ad ea&longs;dem partes concaua. ne­<lb/>que enim refert, &longs;iuè centrum &longs;it in E, &longs;iuè in alio &longs;itu, dum­<lb/>modo intra triangulum exi&longs;tat. demon&longs;tratio enim <expan abbr="eod&etilde;">eodem</expan> mo­<lb/>do &longs;emper concludet punctum H centrum e&longs;&longs;e grauitatis pa <lb/>rallelogrammi AC, quod idem ob&longs;eruandum e&longs;t in <expan abbr="nõnullis">nonnullis</expan> <lb/>alijs demon&longs;trationibus. vt in &longs;ecunda demon&longs;tratione deci­<lb/>mæ tertiæ, hui^{9} & in prima &longs;ecun dilibri. Antequam <expan abbr="aut&etilde;">autem</expan> Ar­<lb/>chimedes centrum grauitatis triangulorum o&longs;tendat, nonnul <lb/>las pr&ecedil;mittit propo&longs;itiones. </s></p>
<p type="margin">
<s><margin.target id="marg91"></margin.target>9. <emph type="italics"/>post hu­<lb/>ius.<emph.end type="italics"/></s></p>
<p type="head">
<s>PROPOSITIO. XI.</s></p>
<p type="main">
<s>Si duo triangula inter&longs;e &longs;imilia fuerint, & in i­<lb/>p&longs;is &longs;int puncta ad triangula &longs;imiliter po&longs;ita & alre <lb/>rum punctum trianguli, in quo e&longs;t, centrum fue­<lb/>rit grauitatis, & alterum punctum trianguli, in <lb/>quo e&longs;t, centrum grauitatis exi&longs;tet. </s></p>
<pb pagenum="87"/>
<p type="main">
<s>Dicimus quidem punctain &longs;imilibus figuris e&longs;&longs;e <lb/>&longs;imiliter po&longs;ita, è quibus ad æquales angulos du­<lb/>ctæ rectæ lineæ, æqual es efficiunt angulos ad ho­<lb/>mologalatera. Vt dictum fuit in &longs;eptimo po&longs;tulato. </s></p>
<figure></figure>
<p type="main">
<s><emph type="italics"/>Sint duo triangula ABC DEF<emph.end type="italics"/> &longs;imilia. <emph type="italics"/>&longs;it què AC ad DE, vt <lb/>AB ad DE, & BC ad EF. & in præfatis triangulis ABC DEF <lb/>&longs;int puncta HN &longs;imiliter po&longs;ita &longs;itquè punctum H centrum grauitatis <lb/>trianguli ABC. Dico & punctum N centrum e&longs;&longs;e grauitatis trianguli <lb/>DEF. non &longs;it quidem, &longs;ed, &longs;i fieripote&longs;t, &longs;it punctum G centrum grauita <lb/>tis trianguli DEF. <expan abbr="connectãturquè">connectanturquè</expan> HA HB HC, DN EN FN, <lb/>DG EG FG. Quoniamigitur &longs;imile e&longs;t triangulum ABC triangulo <lb/>DEF, &<emph.end type="italics"/> ip&longs;orum <emph type="italics"/>centra grauitatum &longs;unt puncta HG. &longs;imi­<lb/>lium autem figurarum centra grauitatum &longs;unt &longs;imiliter po&longs;ita; ita vt<emph.end type="italics"/>
<arrow.to.target n="marg92"></arrow.to.target><lb/>ab ip&longs;is ad &ecedil;quales angulos ductæ rectæ line&ecedil; <emph type="italics"/>æquales faciant <lb/>angulos ad homologa latera, vnumquemquè vnicuiquè; erit angulus <lb/>GDE ip&longs;i HAB aqualis. at verò anguius HAB aqualis est angulo <lb/>EDN, cùm &longs;int puncta HN &longs;imiliter po&longs;ita: angulus igitur EDG <lb/>angulo EDN æqualis existit. maior minori quòd fierinon potest. Non <lb/>igitur punctum G centrum e&longs;t grauitatis trianguli DEF. Quare e&longs;t <lb/>punctum N. quod demonstrare oportebat.<emph.end type="italics"/></s></p>
<pb pagenum="88"/>
<p type="margin">
<s><margin.target id="marg92"></margin.target>6.& 7 <emph type="italics"/>po&longs;t <lb/>huius.<emph.end type="italics"/></s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>In hac propo&longs;itione &longs;upponit Archimedes dari po&longs;&longs;e pun­<lb/>cta in triangulis &longs;imilib^{9} &longs;imiliter po&longs;ita, qd <expan abbr="quid&etilde;">quidem</expan> &longs;ieri po&longs;&longs;e <lb/>o&longs;tendimus in &longs;cholijs &longs;eptimi po&longs;tulati. Præterea idem vide­<lb/>tur Archimedes in triangulis demon&longs;trare, quod in &longs;exto po­<lb/>&longs;tulato vniuer&longs;aliter in figuris &longs;uppo&longs;uit. Nam &longs;i centra gra­<lb/>uitatis &longs;upponuntur in &longs;imilibus figuris e&longs;&longs;e &longs;imiliter po&longs;ita; <lb/>& in &longs;imilibus triangulis quoque erunt &longs;imiliter po&longs;ita. In­<lb/>ter h&ecedil;c tamen maxima e&longs;t differen tia, nam in po&longs;tulato inquit, <lb/>centra grauitatum in &longs;imilibus figuris e&longs;&longs;e &longs;imiliter po&longs;ita; cu <lb/>ius quidem conuer&longs;um, nempè puncta in &longs;imilibus figuris &longs;i­<lb/>militer po&longs;ita e&longs;&longs;e ip&longs;arum centra grauitatis, e&longs;t falium. quod <lb/>e&longs;t quidem manife&longs;tum ab&longs;que alio exemplo. ac propterea <lb/>Archimedes hoc in loco inquit, &longs;i duo erunt pun&longs;ta in &longs;imi­<lb/>libus triangulis &longs;imiliter po&longs;ita, & alterum ip&longs;orum fuerit <expan abbr="c&etilde;-trum">cen­<lb/>trum</expan> grauitatis. & alterum quoque <expan abbr="c&etilde;trum">centrum</expan> grauitatis exi&longs;tet. <lb/>Vnde propo&longs;itio h&ecedil;c potiùs e&longs;t conuer&longs;a po&longs;tulati, quàm <lb/>eadem. </s></p>
<p type="main">
<s>Ob demon&longs;trationem autem noui&longs;&longs;e oportet, quòd &longs;i pun <lb/>ctum G fuerit in linea DN, tuncanguli EDG EDN e&longs;&longs;entin <lb/>ter&longs;e &ecedil;quales, ac propterea demon&longs;tratio nihil ab&longs;urdi conclu <lb/>deret. In hoc autem ca&longs;u o&longs;tendendum e&longs;&longs;et, angulum EFG <lb/>ip&longs;i EFN &ecedil;qualem e&longs;&longs;e, vel FEG ip&longs;i FEN. quæ quidem eo­<lb/>dem pror&longs;us modo o&longs;tendentur. comparando nempè angu­<lb/>los EFG EFN angulo BCH; angulos verò FEG FEN ip&longs;i <lb/>CBH. Quòd &longs;i G fuerit in alio &longs;itu, vtin triangulo EDN, <lb/>tuncanguli FDG FDN o&longs;tendentur &ecedil;quales. & ita in alijs <lb/>ca&longs;ibus, vbicunque &longs;cilicet fuerit punctum G, &longs;emper ali­<lb/>quod inuenietur huiu&longs;modi ab&longs;urdum. quæ quidem omni­<lb/>nò fieri non po&longs;&longs;unt. </s></p>
<pb pagenum="89"/>
<p type="head">
<s>PROPOSITIO. XII.</s></p>
<p type="main">
<s>Si duo triangula &longs;imilia fuerint, alterius verò <lb/>trianguli centrum grauitatis in rectalinea fuerit, <lb/>quæ &longs;it ab aliquo angulo ad dimidiam ba&longs;im du­<lb/>cta; & alrerius trianguli centrum grauitatis erit in <lb/>linea &longs;imiliter ducta. </s></p>
<figure></figure>
<p type="main">
<s><emph type="italics"/>Sint duo triangula ABC DEF<emph.end type="italics"/> &longs;imilia <emph type="italics"/>&longs;itquè AC ad DF, vt <lb/>AB ad DE, & BC ad FE. Diui&longs;aquè AC bifariam in G, iunga <lb/>tur BG. centrum verò grauitatis trianguli ABC &longs;it punctum H in li <lb/>nea BG. Dico centrum grauitatis trianguli EDF e&longs;&longs;e in recta linea &longs;i <lb/>militer ducta. &longs;ecetur DF bifariam in puncto M. & iungatur EM. <lb/>& vt BG ad BH, ita fiat ME ad EN. connectanturquè AH <lb/>HC, DN NF. Quoniam enim<emph.end type="italics"/> e&longs;t BA ad ED, vt AC ad DF, & <lb/><emph type="italics"/>AG dimidia e&longs;t ip&longs;ius AC; ip&longs;ius verò DF dimidiaest DM; erit BA <lb/>ad ED, vt AG ad DM.<emph.end type="italics"/> Quoniam autem ob <expan abbr="triãgulorum">triangulorum</expan>
<arrow.to.target n="marg93"></arrow.to.target><lb/>ABC DEF &longs;imilitudinem angulus BAC angulo EDF e&longs;t &ecedil;­<lb/>qualis. & vt AB ad DE, ita AG ad DM; <expan abbr="permutando&qacute;">permutandoque</expan>; AB ad
<arrow.to.target n="marg94"></arrow.to.target><lb/>AG, vt DE ad DM; erit <expan abbr="triangulũ">triangulum</expan> ABG <expan abbr="triãgulo">triangulo</expan> DEM &longs;imile. <lb/><expan abbr="&longs;imiliũ">&longs;imilium</expan> <expan abbr="ãt">ant</expan> <expan abbr="triãgulorũ">triangulorum</expan> <expan abbr="ãguli">anguli</expan> <expan abbr="sũt">sunt</expan> &ecedil;quales, <emph type="italics"/>et circa æquales <expan abbr="ãgulos">angulos</expan> late<emph.end type="italics"/>
<pb pagenum="90"/><emph type="italics"/>ra sut proportionalia. erit <lb/>igitur angul^{9} AGB angulo<emph.end type="italics"/><lb/>
<arrow.to.target n="fig37"></arrow.to.target><lb/><emph type="italics"/>DME aqualis, et<emph.end type="italics"/> ABG ip <lb/>&longs;i DEM æqualis quare <lb/><emph type="italics"/>vt AG ad DM, ita e&longs;t BG<emph.end type="italics"/><lb/>
<arrow.to.target n="marg95"></arrow.to.target> <emph type="italics"/>ad EM,<emph.end type="italics"/> & vt AB ad DE, <lb/>ita BG ad EM; & pmu­<lb/>tado AB ad BG, vt DE <lb/>ad EM. <emph type="italics"/>e&longs;t autem BG ad<emph.end type="italics"/><lb/>
<arrow.to.target n="marg96"></arrow.to.target> <emph type="italics"/>BH, vt ME ad EN, erit igitur ex æquali<emph.end type="italics"/> AB ad BH, vt DE ad EN. <lb/>
<arrow.to.target n="marg97"></arrow.to.target> rur&longs;u&longs;què permutando <emph type="italics"/>AB ad DE, vt BH ad EN.<emph.end type="italics"/> <expan abbr="quoniã">quoniam</expan> <lb/>
<arrow.to.target n="marg98"></arrow.to.target> autem anguli ABH DEN, quos ip&longs;æ line&ecedil; continent, &longs;unt <lb/>æquales, erit triangulun. ABH triangulo DEN &longs;imile. qua <lb/>re anguli &longs;unt inter&longs;e æquales, <emph type="italics"/>& circa a quales angulos latera &longs;unt <lb/>proportionalia &longs;i autem hoc, angulus BAH angulo EDN est æqualis. <lb/>Vnde & reliquus angulus HAC angulo NDF æquolis exi&longs;tit.<emph.end type="italics"/> <gap/>qui­<lb/>dem totius BAC ip&longs;i EDF e&longs;t æqualis. <emph type="italics"/>Eademquè ratione an-<emph.end type="italics"/><lb/>
<arrow.to.target n="marg99"></arrow.to.target> <emph type="italics"/>gulus BCH ip&longs;i EFN est æqualis. & angulas HCG angulo NFM <lb/>æqualis, o&longs;ten&longs;um est autem angulum ABH ip&longs;i DEM aqualem e&longs;&longs;e.<emph.end type="italics"/><lb/>ob &longs;imilitudinem autem riangulorum ABC DEF totus an <lb/>
<arrow.to.target n="marg100"></arrow.to.target> gulus ABC e&longs;tip&longs;i DEF &ecedil; ualis: <emph type="italics"/>ergo & reliquus angulus HBC <lb/>ip&longs;i NEF æqualis exi&longs;tit. Porrò ex his omnibus patet puncta HN ad <lb/>homologa latera e&longs;&longs;e &longs;imiliter po&longs;ita, &<emph.end type="italics"/> cum ip&longs;is <emph type="italics"/>angulas æquales effi­<lb/>cere. Cùm igitur puncta HN &longs;int &longs;imiliter po&longs;ita; & punctum H cen­<lb/>trum e&longs;t grauitatis trianguli ABC, & puncium N trianguli DEF <expan abbr="c&etilde;-trum">cen­<lb/>trum</expan> grauitatis existet.<emph.end type="italics"/> exi&longs;tente igitur centro grauitatis H in li <lb/>nea BG ab angulo ad dimidiam ba&longs;im ducta. & alterum gra <lb/>uitatis centrum N in linea EM &longs;imiliter ducta reperitur. <lb/>quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg93"></margin.target>16. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg94"></margin.target>6.<emph type="italics"/>&longs;eati.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg95"></margin.target>16. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg96"></margin.target>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg97"></margin.target>16. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg98"></margin.target>6. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg99"></margin.target>7. <emph type="italics"/>post hu <lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg100"></margin.target>11.<emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<figure id="fig37"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>In &longs;equenti Archimedes o&longs;tendet, in qua linea reperitur <expan abbr="c&etilde;">cem</expan> <lb/>trum grauita tis cuiu&longs;libet trianguli. quod quidem duobus a&longs;­<lb/>&longs;equitur medijs. Diligenter autem omnia &longs;unt con&longs;ideranda; <lb/>quoniam in hoc con&longs;i&longs;tit tota per&longs;crutatio centri grauitatis <lb/>triangulorum. Quapropter vt prior demon&longs;tratio appareat <lb/>per&longs;picua, h&ecedil;c antea dem on &longs;trabimus. </s></p>
<pb pagenum="91"/>
<p type="main">
<s>LEMMA. I. </s></p>
<p type="main">
<s>Æquidi&longs;tantes lineæ lineas in eadem proportione di&longs;pe­<lb/>&longs;cunt. </s></p>
<p type="main">
<s>Sintline&ecedil; AB CD, quas &longs;ecent æqui­<lb/>
<arrow.to.target n="fig38"></arrow.to.target><lb/>di&longs;tantes lineæ AC EF BD. Dico ita e&longs;­<lb/>&longs;e BE ad EA, vt DF ad FC. primùm <lb/>quidem AB CD vel &longs;unt &ecedil;quidi&longs;tantes,
<arrow.to.target n="marg101"></arrow.to.target><lb/>vel minùs. &longs;i &longs;unt æquidi&longs;tantes, iam habe <lb/>tur in tentum. Nam BE erit æqualis DF, <lb/>& EA ip&longs;i FC. vnde &longs;equitur ita e&longs;&longs;e BE <lb/>
<arrow.to.target n="fig39"></arrow.to.target><lb/>ad EA, vt DF ad FC. </s></p>
<p type="margin">
<s><margin.target id="marg101"></margin.target>34. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<figure id="fig38"></figure>
<figure id="fig39"></figure>
<p type="main">
<s>Si verò AB CD non fuerint æquidi­<lb/>&longs;tantes, concurrantin G, vt in &longs;ecunda fi­<lb/>
<arrow.to.target n="fig40"></arrow.to.target><lb/>gura, & quoniam BD EF &longs;unt
<arrow.to.target n="marg102"></arrow.to.target> æquidi­<lb/>&longs;tantes, erit GB ad BE, vt GD ad DF.
<arrow.to.target n="marg103"></arrow.to.target><lb/>& <expan abbr="cõponendo">componendo</expan> GE ad EB, vt GF ad FD.
<arrow.to.target n="marg104"></arrow.to.target><lb/>conuertendoquè BE ad EG, vt DF ad <lb/>FG, rur&longs;us quoniam EF AC &longs;unt æquidi <lb/>&longs;tantes; erit GE ad EA, vt GF ad FC, e­<lb/>ritigitur ex æquali BE ad EA, vt DF ad FC. </s></p>
<p type="margin">
<s><margin.target id="marg102"></margin.target>2.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg103"></margin.target>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg104"></margin.target><emph type="italics"/>cor.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="quĩti">quinti</expan><emph.end type="italics"/></s></p>
<figure id="fig40"></figure>
<p type="main">
<s>Secent verò &longs;e&longs;e lineæ AB CD, vt in tertia figura, ob
<arrow.to.target n="marg105"></arrow.to.target> &longs;imi­<lb/>litudinem triangulorum BGD EGF, it a erit BG ad GE, vt
<arrow.to.target n="marg106"></arrow.to.target><lb/>DG ad GF. & componendo BE ad EG, vt DF ad FG. e&longs;t
<arrow.to.target n="marg107"></arrow.to.target><lb/>verò GE ad EA, vt GF ad FC. ergo ex æquali BE ad EA <lb/>erit, vt DF ad FC. quod demon&longs;trare oportebat. </s></p>
<pb pagenum="92"/>
<p type="margin">
<s><margin.target id="marg105"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 4. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg106"></margin.target>18. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg107"></margin.target>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="main">
<s>LEMMA. II. </s></p>
<p type="main">
<s>Sit A ad B, vt C ad D; rur&longs;us A ad E &longs;it, vt C ad F. <lb/>Dico primùm A ad BE &longs;imul ita e&longs;&longs;e, vt C ad DF. </s></p>
<figure></figure>
<p type="main">
<s>
<arrow.to.target n="marg108"></arrow.to.target> Quoniam enim A e&longs;t ad B, vt C ad D, erit conuertendo <lb/>
<arrow.to.target n="marg109"></arrow.to.target> B ad A, vt D ad C. e&longs;t autem A ad E, vt C ad F; ergo ex &ecedil;­<lb/>
<arrow.to.target n="marg110"></arrow.to.target> quali B erit ad E, vt D ad F. quare componendo BE ad <lb/>
<arrow.to.target n="marg111"></arrow.to.target> E, vt DF ad F. quoniam autem A e&longs;t ad E, vt C ad F; e <lb/>
<arrow.to.target n="marg112"></arrow.to.target> rit conuertendo E ad A, vt F ad C. rur&longs;us igitur ex &ecedil;quali <lb/>erit BE ad A, vt DF ad C. ac denique conuertendo A e­<lb/>rit ad BE, vt C ad DF. </s></p>
<p type="margin">
<s><margin.target id="marg108"></margin.target><emph type="italics"/>cor.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="quĩti">quinti</expan><emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg109"></margin.target>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg110"></margin.target>18. <emph type="italics"/>qninti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg111"></margin.target><emph type="italics"/>cor.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="quĩti">quinti</expan><emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg112"></margin.target>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="main">
<s>Si verò fuerint quattuor magnitudines; vt adhue A (in ea­<lb/>dem figura) ad G &longs;it, vt C ad H. &longs;imili­<lb/>
<arrow.to.target n="fig41"></arrow.to.target><lb/>ter o&longs;tendetur A ad omnes BEG &longs;imul <lb/>&longs;umptas ita e&longs;&longs;e, vt C ad omnes &longs;imul <lb/>DFH. &longs;umendo vt in &longs;ecunda figura BE <lb/>pro vna tan ùm magnitudine, & DF pro <lb/>alia; eruntque ex vtraque parte tres <expan abbr="tãtùm">tantùm</expan> <lb/>magnitudines; eritquè A ad BE &longs;imul, <lb/>vt C ad DF &longs;imul, vt o&longs;ten&longs;um e&longs;t, dein <lb/>de A ad G e&longs;t, vt C ad H, erit igitur <lb/>A ad BEG &longs;imul, vt C ad DFH. </s></p>
<pb pagenum="93"/>
<figure id="fig41"></figure>
<p type="main">
<s>Pariquè ratione &longs;i quinque fuerint magnitudines, eodem <lb/>modo tres mediæ <expan abbr="iũgatur">iungatur</expan> &longs;imul, ita vttres &longs;int <expan abbr="dũtaxat">duntaxat</expan> magni <lb/>tudines. & &longs;ic in infinitum. quod demon&longs;trare oportebat. </s></p>
<p type="head">
<s>COROLLARIVM.</s></p>
<p type="main">
<s>Ex hoc elici pote&longs;t. quòd &longs;i fuerint quotcun que magnitudi <lb/>nes proportionales; & ali&ecedil; ip&longs;is numero æquales, & in eadem <lb/>proportione, vt&longs;cilicet &longs;it (vt in prima figura) A ad B, vt C <lb/>ad D, B verò ad E, vt D ad F. deinde vt E ad G, &longs;ic F <lb/>ad H, & ita deinceps, &longs;i plures fuerint magnitudines, &longs;i­<lb/>militer erit A ad omnes BEG &longs;imul &longs;umptas, vt C ad om­<lb/>nes &longs;imul DFH. </s></p>
<p type="main">
<s>Primùm quidem A e&longs;t ad B, vt C ad D. & quoniam ma <lb/>gnitudines &longs;unt proportionales, ex &ecedil;quali erit A ad E, vt C
<arrow.to.target n="marg113"></arrow.to.target><lb/>ad F. &longs;imiliter A ad G, vt C ad H. Ex quibus &longs;equitur <lb/>A ad BE &longs;imul ita e&longs;&longs;e, vt C ad DF. A verò ad omnes <lb/>BEG &longs;imul, vt C ad omnes &longs;imul DFH. & ita &longs;i plures fue <lb/>rint mag nitudines. </s></p>
<p type="margin">
<s><margin.target id="marg113"></margin.target>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="main">
<s>LEMMA. III. </s></p>
<p type="main">
<s>Sit triangulum ABC, cuiuslatus BC in quotcunque di­<lb/>uidatur partes æquales BE ED DF FC. & a punctis EDF <lb/>ip&longs;i AB equidi&longs;tanres ducantur EG DH FK. rur&longs;us à pun <lb/>ctis GHK ip&longs;i BC &ecedil;quidi&longs;tantes ducantur GL HM KN. <lb/>Dico triangulum ABC ad omnia triangula ALG GMH <lb/>HNK KFC &longs;imul&longs;umpta eandem habere proportionem, <lb/>quam habet CA ad AG. </s></p>
<pb pagenum="94"/>
<p type="main">
<s>
<arrow.to.target n="marg114"></arrow.to.target> <expan abbr="Quoniã">Quoniam</expan> enim FK &ecedil;quidi&longs;tans e&longs;tip&longs;i DH; erit CF ad FD, <lb/>vt CK ad KH. <expan abbr="&longs;unt&qacute;">&longs;untque</expan> CF FD æquales; ergo & CK KH in­<lb/>ter&longs;e &longs;unt æquales. &longs;imiliter propter lineas æquidi&longs;tantes FK <lb/>
<arrow.to.target n="marg115"></arrow.to.target> DH EG, ita e&longs;t KH ad HG, vt FD ad DE; e&longs;t autem FD <lb/>æqualis DE; eritigitur KH ip&longs;i HG æqualis. Pariquè ra­<lb/>
<arrow.to.target n="fig42"></arrow.to.target><lb/>tione o&longs;tendetur ob &ecedil;quidi&longs;tantes lineas DH EG BA, <expan abbr="lineã">lineam</expan> <lb/>HG ip&longs;i GA æqualem e&longs;&longs;e. Ex quibus patet CK KH HG <lb/>GA inter &longs;e æquales e&longs;&longs;e. Quoniam autem trianguloru ABC <lb/>kFC angulusad C e&longs;tvtrique communis; & ABC ip&longs;i kFC, <lb/>
<arrow.to.target n="marg116"></arrow.to.target> & BAC ip&longs;i FKC æqualis, cum &longs;it Fk ip&longs;i AB æquidi&longs;tans; <lb/>erit triangulum ABC ip&longs;i KFC &longs;imile. & quonian NK FC, <lb/>& HN KF &longs;unt &ecedil;quidi&longs;tantes, erunt anguli KCFCkF angu <lb/>lis HkN KHN &ecedil;quales; ac propterea reliquus CFK reliquo <lb/>KNH &ecedil;qualis: latus verò CK lateri KH e&longs;t &ecedil;quale; erit igi­<lb/>
<arrow.to.target n="marg117"></arrow.to.target> tur triangulum KFC triangulo HNK &longs;imile, & &ecedil;quale. &longs;imi <lb/>literquè <expan abbr="o&longs;t&etilde;detur">o&longs;tendetur</expan> omnia triangula ALG GMH HNK KFC <lb/>in ter&longs;e&longs;e &longs;imilia, & æqualia e&longs;&longs;e. & obid ip&longs;i ABC &longs;imilia e&longs;&longs;e. <lb/>Fiat igit vt AC ad AG, ita AG ad alia O. &longs;imiliterv AC ad GH, <lb/>ita GH ad P. rur&longs;usvt AC ad Hk, ita HK ad <expan abbr="q.">que</expan> deniquè <lb/>vt AC ad Ck, ita CK ad R. & quoniam AG GH HK KC <lb/>
<arrow.to.target n="marg118"></arrow.to.target> &longs;unt æquales, eadem AC ad vnamquamque ip&longs;arum ean­<lb/>dem habebit proportionem, ergo eandem quoque habebit <lb/>propo&longs;itionem AG ad O, vt GH ad P, & HK ad Q, &
<pb pagenum="95"/>KC ad R. ac propterea lineæ OPQR inter&longs;e&longs;unt æquales. <lb/>Atverò quoniam ita e&longs;t AC ad AG, vt AG ad O, & vt <lb/>AC ad GH, ita GH, hoc e&longs;t AG ip&longs;i &ecedil;qualis, ad P. rur&longs;us <lb/>vt AC ad HK, ita HK, hoc e&longs;t AG ad <expan abbr="q.">que</expan> ac tandem vt <lb/>AC ad KC, ita KC, hoc e&longs;t AG ip&longs;i &ecedil;qualis, ad R. erit AC
<arrow.to.target n="marg119"></arrow.to.target><lb/>ad omnes con&longs;equentes &longs;imul &longs;umptas AG GH HK KC, <lb/>hoc e&longs;t erit AC ad eandem AC, vt AG ad omnes &longs;imul <lb/>OPQR. vnde &longs;equitur omnes &longs;imul OPQR ip&longs;i AG &ecedil;qua <lb/>les e&longs;&longs;e. Itaque quoniam &longs;imilia triangula in dupla &longs;unt
<arrow.to.target n="marg120"></arrow.to.target> pro­<lb/>portione laterum homologorum, erit triangulum ABC ad <lb/>ALG, vt AC ad O. eodemquè modo erit triangulum ABC <lb/>ad GMH, vt AC ad P. rur&longs;us ABC ad HNK, vt AC ad <lb/>Q, & vt idem ABC ad KFC, ita AC ad R. triangulum <lb/>igitur ABC ad omnes con&longs;equentes, videlicet ad omnia <expan abbr="triã">triam</expan>
<arrow.to.target n="marg121"></arrow.to.target><lb/>gula &longs;imul &longs;umpta ALG GMH HNK KFC, eritvt AC ad <lb/>omnes &longs;imul OPQR. hoc e&longs;t ad AG. o&longs;ten&longs;um e&longs;t igitur, <lb/>quod propo&longs;itum fuit. </s></p>
<p type="margin">
<s><margin.target id="marg114"></margin.target>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg115"></margin.target>1. <emph type="italics"/>lemma.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg116"></margin.target>29. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg117"></margin.target>76. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg118"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 17 <emph type="italics"/><expan abbr="quĩi">quini</expan>.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg119"></margin.target><emph type="italics"/>ex <expan abbr="præced&etilde;">præcedem</expan> <lb/>ti lemmate<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg120"></margin.target>19.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg121"></margin.target><emph type="italics"/>ex <expan abbr="præced&etilde;">præcedem</expan> <lb/>ti lemmate<emph.end type="italics"/></s></p>
<figure id="fig42"></figure>
<p type="head">
<s>PROPOSITIO. XIII.</s></p>
<p type="main">
<s>Omnis trianguli centrum grauitatis e&longs;t in recta <lb/>linea ab angulo ad dimidiam ba&longs;im ducta. </s></p>
<p type="main">
<s><emph type="italics"/>Sit triangulum ABC. & in ip&longs;o &longs;it AD<emph.end type="italics"/> ab angulo A <emph type="italics"/>ad dimi­<lb/>diamba&longs;im BC ducta. o&longs;tendendum est, centrum grauitatis trianguli <lb/>ABC e&longs;&longs;e in linea AD. Non &longs;it quidem, &longs;ed &longs;i fieri potest &longs;it punctum <lb/>H. & ab ip&longs;o ducatur HI æquidi&longs;tansip&longs;i BC,<emph.end type="italics"/> quæ ip&longs;am AD &longs;ecet
<arrow.to.target n="marg122"></arrow.to.target><lb/>in I. <emph type="italics"/>Deinde diui&longs;a DC bifariam, idquè &longs;emper fiat, dones relinqua­<lb/>tur linea<emph.end type="italics"/> D<foreign lang="greek">w</foreign> <emph type="italics"/>minor ip&longs;a HI. Diuidaturquè ip&longs;arum vtraque BD DC <lb/>in partes æquales<emph.end type="italics"/> D<foreign lang="greek">w</foreign>; parte&longs;què in DC exr&longs;tentes &longs;int D<foreign lang="greek">w wb <lb/>b</foreign>Z ZC; quibus re&longs;pondeant æquales partes D<foreign lang="greek">aazz</foreign>O OB. <emph type="italics"/>& <lb/>a &longs;ectionum punctis ducantur<emph.end type="italics"/> OE <foreign lang="greek">z</foreign>G <foreign lang="greek">a</foreign>L <foreign lang="greek">w</foreign>M <foreign lang="greek">b</foreign>K ZF <emph type="italics"/>æquidictan <lb/>tes ip&longs;i AD. & connectantur EF G<emph.end type="italics"/>k <emph type="italics"/>LM quæ nimirum ip&longs;i BC <lb/>æquidistantes erunt.<emph.end type="italics"/> cùm enim &longs;int BD DC inter&longs;e equales, iti­<lb/>dem OB ZC æquales; erit DO ip&longs;i DZ &ecedil;qualis. quare DO <lb/>ad OB e&longs;t, vt DZ ad ZC. Quoniam autem EO FZ &longs;unt
<pb pagenum="96"/>1pd AD æquidi&longs;tantes, erit AE ad EB, vt DO ad OB; & vt <lb/>
<arrow.to.target n="marg123"></arrow.to.target> DZ ad ZC, &longs;ic AF ad FC. atque DO ad OB e&longs;t, vt DZ ad <lb/>ZC. erit igitur AE ad EB, vt AF ad FC. quare EF ip&longs;i BC <lb/>
<arrow.to.target n="marg124"></arrow.to.target> e&longs;t æquidi&longs;tans, eodemquè modo o&longs;tendetur, ita e&longs;&longs;e AG ad <lb/>
<arrow.to.target n="fig43"></arrow.to.target><lb/>GB, vt AK ad KC, & AL ad LB, vt AM ad MC. ex quib^{9} <lb/>&longs;equitur LM GK EF non &longs;olùm ip&longs;i BC, verùm etiam inter­<lb/>&longs;e&longs;e parallelas e&longs;&longs;e. &longs;ecct EF lineas G<foreign lang="greek">z</foreign> K<foreign lang="greek">b</foreign> in X<foreign lang="greek">e</foreign>. ip&longs;am verò <lb/>AD in T. lineaquè GK &longs;ecet L<foreign lang="greek">a</foreign> M<foreign lang="greek">w</foreign> in N<foreign lang="greek">d</foreign>, & AD in Y. <lb/>linea deniquè LM ip&longs;am AD in S di&longs;pe&longs;cat. Quoniam au <lb/>tem D<foreign lang="greek">w</foreign> e&longs;t ip&longs;i HI æquidi&longs;tans, e&longs;tquè D<foreign lang="greek">w</foreign> minor <expan abbr="quã">quam</expan> HI, li <lb/>nea <foreign lang="greek">w</foreign>M ip&longs;i AL &ecedil;quidi&longs;tans ip&longs;am HI &longs;ecabir. ac propterea <lb/>punctum H centrum grauitatis trianguli ABC extra paral­<lb/>
<arrow.to.target n="marg125"></arrow.to.target> lelogrammum DM reperitur. At verò quoniam LD DM <lb/>&longs;unt para lelogramma, erunt LS <foreign lang="greek">a</foreign>D inter&longs;e æquales, &longs;imili­<lb/>ter SM D<foreign lang="greek">w</foreign> &ecedil;quales. &longs;untverò <foreign lang="greek">a</foreign>D D<foreign lang="greek">w</foreign> &ecedil;quales: ergo & LS <lb/>SM inter&longs;e &longs;unt &ecedil;quales. eademquè rarione NY Y<foreign lang="greek">d</foreign> inter&longs;e­<lb/>&longs;e, & ip&longs;is LS SM &ecedil;quales exi&longs;tent. quarelinea SY bifariam <lb/>diuiditlatera oppo&longs;ita parallelogrammi MN. pariquè ratio­<lb/>ne o&longs;tendetur lineam YT bifariam diuidere oppo&longs;ita latera <lb/>parallelogrammi KX; lineamquè TD latera oppo&longs;ita paral-
<pb pagenum="97"/>lelogrammi FO bifariam quoque diuidere. <emph type="italics"/>Itaque parallelogrà <lb/>mi MN centrum grauitatis est in linea <foreign lang="greek">*u</foreign>S. parallilogrammi ver<gap/><lb/>KX grouitatis centrum est in linea T<foreign lang="greek">*u</foreign>. parallelogrammi autem FO in <lb/>linea TD; magnitu linis igitur ex<emph.end type="italics"/> his <emph type="italics"/>omnibus<emph.end type="italics"/> parallelogrammi <lb/>MN KX FO <emph type="italics"/>compo&longs;itæ centrum grauitatis e&longs;t in recta linea S D. &longs;iv <lb/>itaque punctum R.<emph.end type="italics"/> quod quidem erit centrum grauitatis figura <lb/>LNGXEOZF <foreign lang="greek">e</foreign>K<foreign lang="greek">d</foreign>M. <emph type="italics"/><expan abbr="lũgatur&qacute;">lungaturque</expan>; RH, & producatur,<emph.end type="italics"/> quæ ipsa <foreign lang="greek">w</foreign>M <lb/>&longs;ecet in P. <emph type="italics"/>ip&longs;iquè AD<emph.end type="italics"/> a puncto C <emph type="italics"/>æqui di&longs;tans ducatur CV,<emph.end type="italics"/> qu<gap/><lb/>ip&longs;i RH occurrat in V. <emph type="italics"/><expan abbr="triangulũ">triangulum</expan> itaque ADC ad omnia triangu <lb/>la ex AM MK<emph.end type="italics"/> k<emph type="italics"/>F FC de&longs;cripta &longs;imiliaip&longs;i ADC,<emph.end type="italics"/> hoc e&longs;t ad tria <lb/>gula ASM M <foreign lang="greek">d</foreign>K K<foreign lang="greek">e</foreign>F FZC &longs;imul &longs;umpta <emph type="italics"/>eandem habet propor <lb/>tionem, quam habet CA ad AM. &longs;iquidem &longs;unt AM MK<emph.end type="italics"/> k<emph type="italics"/>F FC<emph.end type="italics"/>
<arrow.to.target n="marg126"></arrow.to.target><lb/><emph type="italics"/>æquales quia verò & triangulum ADB ad omnia ex AL LG GE <lb/>EB de&longs;cripta triangula &longs;imilia<emph.end type="italics"/> ALS LGN GEX EFO <emph type="italics"/>eandem ha <lb/>bet proportionem, quam ‘BA ad AL<emph.end type="italics"/>: & antecedentes &longs;imul ad
<arrow.to.target n="marg127"></arrow.to.target><lb/>omnes con&longs;equentes, hoc e&longs;t totum triangulum ABC ad on <lb/>nia triangula &longs;imul &longs;umpta, quæ &longs;unt in AB, & in AC con&longs;ti­<lb/>tuta, eandem habebit proportionem, quam habet AC AB &longs;i <lb/>mul ad AM AL &longs;imul, quia verò ob <expan abbr="&longs;imilitudin&etilde;">&longs;imilitudinem</expan> <expan abbr="triangulorũ">triangulorum</expan> <lb/>ABC ALM CA ad AM e&longs;t, vt BA ad AL; erit CA ad AM, vt <lb/>CA BA &longs;imul ad AM AL &longs;imul. <emph type="italics"/>triangulum igitur ABC ad omnia<emph.end type="italics"/>
<arrow.to.target n="marg128"></arrow.to.target><lb/><emph type="italics"/>prædicta triangula eandem habet proportionem quam habet CA ad AM. <lb/>Atque CA ad AM maiorem habet proportionem quàm VR ad RH; e­<lb/>tenim proportio ip&longs;ius CA ad AM e&longs;t eadem, quæ est totius VR <expan abbr="adipsã">adipsam</expan> <lb/>R. p. <expan abbr="quãdoquid&etilde;">quandoquidem</expan> triangula<emph.end type="italics"/> ACD MC<foreign lang="greek">w</foreign> <emph type="italics"/>&longs;unt &longs;imilia.<emph.end type="italics"/> <expan abbr="&longs;int&qacute;">&longs;intque</expan>; AD &
<arrow.to.target n="marg129"></arrow.to.target><lb/>M<foreign lang="greek">w</foreign> &ecedil;quidi&longs;tantes, &longs;itquè propterea CA ad AM, vt CD ad <lb/>D<foreign lang="greek">w</foreign>. & quoniam VR DC àlineis DR <foreign lang="greek">w</foreign>p CV æquidi&longs;tantib^{9}
<arrow.to.target n="marg130"></arrow.to.target><lb/>diuiduntur; erit C<foreign lang="greek">w</foreign> ad <foreign lang="greek">w</foreign>D, vt VP ad PR. & <expan abbr="cõponendo">componendo</expan> CD
<arrow.to.target n="marg131"></arrow.to.target><lb/>ad D<foreign lang="greek">w</foreign>, vt VR ad RP. quare vt CA ad AM, ita VR ad RP.
<arrow.to.target n="marg132"></arrow.to.target><lb/>quia verò VR ad RP maiorem habet proportionem, quàm
<arrow.to.target n="marg133"></arrow.to.target><lb/>ad RH. maiorem quoque habebit proportionem CA ad <lb/>AM, quàm VR ad RH. e&longs;t autem CA ad AM, vt <expan abbr="triangulũ">triangulum</expan> <lb/>ABC ad omnia triangula in lineis AC AB. (vt dictum e&longs;t) <lb/>con&longs;tituta; ergo <emph type="italics"/>& triangulum ABC adprædicta<emph.end type="italics"/> triangula <emph type="italics"/>maio <lb/>rem habet proportionem, quàm VR ad RH. Quare & diuidendo pa-<emph.end type="italics"/>
<arrow.to.target n="marg134"></arrow.to.target><lb/><emph type="italics"/><expan abbr="rallelogrāma">rallelogramma</expan> MN<emph.end type="italics"/> k<emph type="italics"/>X FO<emph.end type="italics"/> hoc e&longs;t figura LNGXEOZF <foreign lang="greek">e</foreign>K <foreign lang="greek">d</foreign>M) <emph type="italics"/>ad <lb/>circumrelicta triangula<emph.end type="italics"/> in lineis AC AB con&longs;tituta <emph type="italics"/>maiorem ha-<emph.end type="italics"/>
<pb pagenum="98"/><emph type="italics"/>bent proportionem, quam NH ad HR.<emph.end type="italics"/> linea igitur, quæ eandem <lb/>habeat proportionem ad HR, quam parallelogramma MN <lb/>kX FO ad circumrelicta triangula, maior erit, quàm VH <lb/><emph type="italics"/>Fiat itaquè in eademproportione QH ad HR, ut parallelogramma ad <lb/>triangula;<emph.end type="italics"/> erit vtique QH maior, quam VH. <emph type="italics"/>Quoniam igitur e&longs;t <lb/>magnitudo ABC, cuius centrum grauitatis est H, & ab ea magnitudo<emph.end type="italics"/><lb/>
<arrow.to.target n="fig44"></arrow.to.target><lb/><emph type="italics"/>auferatur compo&longs;ita ex MN<emph.end type="italics"/> k<emph type="italics"/>X FO parallelogrammis; & magnitudi <lb/>nis ablatæ centrum grauitatis e&longs;t punctum R; magnitudinis reliquæ ex <lb/>circumrelictis triangulis compo&longs;itæ centrum grauitatis erit in recta li-<emph.end type="italics"/><lb/>
<arrow.to.target n="marg135"></arrow.to.target> <emph type="italics"/>nea RH<emph.end type="italics"/> ex parte H <emph type="italics"/>producta, a&longs;&longs;umptaquè aliqua<emph.end type="italics"/> vt, QH, <emph type="italics"/>quæ ad <lb/>HR eam habeat proportionem, quam habet magnnudo<emph.end type="italics"/> ex parallelo­<lb/>grammis MN KX FO con&longs;tans <emph type="italics"/>ad reliquum,<emph.end type="italics"/> hoc e&longs;t ad reli­<lb/>qua triangula, <emph type="italics"/>ergo punctum Q centrum est grauitatis magnitudinis <lb/>ex ip&longs;is circumrelictis<emph.end type="italics"/> triangulis <emph type="italics"/>compo&longs;itæ. quoa fieri non pote&longs;i aucta <lb/>enim recta linea <foreign lang="greek">qk</foreign> per Q ip&longs;i AD æquidistante in<emph.end type="italics"/> ed dem <emph type="italics"/>plano<emph.end type="italics"/> <expan abbr="triã">triam</expan> <lb/>guli ABC, <emph type="italics"/>in ip&longs;a e&longs;&longs;ent omnia centra<emph.end type="italics"/> grauitatis trian­<lb/>gulorum, <emph type="italics"/>hoc est in vtramque partem<emph.end type="italics"/> Q<foreign lang="greek">q</foreign> Q<foreign lang="greek">k</foreign>, centraquè <lb/>grauitatis trianguli ALM, ac centrum magnitudinis ex vtri&longs;­<lb/>què triangulis LGN MK <foreign lang="greek">d</foreign> <expan abbr="cōpo&longs;it&ecedil;">compo&longs;it&ecedil;</expan> in parte Q<foreign lang="greek">q</foreign> e&longs;&longs;e <expan abbr="deber&etilde;t">deberent</expan>.
<pb pagenum="99"/>centra verò grauitatis magnitudinis ex GEX K<foreign lang="greek">e</foreign>F compo­<lb/>&longs;it&ecedil;, ac magnitudinis ex. EBO FZC compo&longs;&longs;tæ, e&longs;&longs;ent in par <lb/>te Q<foreign lang="greek">k</foreign>, ita vt punctum Q magnitudinis ex omnibus trian­<lb/>gulis compo&longs;itæ centrum e&longs;&longs;et grauitatis. quæ <expan abbr="quid&etilde;&longs;unt">quiden&longs;unt</expan> om­<lb/>nino ab&longs;urda. Quòd &longs;i ducta linea per Q, non fuerit etiam <lb/>ip&longs;i AD &ecedil;quidi&longs;tans, eadem &longs;equentur in conuenien tia. <emph type="italics"/>Ma <lb/>ni&longs;estum e&longs;t igitur; quod propo&longs;itum fuerat.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg122"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> t. <emph type="italics"/>deci­<lb/>mi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg123"></margin.target>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg124"></margin.target>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg125"></margin.target>34. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg126"></margin.target>3. <emph type="italics"/>lemma.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg127"></margin.target><emph type="italics"/>ex<emph.end type="italics"/>12.<emph type="italics"/><expan abbr="quĩti">quinti</expan><emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg128"></margin.target><emph type="italics"/>ex<emph.end type="italics"/>12.<emph type="italics"/><expan abbr="quĩti">quinti</expan><emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg129"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 4.<emph type="italics"/>&longs;exti<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg130"></margin.target>1. <emph type="italics"/>lemma.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg131"></margin.target>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg132"></margin.target>11. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg133"></margin.target>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg134"></margin.target>20. <emph type="italics"/>quinti <lb/>add.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg135"></margin.target>8.<emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<figure id="fig43"></figure>
<figure id="fig44"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Id ip&longs;um vult ad huc Archimedes aliter o&longs;tendere. ob <expan abbr="&longs;equ&etilde;">&longs;equem</expan> <lb/>tem verò demon&longs;trationem hoc priùs cogno&longs;cere oportet. </s></p>
<p type="head">
<s>LEMMA.</s></p>
<p type="main">
<s>Si intra triangulum vni lateri &ecedil;quidi&longs;tans ducatur, ab op­<lb/>po&longs;ito autem angulo intra triangulum quoquè recta ducatur <lb/>linea, æquidi&longs;tantes lineas in eadem proportione di&longs;pe&longs;cet. </s></p>
<p type="main">
<s>Hoc in &longs;ecundo no&longs;trorum plani&longs;ph&ecedil;riorum libro in ea <lb/>parte o&longs;tendimus, vbi quomodo conficienda &longs;it ellip&longs;is, in&longs;tru <lb/>mento à nobis inuento demon&longs;trauimus. hoc nempè modo, <lb/>
<arrow.to.target n="fig45"></arrow.to.target><lb/>Sit triangulum ABC, ip&longs;iquè BC in­<lb/>tra triangulum ducatur vtcumquè æ­<lb/>quidi&longs;tans DE. à punctoquè A intra <lb/>triangulum &longs;imiliter quocumque du­<lb/>catur AF; quæ lineam BC &longs;ecet in F; <lb/>lineam verò DE in G. Dico ita o&longs;&longs;e <lb/>CF ad FB, vt EG ad GD. <expan abbr="Quoniã">Quoniam</expan> <lb/>enim GE FC &longs;unt æquidi&longs;tantes, erit <lb/>triangulum AFC triangulo AGE æquiangulum, vt igitur
<arrow.to.target n="marg136"></arrow.to.target><lb/>AF ad AG, ita CF ad EG. ob ean demquè cauíam ita e&longs;t FA <lb/>ad AG, vt FB ad GD. quare vt CF ad EG, ita e&longs;t FB ad GD.
<arrow.to.target n="marg137"></arrow.to.target><lb/>ac permutando, vt CF ad FB, ita EG ad GD. quod demon
<arrow.to.target n="marg138"></arrow.to.target><lb/>&longs;trare oportebat. </s></p>
<pb pagenum="100"/>
<p type="margin">
<s><margin.target id="marg136"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 4.<emph type="italics"/>&longs;exti<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg137"></margin.target>11.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg138"></margin.target>16.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<figure id="fig45"></figure>
<p type="head">
<s><emph type="italics"/>IDEM ALITER.<emph.end type="italics"/></s></p>
<p type="main">
<s><emph type="italics"/>Sit triangulum ABC, ducaturquè AD<emph.end type="italics"/> ab angulo A <emph type="italics"/>ad <expan abbr="dimidiã">dimidiam</expan><emph.end type="italics"/><lb/>ba&longs;im <emph type="italics"/>BC. Dico in linea AD centrum e&longs;&longs;e grauitatis trianguli ABC. <lb/>N on &longs;it autem, &longs;ed &longs;i fieri pote&longs;t; &longs;it H. iunganturquè AH HB HC, & <lb/>ED<emph.end type="italics"/> DF <emph type="italics"/>FE ad dimidias BA<emph.end type="italics"/> BC <emph type="italics"/>AC<emph.end type="italics"/> ducantur, &longs;ecetquè EF ip­<lb/>&longs;am AD in M. & <emph type="italics"/>ip&longs;i AH æquidistantes ducantur EK FL. &<emph.end type="italics"/><lb/>
<arrow.to.target n="fig46"></arrow.to.target><lb/><emph type="italics"/>iungantur KL LD Dk DH<emph.end type="italics"/>; &longs;ecetquè DH ip&longs;am KL in N. <lb/>iungaturquè <emph type="italics"/>MN. Quoniam igitur triangulum ABC &longs;imile est <expan abbr="triã">triam</expan> <lb/>gulo DFC, cùm &longs;it BA ip&longs;i FD æquidistans<emph.end type="italics"/>; &longs;iquidem &longs;unt late­<lb/>
<arrow.to.target n="marg139"></arrow.to.target> ra CA CB bifariam diui&longs;a, ideoquè &longs;it CF ad FA, vt CD <lb/>ad DB. <emph type="italics"/>trianguliquè ABC centrum grauitatis est punctum H; &<emph.end type="italics"/><lb/>
<arrow.to.target n="marg140"></arrow.to.target> <emph type="italics"/>trianguli FDC centrum grauitatis erit punctum L. puncta enim HB <lb/>intra vtrumquè triangulum &longs;unt &longs;imiliter po&longs;ita. etenim ad homologa <lb/>latera angulos efficiunt æquales. hoc enim per&longs;picuum. est<emph.end type="italics"/> cùm enim <lb/>&longs;int triangulorum ABC DFC homologa latera AC FC, <lb/>
<arrow.to.target n="marg141"></arrow.to.target> AB FD, BC DC, &longs;intquè AH FL æquidi&longs;tantes; erit an­<lb/>gulus LFC angulo HAC &ecedil;qualis. &longs;ed angulus CFD e&longs;t ip&longs;i
<pb pagenum="101"/>CAB æqualis; reliquus igitur angulus LFD reliquo HAB <lb/>æqualis exi&longs;tit. & quoniam ita e&longs;t CF ad FA, vt CL ad LH,
<arrow.to.target n="marg142"></arrow.to.target><lb/>cùm &longs;int FL AH &ecedil;quidi&longs;tantes. CF verò dimidia e&longs;t ip&longs;ius <lb/>CA, erit & CL ip&longs;ius quoque CH dimidia. at CD ip&longs;ius <lb/>CB dimidia exi&longs;tit; erit igitur DL ip&longs;i BH &ecedil;quidi&longs;tans. ac
<arrow.to.target n="marg143"></arrow.to.target><lb/>propterea angulus LDC e&longs;t ip&longs;i HBC &ecedil;qualis, & LDF ip&longs;i
<arrow.to.target n="marg144"></arrow.to.target><lb/>HBA &ecedil;qualis. cùm &longs;ittotus CDF toti CBA &ecedil;qualis; anguli <lb/>verò ACH & HCB tam &longs;unt trianguli ABC, quàm FDC. <lb/><emph type="italics"/>Obeandem autem rationem trianguli EBD centrum grauitatis est <expan abbr="pũ-">pun-</expan><emph.end type="italics"/>
<arrow.to.target n="marg145"></arrow.to.target><lb/><emph type="italics"/>ctum K.<emph.end type="italics"/> &longs;imiliter enim o&longs;tendetur punctum K in triangu­<lb/>lo EBD e&longs;&longs;e &longs;imiliter po&longs;itum, vt H in triangulo ABC. <lb/><emph type="italics"/>Quare magnitudinis ex vtri&longs;què triangulis EBD FDC compo&longs;itæ <lb/>centrum grauitatis e&longs;t in medietate lineæ<emph.end type="italics"/> k<emph type="italics"/>L. cum triangula EBD<emph.end type="italics"/>
<arrow.to.target n="marg146"></arrow.to.target><lb/><emph type="italics"/>FDC &longs;int æqualia.<emph.end type="italics"/> &longs;unt enim in &ecedil;qualibus ba&longs;ibus BD DC,
<arrow.to.target n="marg147"></arrow.to.target><lb/>& in ij&longs;dem parallelis EF BC, &longs;iquidem e&longs;t AE ad EB, vt
<arrow.to.target n="marg148"></arrow.to.target><lb/>AF ad FC. quippè cùm latera AB AC &longs;int bifariam diui­<lb/>&longs;a. <emph type="italics"/>medium veròip&longs;ius<emph.end type="italics"/> k<emph type="italics"/>L e&longs;t punctum N; cùm &longs;it<emph.end type="italics"/> KE ip&longs;i AH <lb/>&ecedil;quidi&longs;tans, & ob id &longs;it <emph type="italics"/>BE ad EA, vt B<emph.end type="italics"/>k <emph type="italics"/>ad<emph.end type="italics"/> k<emph type="italics"/>H.<emph.end type="italics"/> & vt BE
<arrow.to.target n="marg149"></arrow.to.target><lb/>ad EA, ita CF ad FA; <emph type="italics"/>vt autem CF ad FA, &longs;ic CL ad LH.<emph.end type="italics"/><lb/>quare vt BK ad KH, ita CL ad LH. <emph type="italics"/>Si autem hoc. æquidi-<emph.end type="italics"/>
<arrow.to.target n="marg150"></arrow.to.target><lb/><emph type="italics"/>&longs;tans est BC ip&longs;i<emph.end type="italics"/> k<emph type="italics"/>L, & iuncta est DH, erit igitur BD ad DC, vt<emph.end type="italics"/>
<arrow.to.target n="marg151"></arrow.to.target><lb/><emph type="italics"/>KN ad NL.<emph.end type="italics"/> D verò medium e&longs;t ip&longs;ius BC. ergo & N
<arrow.to.target n="marg152"></arrow.to.target> me­<lb/>dium e&longs;t ip&longs;ius KL. <emph type="italics"/>Quare magnitudinis ex vtri&longs;què <expan abbr="dictorũ">dictorum</expan> trian <lb/>gulorum<emph.end type="italics"/> EBD & FDC <emph type="italics"/>compo&longs;itæ centrum<emph.end type="italics"/> grauitatis <emph type="italics"/>est punctum<emph.end type="italics"/>
<arrow.to.target n="marg153"></arrow.to.target><lb/><emph type="italics"/>N. parallelogrammi verò AEDF centrum grauitatis e&longs;t punctum M,<emph.end type="italics"/><lb/>vbi &longs;imiliter diametri concurrunt, <emph type="italics"/>ac propterea magnitudinis ex<emph.end type="italics"/>
<arrow.to.target n="marg154"></arrow.to.target><lb/><emph type="italics"/>omnibus<emph.end type="italics"/> triangulis EBD FDC vna <expan abbr="cũ">cum</expan> parallelogramo AEDF <lb/><emph type="italics"/>compo&longs;itæ centrum grauitatis e&longs;t in linea MN. Verùm<emph.end type="italics"/> <expan abbr="triangulorũ">triangulorum</expan> <lb/>EBD FDC, &longs;imulquè parallelogrammi AEDF, hoc e&longs;t totius <lb/><emph type="italics"/>trianguli ABC grauitatis centrum est punctum H; linea igitur MN pro<emph.end type="italics"/>
<arrow.to.target n="marg155"></arrow.to.target><lb/><emph type="italics"/>ducta tran&longs;ibit per punctum H. quod e&longs;&longs;e non pote&longs;t.<emph.end type="italics"/> etenim cùm &longs;it <lb/>KN ip&longs;i BD æquidi&longs;tans; erit BK ad KH, vt DN ad <lb/>NH: vtautem BK ad KH, ita e&longs;t BE ad EA, & vt BE ad <lb/>EA, ita e&longs;t DM ad MA, cùm &longs;it EM ip&longs;i BD æquidi&longs;tans. <lb/>erit igitur DM ad MA, vt DN ad NH. quare MN ip&longs;i AH <lb/>e&longs;t &ecedil;quidi&longs;tans; ideoquè MN numquam cùm AH conueni­<lb/>re pote&longs;t. <emph type="italics"/>Non est igitur<emph.end type="italics"/> punctum <emph type="italics"/>H centrum grauitatis trianguli<emph.end type="italics"/>
<pb pagenum="102"/><emph type="italics"/>ABC. quare non e&longs;t extra lineam AD. in ip&longs;i igitur exi&longs;tit.<emph.end type="italics"/> Quod <lb/>demonitrare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg139"></margin.target>2.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg140"></margin.target>11.<emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg141"></margin.target>29. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg142"></margin.target>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg143"></margin.target>2.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg144"></margin.target>29. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg145"></margin.target>11. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg146"></margin.target>4.<emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg147"></margin.target>38. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg148"></margin.target>2.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg149"></margin.target>2.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg150"></margin.target>11.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg151"></margin.target>2.<emph type="italics"/>&longs;exti. <lb/>lemma.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg152"></margin.target>*</s></p>
<p type="margin">
<s><margin.target id="marg153"></margin.target>11.<emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg154"></margin.target>*</s></p>
<p type="margin">
<s><margin.target id="marg155"></margin.target>A</s></p>
<figure id="fig46"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>
<arrow.to.target n="marg156"></arrow.to.target> Inquit Archimedes <emph type="italics"/>linea igitur MN producta tran&longs;ibit per pun­<lb/>ctum H. quod e&longs;&longs;e non pote&longs;t,<emph.end type="italics"/> nempè, vt non ip&longs;amet linea MN, <lb/>&longs;ed eius pars, &longs;iuead M, &longs;iue ad N producta cum H conue <lb/>nireoporteat. cùm tamen ip&longs;amet linea MN per punctum <lb/>H tran&longs;ire debeat. ita vt punctum H &longs;it inter puncta MN; <lb/>hoc e&longs;t in linea MN, & non in eius parte producta. Nam &longs;i <lb/>punctum H centrum e&longs;t grauitatis totius trianguli ABC. <lb/>punctum verò N centrum grauitatis magnitudinis ex <expan abbr="triãgu">triangu</expan> <lb/>lis EBD FDC compo&longs;it&ecedil;; atque punctum M centrum gra­<lb/>uitatis parallelogrammi AEDF; oportet vt punctum H ita li­<lb/>neam diuidat MN; vt eius partes magnitudinibus permuta­<lb/>tim re&longs;pondeant. vt nimirum pars ad M ad partem ad N &longs;it, <lb/>vt magnitudo ex triangulis EBD FDC con&longs;tans ad parallelo <lb/>grammum AEDF. vt ex &longs;exta, & octaua huius propo&longs;itione <lb/>per&longs;picuum e&longs;t. Quare punctum H in linea MN e&longs;&longs;e debe­<lb/>ret; vt ip&longs;emet Atchimedes paulò &longs;uperiùs affirmauit; cùm in­<lb/>
<arrow.to.target n="marg157"></arrow.to.target> quit. <emph type="italics"/>ac propterea magnitudinis ex omnibus compo&longs;itæ contrum grauita­<lb/>tis e&longs;t in linea MN.<emph.end type="italics"/> & non dixit in eius parte producta. Quodiv <lb/>ca vel del<gap/>dum e&longs;t verbum illud <emph type="italics"/>producta,<emph.end type="italics"/> tanquam ab aliquo <lb/>additum, vel ideo tamen hoc dixi&longs;&longs;e voluit Archimedes, vt o­<lb/>&longs;tenderet lineam MN nullo modo (etiam &longs;i produceretur) <expan abbr="cũ">cum</expan> <lb/>H conuenire po&longs;&longs;e. </s></p>
<p type="margin">
<s><margin.target id="marg156"></margin.target>A</s></p>
<p type="margin">
<s><margin.target id="marg157"></margin.target>*</s></p>
<p type="head">
<s>PROPOSITIO. XIIII.</s></p>
<p type="main">
<s>Omnis trianguli centrum grauitatis e&longs;t <expan abbr="punctũ">punctum</expan> <lb/>in quo rectæ lineæ ab angulis trianguli ad dimidia <lb/>later a ductæ concurrunt. </s></p>
<pb pagenum="103"/>
<p type="main">
<s><emph type="italics"/>Sit triangulum ABC, &<emph.end type="italics"/> ab angulo A <emph type="italics"/>ducatur AD ad dimi­<lb/>diam BC. BE verò<emph.end type="italics"/> ab angulo B <emph type="italics"/>ad dimidiam AC.<emph.end type="italics"/> qu&ecedil; quidem <lb/>line&ecedil; AD BE &longs;einuicem &longs;ecent in <expan abbr="pū">pum</expan> <lb/>
<arrow.to.target n="fig47"></arrow.to.target><lb/>cto H. <emph type="italics"/>Quoniam igitur centrum grauita­<lb/>tis trianguli ABC est in vtraque linea <lb/>AD BE; hoc enim demonstratum e&longs;t<emph.end type="italics"/> in <lb/>pr&ecedil;cedenti. erit vtique centrum graui­<lb/>tatis, vbilineç AD BE &longs;e <expan abbr="inuic&etilde;">inuicem</expan> <expan abbr="&longs;ecãt">&longs;ecant</expan>. <lb/>&longs;ecant verò &longs;e&longs;e in H. <emph type="italics"/>ergo punctum <lb/>H centrum e&longs;t grauitatis<emph.end type="italics"/> trianguli ABC. <lb/>quod demon&longs;trare oportebat. </s></p>
<figure id="fig47"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Similiter &longs;i ducta fuerit CH, & producta, bifariam &longs;ecaret <lb/>AB. In hac enim linea e&longs;&longs;et centrum grauitatis trianguli; <expan abbr="c&etilde;">cem</expan> <lb/>trum verò e&longs;t in linea ab angulo ad dimidiam ba&longs;im ducta: <lb/>ergo hæc linea ab angulo C ad dimidiam AB ducta e&longs;&longs;et. <lb/>Præterea &longs;i linea à puncto C ad dimidiam AB ducta <expan abbr="nõ">non</expan> tran <lb/>&longs;iret per H; e&longs;&longs;et vtique in hac linea centrum grauitatis; &longs;ed
<arrow.to.target n="marg158"></arrow.to.target> <expan abbr="c&etilde;-trum">cen­<lb/>trum</expan> quoque grauitatis e&longs;t in linea AD, & in linea BE, ut in <lb/>H; vnius igitur figur&ecedil; plura darentur centra grauitatis. quod <lb/>fieri non pote&longs;t. quod quidem, cùm &longs;it in con ueniens, nos in <lb/>no&longs;tro Mechanicorum libro dari non po&longs;&longs;e &longs;uppo&longs;uimus. <lb/>Quare linea CH in directum ducta, bifariam &longs;ecaret AB. <lb/>quod quidem paulò infra aliter quoque o&longs;tendemus, <expan abbr="nõnul">nonnul</expan> <lb/>lis prius demon&longs;tratis; quæ Archimedes ob &longs;equentem <expan abbr="demõ-&longs;trationem">demon­<lb/>&longs;trationem</expan>, tanquam demon&longs;trata &longs;upponit. Vult enim Ar­<lb/>chimedes, po&longs;tquam inuenit centrum grauitatis cuiu&longs;libet <lb/>trianguli, centrum quoque grauitatis quærere trapetij duo la­<lb/>tera &ecedil;quidi&longs;tantia habentis. quod e&longs;t quidem pars trianguli, <lb/>& tanquam fru&longs;tum a triangulo ab&longs;ci&longs;&longs;um. &longs;upponitquè den <lb/>trum grauitatis cuiu&longs;libet trianguli e&longs;&longs;e in recta linea ba&longs;i du <lb/>cta &ecedil;quidi&longs;tante, quæ latera ita diuidat, vt partes ad uerticem <lb/>&longs;int reliquarum partium duplæ. quod quidem ortum ducit <lb/>ex cognitione alterius theorematis o&longs;tendentis centrum gra-
<pb pagenum="104"/>uitatis cuiu&longs;libet trianguli e&longs;&longs;e in recta linea ab angulo ad di­<lb/>midiam ba&longs;im ducta (vt Archimedes demon&longs;trauit) & in&longs;u­<lb/>per in eo puncto, quod dictam lineam diuidatita, vt pars ad <lb/>angulum reliqu&ecedil; ad ba&longs;im &longs;it dupla. Quare hoc prius ita <expan abbr="o&longs;t&etilde;">o&longs;tem</expan> <lb/>demus. </s></p>
<p type="margin">
<s><margin.target id="marg158"></margin.target>13.<emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="head">
<s>PROPOSITIO.</s></p>
<p type="main">
<s>Omnis trianguli centrum grauitatis e&longs;t punctum in recta <lb/>linea ab angulo ad dimidiam ba&longs;im ducta exi&longs;tens, quod li­<lb/>neam diuidat, ita vt poitio ad angulum reliquæ ad ba&longs;im, &longs;it <lb/>dupla. </s></p>
<p type="main">
<s>Sit triangulum ABC, in quo ab an <lb/>
<arrow.to.target n="fig48"></arrow.to.target><lb/>gulo A ad dimidiam ba&longs;im BC re­<lb/>cta ducatur linea AD. Ducaturquè <lb/>ab angulo B ad dimidiom ba&longs;im <lb/>AC linea BE, quæ&longs;ecet AD in F. Et <lb/>quoniam centrum grauitatis <expan abbr="triãgu-">triangu­<lb/></expan>
<arrow.to.target n="marg159"></arrow.to.target> li ABC e&longs;t punctum F; <expan abbr="o&longs;tendendũ">o&longs;tendendum</expan> <lb/>e&longs;t lineam FA ip&longs;ius FD duplam e&longs;­<lb/>&longs;e. iungatur FC. quoniam enim AE <lb/>e&longs;t equalis ip&longs;i EC, erit triangulum <lb/>
<arrow.to.target n="marg160"></arrow.to.target> ABE triangulo EBC æquale, cùm <lb/>&longs;int &longs;ub eadem altitudine. Ob eandemquè cau&longs;am <expan abbr="triangulũ">triangulum</expan> <lb/>AFE triangulo EFC exi&longs;tit æquale. &longs;i igitur à triangulo ABE <lb/>auferatur triangulum AFE, & à triangulo EBC triangulum <lb/>auferatur EFC; relinquetur triangulum ABF triangulo BFC <lb/>æquale. Rur&longs;us quoniam BD e&longs;t æqualis ip&longs;i DC; erit trian­<lb/>
<arrow.to.target n="marg161"></arrow.to.target> gulum BFD triangulo DFC æquale, &longs;iquidem candem ha­<lb/>bentaltitudinem. duplum igitur e&longs;t triangulum BFC <expan abbr="triãgu-li">triangu­<lb/>li</expan> BFD. Quare & triangulum ABF trianguli BFD duplum <lb/>
<arrow.to.target n="marg162"></arrow.to.target> exi&longs;tit. quia verò triangula ABF FBD in eadem &longs;unt altitudi <lb/>ne, idcirco &longs;e&longs;e habebunt, vt ba&longs;es AF FD. atque triangulum <lb/>ABF. duplum e&longs;t ip&longs;ius FBD; ergo portio AF ip&longs;ius FD dupla <lb/>exi&longs;tit. quod demon&longs;trare oportebat. </s></p>
<pb pagenum="105"/>
<p type="margin">
<s><margin.target id="marg159"></margin.target>14.<emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg160"></margin.target>1.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg161"></margin.target>1.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg162"></margin.target>1.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<figure id="fig48"></figure>
<p type="main">
<s>ALITER. </s></p>
<p type="main">
<s>Sit rur&longs;us triangulum ABC, & AD BE ab angulis ad di <lb/>midias ba&longs;es ductæ &longs;int erit vtique punctum, F (vbi &longs;e in ui
<arrow.to.target n="marg163"></arrow.to.target><lb/>cen fecant) centrum grauita tis triangulb ABC. Drco AF a­<lb/>p&longs;ius FD duplam e&longs;&longs;e. Iungatur DE. Quoniam enim BC <lb/>
<arrow.to.target n="fig49"></arrow.to.target><lb/>AC in punctis DE bifariam &longs;ecantur; erit <lb/>CD ad DB, vt CE ad EA. linea igitur <lb/>DE ip&longs;i AB e&longs;t æquidi&longs;tans. quare
<arrow.to.target n="marg164"></arrow.to.target> trian­<lb/>gulum ABC &longs;imile e&longs;t triangulo EDC.
<arrow.to.target n="marg165"></arrow.to.target><lb/>ac propterea ita e&longs;t BC ad CD, vt AB <lb/>ad DE. e&longs;t autem. BC dupla ip&longs;ius CD <lb/>(&longs;iquidem punctum D bifariam diuidit <lb/>BC) erit igitur AB dupla ip&longs;ius DE. At <lb/>vero quoniam AB DE &longs;unt parallelæ, erit triangulum AFB <lb/>triangulo EFD &longs;imile. & vt AB ad ED, ita AF ad FD, e&longs;t
<arrow.to.target n="marg166"></arrow.to.target><lb/>autem AB ip&longs;ius ED dupla, ergo AF ip&longs;ius FD dupla <lb/>exi&longs;tit. quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg163"></margin.target>14. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg164"></margin.target>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg165"></margin.target>4. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg166"></margin.target>4.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<figure id="fig49"></figure>
<p type="main">
<s>Exijs, quæ demon&longs;trata &longs;unt, o&longs;tendemus, quod paulò an <lb/>te propoiuimus, nempè cùm lineæ AD BE bifariam &longs;ecent <lb/>BC CA. Dico lineam CF productam bifariam quoque &longs;e­<lb/>care ip&longs;am AB. </s></p>
<p type="main">
<s>Producatur enim (ijsdem po&longs;itis) CFGH; quæ lineam <lb/>
<arrow.to.target n="fig50"></arrow.to.target><lb/>AB &longs;ecetin G. & à puncto B <lb/>ip&longs;i AD æquidi&longs;tans ducatur <lb/>BH. quæ ip&longs;i CG occuriat in <lb/>H. Quoniam igitur FD, e&longs;t i­<lb/>p&longs;i BH &ecedil;quidi&longs;tans, erit CD <lb/>ad DB, vt CF ad FH. CD
<arrow.to.target n="marg167"></arrow.to.target> ve­<lb/>rò e&longs;t æqualis BD; ergo CF ip&longs;i <lb/>FH æqualis exi&longs;tit. ac propterea <lb/>CH dupla e&longs;t ip&longs;ius (F. At ve­<lb/>rò quoniam ob &longs;imilitudinem <lb/><expan abbr="triangulorũ">triangulorum</expan> CBH CDF, ita e&longs;t <lb/>HC ad CF, vt BH ad DF; erit & BH ip&longs;ius FD duplex.
<pb pagenum="106"/>verùm & AF (ex proximè demon&longs;tratis) ip&longs;ius FD duplex <lb/>exi&longs;tit. erunt igitur BH FA inter&longs;e &ecedil;quales. Quoniam autem <lb/>BH e&longs;t &ecedil;quidi&longs;tans ip&longs;i AF, æquiangula erunt triagula GBH <lb/>
<arrow.to.target n="marg168"></arrow.to.target> GAF. quare vt BH ad AF, ita BG ad GA, quia verò BH e&longs;t <lb/>ip&longs;i AF æqualis; erit & BG ip&longs;i GA æqualis. ergo recta li­<lb/>nea EFG bifariam diuidit AB. quod demon&longs;trare oporte­<lb/>bat. </s></p>
<p type="margin">
<s><margin.target id="marg167"></margin.target>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg168"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 4.<emph type="italics"/>&longs;exti<emph.end type="italics"/></s></p>
<figure id="fig50"></figure>
<p type="main">
<s>Reliquum e&longs;t, vt ob &longs;equentem dem on&longs;trationem alteram <lb/>propo&longs;itionem o&longs;tendamus. </s></p>
<p type="head">
<s>PROPOSITIO.</s></p>
<p type="main">
<s>Centrum grauitatis cuiu&longs;libet trianguli e&longs;t in recta linea <lb/>ba&longs;i ducta æquidi&longs;tante, quæ latus ita diuidat, vt pars ad an­<lb/>gulum reliquæ ad ba&longs;im &longs;it dupla. </s></p>
<p type="main">
<s>In trianagulo enim ABC ducta <lb/>&longs;it DE ba&longs;i BC æquidi&longs;tans, quæ <lb/>
<arrow.to.target n="fig51"></arrow.to.target><lb/>latus AB diuidatin D, ita vt DA <lb/>ip&longs;ius DB &longs;it duplex. Dico in linea <lb/>DE centrum e&longs;&longs;e grauitatis triangu <lb/>li ABC. Ducatur ab angulo A ad <lb/>dimidiam BC linea AF, quæ di­<lb/>
<arrow.to.target n="marg169"></arrow.to.target> uidat DE in G. erit AD ad DB, <lb/>vt AG ad GF, ac propterea erit <lb/>AG ip&longs;ius GF dupla. punctum er <lb/>go G centrum e&longs;t grauitatis trian­<lb/>guli ABC. Quare con&longs;tat <expan abbr="centrũ">centrum</expan> <lb/>e&longs;&longs;e in linea DE. quod demon&longs;tra­<lb/>re oportebat </s></p>
<pb pagenum="107"/>
<p type="margin">
<s><margin.target id="marg169"></margin.target>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<figure id="fig51"></figure>
<p type="head">
<s>COROLLARIVM.</s></p>
<p type="main">
<s>Ex hoc elici pote&longs;t centrum grauita tis cuiu&longs;libet trianguli <lb/>e&longs;&longs;e in medio ductæ lineæ ba&longs;i æquidi&longs;tantis, qu&ecedil; latus diui­<lb/>datita, vt portio ad verticem &longs;it reliqu&ecedil; ad ba&longs;im dupla. </s></p>
<p type="main">
<s>E&longs;t enim DG ad GE, vt BF ad FC. &longs;unt verò BF FC
<arrow.to.target n="marg170"></arrow.to.target> æ­<lb/>quales; ergo & DG GE inter&longs;e &longs;unt æquales. quare grauita­<lb/>tis centrum G e&longs;t medium line&ecedil; DE. </s></p>
<p type="margin">
<s><margin.target id="marg170"></margin.target><emph type="italics"/>lemm.<emph.end type="italics"/><lb/>2. <emph type="italics"/>der <lb/>&longs;tratic<emph.end type="italics"/><lb/>13.<emph type="italics"/>hi<emph.end type="italics"/></s></p>
<p type="head">
<s>PROPOSITIO. XV.</s></p>
<p type="main">
<s>Omnis trapezij duo latera inuicem habentis æ­<lb/>quidi&longs;tantia centrum grauitatis e&longs;t in recta linea, <lb/>quæ latera æquidi&longs;tantia bifariam &longs;ecta <expan abbr="cõiungit">coniungit</expan>; <lb/>ita diui&longs;a, vt ip&longs;ius portio terminum habens mino <lb/>rem parallelam bifariam diui&longs;am ad <expan abbr="reliquã">reliquam</expan> por­<lb/>tionem eandem habeat proportionem, quam ha <lb/>bet vtraque &longs;imul, quæ &longs;it æqualis duplæ maioris <lb/>parallelarum cum minore ad <expan abbr="duplã">duplam</expan> minoris cum <lb/>maiore. </s></p>
<p type="main">
<s><emph type="italics"/>Sit trapezium ABCD habens latera AD BC parallela. linea <lb/>verò EF bifariam diuidat AD BC. Quòd igitur in linea EF &longs;it cen <lb/>trum grauitatis trapezii, per&longs;picuum est. productis enim CDG FEG <lb/>BAG, liquet in idem punctum,<emph.end type="italics"/> putà G <emph type="italics"/>concurrere.<emph.end type="italics"/> propterea quòd <lb/>cùm &longs;it AD æquidi&longs;tans ip&longs;i BC, nece&longs;&longs;e e&longs;t proportionem
<arrow.to.target n="marg171"></arrow.to.target><lb/>ip&longs;ius BA ad AG, ip&longs;iusquè FE ad EG, & CD ad DG, quæ <expan abbr="ni-mirũ">ni­<lb/>mirum</expan> in omnibus <expan abbr="ead&etilde;">eadem</expan> e&longs;t, in <expan abbr="vnũ">vnum</expan> & <expan abbr="id&etilde;">idem</expan> <expan abbr="pũctũ">punctum</expan> terminare. <emph type="italics"/><expan abbr="erit&qacute;">eritque</expan>; <lb/>trianguli GBC centrum grauitatis in linea GF. &longs;imiliterque trianguli<emph.end type="italics"/>
<arrow.to.target n="marg172"></arrow.to.target>
<pb pagenum="108"/>
<arrow.to.target n="marg173"></arrow.to.target> <emph type="italics"/>AG D centrum grauitatis in linea EG. ergo reliqui trapezii ABC <lb/>centrum grauitatis erit in linea EF. iungatur itaque BD, quæ int <lb/>æquain punctis<emph.end type="italics"/> K<emph type="italics"/>H diuidatur. ac per ea <expan abbr="ducãtur">ducantur</expan> LHM N<emph.end type="italics"/>k<emph type="italics"/>T<gap/><lb/>BC æquidi&longs;tantes<emph.end type="italics"/>; quæ lineam EF in punctis RS di&longs;pe&longs;cant <lb/><emph type="italics"/>lunganturque DF BE,<emph.end type="italics"/> &longs;ecetquè DF lineam LM in X. ip <lb/>verò EB &longs;ecet NT in O. Iungaturquè <emph type="italics"/>OX<emph.end type="italics"/>, quæ lineam EF <lb/>
<arrow.to.target n="fig52"></arrow.to.target><lb/>
<arrow.to.target n="marg174"></arrow.to.target> P &longs;ecet. <emph type="italics"/>erit itaque trianguli DBC centrum grauitatis in linea H <lb/>cùm &longs;it HB tertia pars ip&longs;ius B D<emph.end type="italics"/>; &longs;itquè propterea DH ip&longs;i <lb/>HB dupla. <emph type="italics"/>& per punctum H ducta &longs;it ba&longs;i<emph.end type="italics"/> BC <emph type="italics"/>æquidi&longs;tans M<emph.end type="italics"/><lb/>
<arrow.to.target n="marg175"></arrow.to.target> <emph type="italics"/>e&longs;t autem centrum quoque grauitatis trianguli DBC in linea DF<emph.end type="italics"/>; q <lb/>e&longs;t ab angulo D ad dimidiam BC ducta. <emph type="italics"/>Quare dicti triang <lb/>centrum grauitatis est punctum X. Eademquè ratione<emph.end type="italics"/> cùm &longs;it D <lb/>tertia pars ip&longs;ius DB, ac proptcrea &longs;it BK ip&longs;ius KD dup <lb/>&longs;itquè KN æquidi&longs;tans ip&longs;i AD; erit centrum grauitatis tri <lb/>guli ABD in linea KN; idem verò centrum reperitur quo <lb/>in linea BE, cùm &longs;it ab angulo B ad dimidiam AD duc <lb/>ergo <emph type="italics"/>punctum O<emph.end type="italics"/>, vbi &longs;e inuicem &longs;ecant, <emph type="italics"/>centrum e&longs;t grauitatist <lb/>guli ABD. magnitudinis igitur ex vtri&longs;que triangulis ABD BI <lb/>compo&longs;itæ, quæ e&longs;t trapezium<emph.end type="italics"/> ABCD, <emph type="italics"/>centrum grauitatis est in rect<emph.end type="italics"/>
<pb pagenum="109"/><emph type="italics"/>nea OX. dicti autem trapezii centrum gauitatis est etiam in li­<lb/>nea EF, quare trapezii ABCD centrum grauitatis est punctum <lb/>P. At verò triangulum BCD ad ABD proportionem habet eam, quam
<arrow.to.target n="marg176"></arrow.to.target><lb/>OP ad P<emph.end type="italics"/>X. cùm &longs;int puncta OX triangulorum centla graui <lb/>tatis, ac punctum P vtrorumque commune centrum. <emph type="italics"/>Sed vt <lb/>triangulum BDC adtriangulum ABD, ita e&longs;t<emph.end type="italics"/> quoque ba&longs;is <emph type="italics"/>BC<emph.end type="italics"/>
<arrow.to.target n="marg177"></arrow.to.target><lb/><emph type="italics"/>ad<emph.end type="italics"/> ba&longs;im <emph type="italics"/>AD.<emph.end type="italics"/> cùm triangula eandem habeant altitudinem, <lb/>&longs;iquidem &longs;unt in ijsdem parallelis AD BC. quare vt BC ad <lb/>AD, ita OP ad PX. <emph type="italics"/>Sed<emph.end type="italics"/> quoniam anguli RPO SPX ad
<arrow.to.target n="marg178"></arrow.to.target> ver­<lb/>ticem &longs;unt &ecedil;quales, & angulus PRO ip&longs;i PSX, veluti angulus
<arrow.to.target n="marg179"></arrow.to.target><lb/>ROP angulo PXS e&longs;t &ecedil;qualis, erit triangulum OPR triangu <lb/>lo XPS &longs;imile; quare <emph type="italics"/>vt OP ad PX, &longs;ic PR ad PS.<emph.end type="italics"/> e&longs;t autem
<arrow.to.target n="marg180"></arrow.to.target><lb/>BC ad AD, vt OP ad PX<emph type="italics"/>; vt igitur BC ad AD, ita RP ad PS.<emph.end type="italics"/>
<arrow.to.target n="marg181"></arrow.to.target><lb/>& antecedentium dupla, duæ &longs;cilicet BC ad AD, vt duæ PR <lb/>ad PS. & componendo duæ BC cum AD ad AD; vt duæ
<arrow.to.target n="marg182"></arrow.to.target><lb/>PR cum PS ad PS. & ad con&longs;equentium dupla, vt &longs;cilicet <lb/>duæ BC cum AD ad duas AD, ita duæ PR cum PS ad duas <lb/>PS. dictum e&longs;t autem BC ad AD ita e&longs;&longs;e, vt PR ad PS. quare <lb/>conuerrendo AD ad BC erit, vt PS ad PR. & antecedentium
<arrow.to.target n="marg183"></arrow.to.target><lb/>dupla. hoc e&longs;t duæ AD ad BC, vt duæ PS ad PR. Itaque in <lb/>eadem &longs;unt proportione duç BC cum AD ad duas AD, vt <lb/>du&ecedil; PR <expan abbr="cũ">cum</expan> PS ad duas PS. &longs;icut verò du&ecedil; AD ad BC, ita du&ecedil; <lb/>PS ad PR. antecedentes igitur ad &longs;uas &longs;imul con&longs;equentes in
<arrow.to.target n="marg184"></arrow.to.target><lb/>eadem erunt proportione. <emph type="italics"/>Quare &longs;icut duæ BC cum AD ad duas <lb/>AD cum BC, ita duæ RP cum PS ad duas P S cum PR, <lb/>verùm duæ quidem RP cum PS e&longs;t vtraque &longs;imul SR RP.<emph.end type="italics"/> bis <lb/>enim a&longs;&longs;umitur PR, &longs;emel verò PS. Cum autem lineæ DH ES <lb/>à lineis diuidantur &ecedil;quidi&longs;tantibus ED OT HM, erit DK ad
<arrow.to.target n="marg185"></arrow.to.target><lb/>KH, vt ER ad CS; kD verò e&longs;t æqualis KH, erit ER ip&longs;i <lb/>RS &ecedil;qualis. erit igitur ER cum RP, <emph type="italics"/>hoc est PE<emph.end type="italics"/> ip&longs;is SR RP <lb/>&ecedil;qualis. <emph type="italics"/>duæ verò PS cum PR e&longs;t vtraque PS SR.<emph.end type="italics"/> bis enim a&longs;­<lb/>&longs;umitur PS, &longs;emel què PR. & quoniam FS e&longs;t &ecedil;qualis ip&longs;i SR. <lb/>quod quidem eodem modo o&longs;tendetur, cùm &longs;it FS ad SR, vt <lb/>BH ad Hk. erit FS cum SP, <emph type="italics"/>hoc est PF<emph.end type="italics"/> ip&longs;is PS SR æqualis. <lb/>Quare ita &longs;ehabet PE ad PF, vt duæ BC cum AD ad duas <lb/>AD cum BC. Centrum igitur grauitatis P trapezij ABCD <lb/>in linea e&longs;t EF, quæ <expan abbr="cõiungit">coniungit</expan> parallelas AD BC bifariam di
<pb pagenum="110"/>ui&longs;as; ita vt pars PE, quæ e&longs;t ad minorem parallelam AD <lb/>reliquampartem PF eam habet proportionem, quam du <lb/>ip&longs;ius BC, quæ e&longs;t maior æquidi&longs;tautium, vna cum min <lb/>AD, ad duplam minoris AD cum maiore BC, <emph type="italics"/>ergo demons<gap/><lb/>ta &longs;unt, quæ propo&longs;ita fuerant.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg171"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 2.<emph type="italics"/>&longs;<gap/><emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg172"></margin.target>13.<emph type="italics"/>hu<gap/><emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg173"></margin.target>8. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg174"></margin.target><emph type="italics"/>ex proxi­<lb/>me demon <lb/>&longs;tratis.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg175"></margin.target>* <lb/>13. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg176"></margin.target>6. <emph type="italics"/>hu<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg177"></margin.target>1. <emph type="italics"/>&longs;e.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg178"></margin.target>15. <emph type="italics"/>p<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg179"></margin.target>29. <emph type="italics"/>p<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg180"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 4.</s></p>
<p type="margin">
<s><margin.target id="marg181"></margin.target>11. <emph type="italics"/><expan abbr="q.">que</expan><emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg182"></margin.target>18. <gap/></s></p>
<p type="margin">
<s><margin.target id="marg183"></margin.target><emph type="italics"/>corol <lb/>quint<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg184"></margin.target><emph type="italics"/>cor.<emph.end type="italics"/>2 <lb/><emph type="italics"/>ma a<gap/><lb/>huius<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg185"></margin.target>1. <emph type="italics"/>l. <lb/>in<emph.end type="italics"/> 13</s></p>
<figure id="fig52"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>
<arrow.to.target n="marg186"></arrow.to.target> Græcus codex po&longs;t ea verba, <emph type="italics"/>cùm &longs;it HB tertia pars ip&longs;ius<emph.end type="italics"/> Z <lb/>habet <foreign lang="greek">kai dia tou_ q sam<gap/>iou w_aza)ll<gap/>los ta) ba\sei o)ux ta_s a( mq</foreign>, qu<gap/><lb/>quidem verba illa <foreign lang="greek">ou)k ta\s</foreign> perperam leguntur; quorum l<gap/><lb/>ponerem <foreign lang="greek">a<gap/>omi\na e)si\</foreign>, ita vt &longs;int hoc modo re&longs;tituenda, <foreign lang="greek">ka<gap/> dia <lb/><gap/> same_iou w_aza/ll<gap/>lws ta_ ba\sei a<gap/>ome\na isi\ a( mq. </foreign></s></p>
<p type="margin">
<s><margin.target id="marg186"></margin.target>*</s></p>
<p type="main">
<s>Hæc &longs;unt, quæ de centro grauitatis figurarum rectiline <lb/>Archimedes &longs;cripta reliquit. Ex quibus maxima certè vtil <lb/>habetur; neque ampliùs de rectilineis figuris Archimedes p <lb/>tractare voluit. ex dictis enim alia omnia dependent. Nan <lb/>tra grauitatis rectilinearum figurarum, quæ æquales angu <lb/>lateraque æqualia habent, ex his in uenire poterimus. quæ <lb/>dem figur&ecedil; in circulo in&longs;cribi po&longs;&longs;unt. Quod &longs;anè Federi <lb/>Comandinus in eius libro de centro grauitatis &longs;olidorum <lb/>prioribus propo&longs;itionibus præ&longs;titit. aliaquè nonnulla, vt<gap/><lb/>tragrauitatis rectilinearum figurarum in ellip&longs;i, deindè ip<gap/><lb/>circuli, & ellip&longs;is centra grauitatis in uenit. omne&longs;què dem <lb/>&longs;trationes in ijs, quæ in hoc libro iam demon&longs;trata &longs;unt, <lb/>dauit. præterea ex his etiam idem Commandinus in com <lb/>tarijs libri Archimedis de quadratura paraboles, (quo ad p <lb/>xim) grauitatis centrum cuiu&longs;libet figur&ecedil; rectilineæ adin <lb/>nit. Quod quidem nos quoque, vt initio polliciti fuimus, <lb/>nullis mutatis idem o&longs;tendemus. hoc prius &longs;uppo&longs;ito. </s></p>
<p type="main">
<s>Triangula in eadem ba&longs;i con&longs;tituta eam inter &longs;e propo<gap/><lb/>nem habent, quam eorum altitudines. </s></p>
<p type="main">
<s>Hoc autem demon&longs;tratum e&longs;t ab excell enti&longs;simis viris, <lb/>ri&longs;què Euclidis interpretibus, Federico <expan abbr="Cõmandino">Commandino</expan>, & Cl <lb/>&longs;tophoro Clauio; qui hanc propo&longs;itionem po&longs;t primam <lb/>ti libri Euclidis demon&longs;trarunt. </s></p>
<pb pagenum="111"/>
<p type="head">
<s>PROBLEMA.</s></p>
<p type="main">
<s>Cuiu&longs;libet rectiline&ecedil; figur&ecedil; centrum grauitatis inuenire. </s></p>
<p type="main">
<s>Triangulorum centrum grauitatis iam ab Archimede de­<lb/>mon&longs;tratum e&longs;t. </s></p>
<p type="main">
<s>Sit itaque primùm quadri <lb/>
<arrow.to.target n="fig53"></arrow.to.target><lb/>laterum ABCD, cuius opor­<lb/>teat centrum grauitatis inue <lb/>nire. Ducatur AC, quæ qua <lb/>drilaterum in duo triangula <lb/>ABC ACD diuidet. à <expan abbr="pũcti&longs;-què">puncti&longs;­<lb/>què</expan> BD ad AC perpendicu <lb/>lares ducantur BE DF. In­<lb/>ueniantur deinde ex dictis <expan abbr="c&etilde;">cem</expan> <lb/>tra grauitatis triangulorum <lb/>ABC ACD. &longs;intquè puncta <lb/>GH. iungaturquè GH, quæ diuidatur in K, ita vt GK <lb/>ad KH &longs;it, vt DF ad BE. Dico punctum K centrum <lb/>e&longs;&longs;e grauitatis quadrilateri ABCD. Quoniam enim triangu­<lb/>la ABC ACD in eadem &longs;unt ba&longs;i AC, erunt inter&longs;e&longs;e, vt al­<lb/>titudines. quare triangulum ACD ita &longs;e habet ad <expan abbr="triangulũ">triangulum</expan> <lb/>ABC, vt DF ad BE. hoc e&longs;t GK ad KH. <expan abbr="punctũ">punctum</expan> ergo K <expan abbr="c&etilde;">cem</expan> <lb/>trum e&longs;t grauitatis magnitudinisex vtril què triangulis ABC
<arrow.to.target n="marg187"></arrow.to.target><lb/>ACD compo&longs;itæ; hoc e&longs;t quadrilateri ABCD. </s></p>
<p type="margin">
<s><margin.target id="marg187"></margin.target><emph type="italics"/>ex 6.h<emph.end type="italics"/></s></p>
<figure id="fig53"></figure>
<p type="main">
<s>Sit autem pentagonum <lb/>
<arrow.to.target n="fig54"></arrow.to.target><lb/>ABCDE. <expan abbr="iungãturquè">iunganturquè</expan> AC <lb/>AD. inueniaturquè <expan abbr="triãgu">triangu</expan> <lb/>li ABC centrum grauitatis <lb/>H. quadrilateri verò ACDE <lb/>ex proximè <expan abbr="demõ">demom</expan> &longs;tra tis <expan abbr="c&etilde;-trum">cen­<lb/>trum</expan> grauitatis inueniatur <lb/>Iam vtique con&longs;tat (du­<lb/>cta HK) centrum grauita <lb/>tis totius ABCDE in linea
<pb pagenum="112"/>HK exi&longs;tere. Rurilus trianguli ADE centrum inueniatur F <lb/>quadrilateri verò ADCB punctum G. iungaturquè GF. e<gap/><lb/>eodem modo centrum grauitatis totius ABCDE in linea F<gap/><lb/>&longs;ed e&longs;t quoque in linea HK, ergo vbr&longs;e inuicem &longs;ecant, vt <lb/>L, centrum erit grauitatis pentagoni ABCDE. </s></p>
<figure id="fig54"></figure>
<p type="main">
<s>In hexagonis &longs;imiliter. <lb/>
<arrow.to.target n="fig55"></arrow.to.target><lb/>vt ABCDEF iungantur <lb/>AC AE, deinceps inuenia <lb/>tur trianguli ABC <expan abbr="c&etilde;trum">centrum</expan> <lb/>grauitatis G, pentagoni <lb/>verò ACDEF ex dictis cen <lb/>trum &longs;it H. ductaquè GH <lb/>centrum grauitatis totius <lb/>ABCDEF erit in linea GH <lb/>&longs;imiliter centrum grauita­<lb/>tis trianguli AFE &longs;it K, <expan abbr="p&etilde;">pem</expan> <lb/>tagoni verò AEDCB &longs;it L, iunctaquè KL, erit centrum gr <lb/>uitatis totius hexagoni in linea KL. verùm e&longs;t etiam in lin <lb/>GH. ergo errt in M. in quo GH <emph type="italics"/>K<emph.end type="italics"/>L &longs;e inuicem &longs;ecant. </s></p>
<figure id="fig55"></figure>
<p type="main">
<s>Nequè aliter in heptago <lb/>
<arrow.to.target n="fig56"></arrow.to.target><lb/>no ABCDEFG, in quo du <lb/>cantur BG CE. trianguli <lb/>verò ABG centrum graui­<lb/>tatis &longs;it H. hexagoni <expan abbr="aut&etilde;">autem</expan> <lb/>GBCDEF, &longs;it K. deinde <lb/>trianguli CDE <expan abbr="centrũ">centrum</expan> gra <lb/>uitatis &longs;it L, hexagoni ve­<lb/>rò CEFGAB &longs;it M. iun­<lb/>cti&longs;què HK ML, eadem ra <lb/>tione centrum grauitatis <lb/>
<arrow.to.target n="marg188"></arrow.to.target> totius heptagoni erit in vtraquè linea Hk LM. ergo erit in </s></p>
<p type="margin">
<s><margin.target id="marg188"></margin.target>*</s></p>
<figure id="fig56"></figure>
<p type="main">
<s>Eodemquè pror&longs;us modo in octagono, & in alijs demc<gap/><lb/>figuris centrum graui ta tis inuenietur. quæ quidem facere <lb/>portebat. </s></p>
<pb pagenum="113"/>
<p type="main">
<s>Curautem hoc modo centra grauitatum in præfatis figu­<lb/>ris po&longs;itione tantùm, & non determinatè ea in determinata, <lb/>linea, & in tali &longs;itu exi&longs;tere inuenerimus, vt in parallelogram <lb/>mis & in triangulis factum fuitab Archimede; explicabitur in <lb/>&longs;ecundo libro po&longs;t tertiam proportionem; vbi o&longs;tendemus, <lb/>in quibus figuris determinatè inueniri pote&longs;t centrum graui­<lb/>tatis. </s></p>
<p type="main">
<s>Antequam autem finem primolibro imponamus, <expan abbr="reliquũ">reliquum</expan> <lb/>e&longs;t; vt ea quæ in præfatione &longs;uppo&longs;uimus, o&longs;tendamus. pri­<lb/>mùm què quando &longs;ecundùm rectam lineam aliqua diuiditur <lb/>figura per centrum grauitatis, aliquando diuidi in partes &longs;em <lb/>per &ecedil;quales, & aliquando in partes inæquales. </s></p>
<p type="head">
<s>PROPOSITIO.</s></p>
<p type="main">
<s>Figura dari pote&longs;t, qu&ecedil; per centrum grauitatis recta li­<lb/>nea diui&longs;a, &longs;emperin partes diuidatur æquales. </s></p>
<p type="main">
<s>Sit <expan abbr="parallelogrammũ">parallelogrammum</expan> <lb/>
<arrow.to.target n="fig57"></arrow.to.target><lb/>ABCD, cuius <expan abbr="centrũ">centrum</expan> gra­<lb/>uitatis E. Ducaturquè per <lb/>E <expan abbr="vtcun&qacute;">vtcunque</expan>; linea GEF, qu&ecedil; <lb/>vel diameter e&longs;t, vel min^{9}. <lb/>&longs;i e&longs;t diameter, iam <expan abbr="cõ&longs;tat">con&longs;tat</expan> <lb/><expan abbr="parallelogrãmum">parallelogrammum</expan> in duo <lb/>&ecedil;qua e&longs;&longs;e diui&longs;um. Si verò non e&longs;t diameter, <expan abbr="ducãtur">ducantur</expan> diametri
<arrow.to.target n="marg189"></arrow.to.target><lb/>AC BD, quæ per E tran&longs;ibunt. Quoniam igitur AF e&longs;t æqui­<lb/>diftans ip&longs;i CG, eritangulus EAF ip&longs;i ECG, & EFA ip&longs;i EGC
<arrow.to.target n="marg190"></arrow.to.target><lb/>æqualis, e&longs;t autem AEF ip&longs;i GEC ad verticem æqualis, <expan abbr="latus&qacute;">latusque</expan>;
<arrow.to.target n="marg191"></arrow.to.target><lb/>AE ip&longs;i EC æquale; erit triangulum AEF triangulo CEG &ecedil;qua <lb/>le. eodemquè modo o&longs;tendetur triangulum FEB triangulo <lb/>EGD. & triangulum AED ip&longs;i BEC æquale. Ex quibus patet. <lb/>figuram ex tribus triangulis compo&longs;itam, hoc e&longs;t figuram <lb/>FGDA ip&longs;i FGCB æqualem e&longs;&longs;e. diuiditurergo <expan abbr="parallelogrã-mum">parallelogran­<lb/>mum</expan> à linea per centrum grauitatis ducta in partes &longs;em perç­<lb/>quales. quod demon&longs;trare oportebat. </s></p>
<pb pagenum="114"/>
<p type="margin">
<s><margin.target id="marg189"></margin.target>34.<emph type="italics"/>primi<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg190"></margin.target>29. <emph type="italics"/>primi<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg191"></margin.target>15. <emph type="italics"/>primi<emph.end type="italics"/></s></p>
<figure id="fig57"></figure>
<p type="main">
<s>Hoc idem multis alijs figuris accidet, vt pentagonis, he <lb/>gonisæquiangulis, & æquilateris, & alijs. </s></p>
<p type="head">
<s>PROPOSITIO.</s></p>
<p type="main">
<s>Figura dari pote&longs;t, quæ per centrum grauitatis recta li <lb/>diui&longs;a, non &longs;emper in partes diuidatur &ecedil;quales. </s></p>
<p type="main">
<s>Habeat triangulum ABC <lb/>
<arrow.to.target n="fig58"></arrow.to.target><lb/>latera AB AC æqualia. trian <lb/>guliverò centrum grauitatis &longs;it <lb/>D. à quo ip&longs;i BC &ecedil;quidi&longs;tans <lb/>Ducatur FDG. Dico partem <lb/>AFG <expan abbr="minor&etilde;">minorem</expan> e&longs;&longs;e parte BFGC. <lb/>ducatur ADE, quæ bifariam <lb/>
<arrow.to.target n="marg192"></arrow.to.target> BC diuidet. & à puncto G <lb/>ip&longs;i AE &ecedil;quidi&longs;tans ducatur <lb/>HGK. compleanturque figur&ecedil; <lb/>EH KF. Quoniam enim FG <lb/>
<arrow.to.target n="marg193"></arrow.to.target> &ecedil;quidi&longs;tans e&longs;t ip&longs;i BC, erit FD ad DG, vt BE ad E<gap/><lb/>& e&longs;t BE ip&longs;i EC æqualis. eritigitur FD ip&longs;i DG &ecedil;qua <lb/>vt etiam paulò ante 15. huius o&longs;tendimus. quare FG ip <lb/>DG dupla. e&longs;t. ac propterea <expan abbr="parallelogrãmum">parallelogrammum</expan> FK dupi <lb/>e&longs;t parallelogrammi DK. quia verò AD ip&longs;ius DE du <lb/>exi&longs;tit, erit quoquè parallelogrammum DH ip&longs;ius DK <lb/>plum. Quare DH ip&longs;i FK e&longs;t æquale. At verò quoni <lb/>
<arrow.to.target n="marg194"></arrow.to.target> FG dupla e&longs;t ip&longs;ius DG. erit triangulum AFG parallelog <lb/>mo DH æquale. triangulum igitur AFG parallelog<gap/><lb/>FK e&longs;t æquale. Quare pars AFG parte BFGC minor <gap/><lb/>&longs;tit. quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg192"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 13. <emph type="italics"/>hui'<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg193"></margin.target><emph type="italics"/>lemma an­<lb/>te <expan abbr="&longs;ecundã">&longs;ecundam</expan> <lb/><expan abbr="demon&longs;tra-tion&etilde;">demon&longs;tra­<lb/>tionem</expan><emph.end type="italics"/> 13 <emph type="italics"/>bu <lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg194"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 41.<emph type="italics"/>pri. <lb/>mi.<emph.end type="italics"/></s></p>
<figure id="fig58"></figure>
<p type="main">
<s>Hinc per&longs;picuum e&longs;t, eandem figuram per centrum gra <lb/>tatis diui&longs;am, aliquando in partes in æquales, aliquando in <lb/>tes æquales diuidi po&longs;&longs;e. in partes in&ecedil;quales iam o&longs;ten&longs;um <lb/>hocaccidere <expan abbr="perlineã">perlineam</expan> FG. in partes verò æquales patet pe <lb/>neam ADE, quæ triangulum ABC in duo &ecedil;qua diuidi<gap/>. t<gap/><lb/>
<arrow.to.target n="marg195"></arrow.to.target> gulum enim ABE triangulo: AEC e&longs;t &ecedil;quale, cùm &longs;int<gap/><lb/>eadem altitudine, ba&longs;e&longs;què BE EC inter&longs;e &longs;int æquales. </s></p>
<pb pagenum="115"/>
<p type="margin">
<s><margin.target id="marg195"></margin.target>1. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="main">
<s>Adhuc (veluti initio quoque diximus) &longs;i fuerit prisma, vt <lb/>AB, cuius altera ba&longs;is &longs;it AC. tale verò &longs;it prisma, vt pl mum <lb/>AC planis CH CK &c. &longs;it erectum. &longs;it autem ip&longs;ius ba&longs;is <lb/>AC centrum grauitatis E. Dico &longs;i prima &longs;u&longs;pendatur ex pu­<lb/>
<arrow.to.target n="fig59"></arrow.to.target><lb/>cto E, ba&longs;im AC horizonti æquidi&longs;tantem permanere. vt co <lb/>gno&longs;camusea, quæ his libris pertractantur, ad praxim po&longs;&longs;e <lb/>reduci. & ne aliquid ab&longs;que demon&longs;tratione confirmatum re <lb/>linquamus. hoc quoque o&longs;tendemus. hoc pacto. </s></p>
<figure id="fig59"></figure>
<p type="main">
<s>Primùm quidem exijs, quæ demon&longs;trata &longs;unt, rectilineæ <lb/>figuræ AC centrum granitatis inueniatur E. eodemquè mo <lb/>do figuræ BD centrum grauitatis &longs;it F. Iungaturquè EF, <lb/>quæ bifariam diuidatur in G. Iam patet punctum G cen­<lb/>trum e&longs;&longs;e grauitatis pri&longs;matis AB, ex octaua propo&longs;itione Fe­<lb/>derici <expan abbr="Cõmandini">Commandini</expan> de centro grauitatis &longs;olidorum, & ex corol <lb/>lario quintæ propo&longs;itionis eiu&longs;dem libri, lineam EF late­<lb/>ribus AD CB &ecedil;quidi&longs;tantem e&longs;&longs;e. quoniam <expan abbr="aut&etilde;">autem</expan> plana CH <lb/>CK ad rectos &longs;untangulos plano AC, erit CB eorum commu
<arrow.to.target n="marg196"></arrow.to.target><lb/>nis&longs;ectio eidem plano AC perpendicularis. acpropterea EF <lb/>ip&longs;i CB æquidi&longs;tans plano AC perpendicularis exi&longs;tit.
<pb pagenum="116"/>Itaque intelligatur &longs;olidum AB ex E &longs;u&longs;pen&longs;um; tunc ex <lb/>ma propo&longs;itione de libra no&longs;trorum mechanicorum pon <lb/>AB ex E &longs;u&longs;pen&longs;um <expan abbr="numquã">numquam</expan> manebit, ni&longs;i recta EG fu <lb/>horizonti perpendicularis. Quando autem EF erit horizc <lb/>ti perpendicularis, erit planum AC horizonti æquidi&longs;tan <lb/>
<arrow.to.target n="marg197"></arrow.to.target> tunc. n. EF tum horizonti, tum plano AC perpendicul<gap/><lb/>exi&longs;tet. Inuento igitur centro grauitatis E ip&longs;ius ba&longs;is A <lb/>&longs;i AB &longs;u&longs;pendatur ex E, linea EGF in centrum mundi to <lb/>det; planumquè AC horizonti erit æquidi&longs;tans. quod de<gap/><lb/>&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg196"></margin.target>19. <emph type="italics"/>v <lb/>mi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg197"></margin.target>14.<emph type="italics"/>vndeci <lb/>mi.<emph.end type="italics"/></s></p>
<p type="main">
<s>PRIMI LIBRI FINIS. </s></p>
<pb pagenum="117"/>
<p type="head">
<s>GVIDIVBALDI <lb/>E MARCHIONIBVS <lb/>MONTIS.</s></p>
<p type="head">
<s>In Secundum Archimedis æqueponderan­<lb/>tium Librum.</s></p>
<p type="head">
<s>PRÆFATIO.</s></p>
<p type="main">
<s>Secundus Archimedisliber, vtinitio primi <lb/>libri præfati &longs;umus, &longs;ubtili&longs;&longs;ima theo­<lb/>remata &longs;peculatur. Vultenim Archimedes <lb/>inue&longs;tigare centrum grauita tis plani coni­<lb/>cæ&longs;ectionis, quæ parabole pa&longs;&longs;im vocatur. <lb/>quamuis Archimedes alio nomine, ac po­<lb/>tiùs de&longs;criptione quadam <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> <expan abbr="hãc">hanc</expan> <expan abbr="nũ-cuparit">nun­<lb/>cuparit</expan>: veluti portio recta linea <expan abbr="rectãguli&qacute;">rectangulique</expan>; coni&longs;ectione <expan abbr="cõ">com</expan> <expan abbr="t&etilde;">tem</expan> <lb/>ta. Refert enim Eutocius A&longs;calonita in principio &longs;ui <expan abbr="comm&etilde;-tarij">commen­<lb/>tarij</expan> in libros conicorum Apollonij Perg&ecedil;i, ex &longs;ententia Ge­<lb/>mini (cui Pappus etiam ex Ari&longs;t&ecedil;i &longs;ententia a&longs;&longs;entire videtur) <lb/>quòd qui ante Apoll onium fuerunt, perfectam, & ab&longs;olutam <lb/>conorum <expan abbr="cognition&etilde;">cognitionem</expan> <lb/>
<arrow.to.target n="fig60"></arrow.to.target><lb/>non habuerunt; inter <lb/>quos re&longs;po&longs;uit Archime <lb/>de. <expan abbr="Nã">Nam</expan> inquit <expan abbr="conũ">conum</expan> de&longs;i <lb/>nientes, ip&longs;um per <expan abbr="rectã">rectam</expan> <lb/>guli <expan abbr="triãguli">trianguli</expan> circumuo­<lb/>lutionem manente vno <lb/>eorum, quæ circa <expan abbr="rectũ">rectum</expan> <lb/><expan abbr="angulũ">angulum</expan> &longs;unt, latere <expan abbr="cõ&longs;i-derarunt">con&longs;i­<lb/>derarunt</expan>. vt habetur in <lb/>definitionibus Euclidis <lb/>vndecimi libri elem <expan abbr="en-torũ">en­<lb/>torum</expan>. vt Conus ABC fit <lb/>ex <expan abbr="circũuoluto">circunuoluto</expan> triangulo rectangulo ADC. conus verò EBC <lb/>ex triangulo EDC, & conus FBC ex rectangulo triangulo
<pb pagenum="118"/>FDC. & &longs;i AD fuerit i­<lb/>
<arrow.to.target n="fig61"></arrow.to.target><lb/>p&longs;i DC æqualis, conus <lb/>ABC vocabit rectan­<lb/>gulus. nam vtcumquè <lb/>ducto plano per axem, <lb/>
<arrow.to.target n="marg198"></arrow.to.target> quod triangulum faciat <lb/>ABC; erit angulus BAC <lb/>ad coniverticem rectus: <lb/>&longs;iquidem DAC recti di <lb/>midius exi&longs;tit, veluti <lb/>DAB. pari ratione &longs;i ED <lb/>fuerit ip&longs;a DC minor; <lb/>erit conus EBC obtu&longs;i <lb/>angulus:nam ducto per axem plano, habebit triangulum <lb/>EBC angulum ad verticem coni BEC obtu&longs;um; cùm &longs;it <lb/>
<arrow.to.target n="marg199"></arrow.to.target> BEC maior BAC. exi&longs;tenteautem FD ip&longs;a DC maiori, co <lb/>nus FBC acutiangulus nuncupabitur; quoniam <expan abbr="triangulũ">triangulum</expan> <lb/>per axem FBC angulum ad verticem coni F acutum po&longs;&longs;ide <lb/>bit; &longs;iquidem minor e&longs;t BFC, quam BAC. Refert deinde, <lb/>quòd vnumquemquè <lb/>horum conorum <expan abbr="eo-d&etilde;">eo­<lb/>dem</expan> modo pi&longs;ci &longs;ecue­<lb/>
<arrow.to.target n="fig62"></arrow.to.target><lb/>runt; vt &longs;it rectangu­<lb/>lus conus ABC; trian <lb/>gulum verò per axem <lb/>&longs;it ABC. in latere au­<lb/>tem AC quoduis &longs;u­<lb/>matur punctum D; <lb/>ducaturquè DE ad <lb/>AC perpendicularis; <lb/>& per DE ducatur pla <lb/>num plano ABC ere <lb/>ctum, quod quidem conum &longs;ecet, &longs;ectio autem &longs;it FDG. qu&ecedil; <lb/>&longs;anè e&longs;t &longs;e ctio, quæ abip&longs;is vocatur rectanguli coni &longs;ectio, <lb/>quippè quæ &longs;i intelligatur terminata recta linea FG, nuncupa <lb/>tur portio recta linea, rectangulique coni &longs;ectione contenta. </s></p>
<pb pagenum="119"/>
<p type="margin">
<s><margin.target id="marg198"></margin.target>3. <emph type="italics"/>primi co <lb/>mcorum A <lb/>pol.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg199"></margin.target>21. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<figure id="fig60"></figure>
<figure id="fig61"></figure>
<figure id="fig62"></figure>
<p type="main">
<s>Si verò conus <lb/>
<arrow.to.target n="fig63"></arrow.to.target><lb/>ABC fuerit obtu <lb/>&longs;iangulus, &longs;itquè <lb/>triangulum per <lb/>axem ABC, <expan abbr="eo-d&etilde;">eo­<lb/>dem</expan> modoà quo­<lb/>uis puncto D, du <lb/>cta DE ad re­<lb/>ctos angulos ip&longs;i <lb/>AC, acper DE <lb/>ducto plano ad <lb/>planum ABC erecto, quod conum &longs;ecet, vt FDG; erit FDG <lb/>obtu&longs;ianguli coni &longs;ectio, quæ vnà cum recta FG vocatur por­<lb/>tio recta linea, obtu&longs;ianguliquè coni &longs;ectione contenta. </s></p>
<figure id="fig63"></figure>
<p type="main">
<s>Similiter <expan abbr="exi&longs;t&etilde;te">exi&longs;tente</expan> co­<lb/>
<arrow.to.target n="fig64"></arrow.to.target><lb/>no acutiangulo ABC, <lb/>cuius triangulum per a­<lb/>xem &longs;it ABC. & à <expan abbr="pũcto">puncto</expan> <lb/>D ducta &longs;it DE perpen­<lb/>dicularis ip&longs;i AC, du­<lb/>ctoquè plano per DE ad <lb/>planum ABC erecto, e­<lb/>rit DFEG acutianguli <lb/>coni &longs;ectio. </s></p>
<figure id="fig64"></figure>
<p type="main">
<s>Apollonius au-­<lb/>tem Perg&ecedil;us, qui ab­<lb/>&longs;oluti&longs;&longs;ima commenta­<lb/>ria de conicis &longs;crip&longs;it, <lb/>huiu&longs;modi conos omnesvocauit rectos; ad differentiam coni <lb/>&longs;caleni. coni enim rectiaxes habent ba&longs;ibus erectos. &longs;caleni ve <lb/>rò nequaquam. & in &longs;calenis latera triangulorum per axem <lb/>non &longs;unt &longs;emper æqualia. quod &longs;emper conis rectis contingit. </s></p>
<p type="main">
<s>Preterea &longs;ectionem rectanguli coni parabolen nominauit; <lb/>obtu&longs;ianguli verò coni &longs;ectionem hyperbolen; &longs;ectionem au <lb/>tem acutianguli coni ellip&longs;im nuncupauit. & in vnoquoque <lb/>cono tàm recto, quàm &longs;caleno has tres ine&longs;&longs;e &longs;ectiones <expan abbr="demõ">demom</expan>
<pb pagenum="120"/>&longs;trauit. Ex quibus colligit Geminus (quem Eutocius, alijquè <lb/>complures &longs;ecuti &longs;unt) eos, qui ante Apollonium extitere, <lb/>conostantùm rectos cognoui&longs;&longs;e. & in vnoquoque cono <expan abbr="vnã">vnam</expan> <lb/>tantùm &longs;ectionem animaduerti&longs;&longs;e. quod quidem &longs;i de ijs, qui <lb/>ante Archimedem fuere intelligatur; ad mitti forta&longs;&longs;e poterit; <lb/>ac præ&longs;ertim de Euclide. vt patet ex definitione coni abeo <lb/>tradita. At verò de Archimede, qui po&longs;t Euclidem, ante verò <lb/>Apollonium fuit, non ita facilè concedendum videtur. <expan abbr="Nã">Nam</expan> ex <lb/>ijs, quæ &longs;cripta reliquit. eum non &longs;olùm notitiam ha-­<lb/>bui&longs;&longs;e de conis rectis; verùm <expan abbr="etiã">etiam</expan> de &longs;calenis facilè ex i-­<lb/>p&longs;ius &longs;criptis conijci pote&longs;t. In primo enim librode &longs;phæ­<lb/>ra, & cylindro multis in locis, vtin &longs;eptima, octaua, no <lb/>na, decimaquarta, decimaquinta propo&longs;itione; alijsquè in <lb/>locis conos nominat &ecedil;quicrures, quod quidem &longs;ecundum i­<lb/>p&longs;um &longs;unt, qui in eius &longs;uperficie æquales habent rectas lineas <lb/>à vertice coni ad ba&longs;im ductas. item in epi&longs;tola quoque libri <lb/>de conoidibus & &longs;ph&ecedil;roidibus, quam Archimedes De&longs;itheo <lb/>&longs;cribit. cùm de obtu&longs;iangulo conoideverba facit, conum vo­<lb/>catæquicrurem. Quòd &longs;i Archimedes hos conos vocauit æ­<lb/>quicrures, cui dubium, ip&longs;um eosad differentiam eorum, qui <lb/>non &longs;unt æquicrures ita nuncupa&longs;&longs;e? qui verò non &longs;unt æ­<lb/>quicrures ex ip&longs;omet Apollonio &longs;unt &longs;caleni; nam æquicrures <lb/>hoc modo coni axes habent ba&longs;ibus erectos. qui igitur non <lb/>erunt æquicrures, eorum axes &longs;uis ba&longs;ibus nunquàm erunt e­<lb/>recti. Præterea idem quoque confirmari pote&longs;t ex demon­<lb/>&longs;tratione vige&longs;imæquintæ propo&longs;itionis eiu<gap/>dem libri, in qua <lb/>cùm nominet Archimehes conum rectum proculdubiò ad <lb/>differentiam eorum, qui non &longs;untrecti ita eum nuncupauit. <lb/>nam &longs;i Aichimedes (ex illorum &longs;ententia) conos tan ùm re­<lb/>ctos cognoui&longs;&longs;et; quor&longs;um his in locis conum rectum, vel æ­<lb/>quicrurem nomina&longs;&longs;et? &longs;at &longs;ibi fui&longs;&longs;et conum tan tum dixi&longs;&longs;e. <lb/>Neque verò dicendum e&longs;t Archimedem per cono recto intel <lb/>lexi&longs;&longs;e conum rectangulum eo modo, quem &longs;upra expo&longs;ui­<lb/>mus. nam in ea propo&longs;itione, dum con&longs;tituit hunc conum, <lb/>non con&longs;urgit conus rectangulus, &longs;ed obtu&longs;iangulus quapro <lb/>pter conum rectum nominatad differentiam coni &longs;caleni. C&ecedil; <lb/>terùm ut manife&longs;tè o&longs;ten damus Archimedem conos cogno-
<pb pagenum="121"/>ui&longs;&longs;e &longs;calenos, con&longs;ideranda e&longs;t octaua propo&longs;itio libri de co­<lb/>noidibus, & &longs;ph æroidibus, in qua proponit Archimedes co­<lb/>num con&longs;tituere, & inuenire, in quo &longs;it&longs;ectio ellip&longs;is data, ver <lb/>tex autem coni in linea exi&longs;tat a centro ellip&longs;is ad<gap/>ectos angu <lb/>los ellip&longs;is plano erecta. Exqua con&longs;tructione planè apparet, <lb/>Archimedem (vt ex eius demon&longs;tratione con&longs;tat) hocin lo­<lb/>co querere, & inuenire conum proculdubio &longs;calenum. vt <expan abbr="etiã">etiam</expan> <lb/>ex nona eiu&longs;dem libri propo&longs;itione per&longs;picuum e&longs;&longs;e pote&longs;t; in <lb/>qua vt plurimùm conus inuenitur &longs;calenus. Ex quibus mani­<lb/>fe&longs;ti&longs;&longs;imè patet Archimedem non &longs;olùm de conis rectis, <expan abbr="verũ">verum</expan> <lb/>etiam de conis &longs;calenis notitiam habui&longs;&longs;e. Porrò ea verba, qu&ecedil; <lb/>refert Eutocius ex &longs;ententia Heraclij, qui Archimedis vitam <lb/>literis mandauit; idip&longs;um &longs;atis manife&longs;tant. Heraclius enim <lb/>inquit Archimedem quidem <expan abbr="primũ">primum</expan> conica theoremata fui&longs;&longs;e <lb/>aggre&longs;&longs;um; Apollonium verò, cùm ea inueni&longs;&longs;etab Archime <lb/>de nondum edita; tanquam eius propria edidi&longs;&longs;e. quod qui­<lb/>dem etiam exip&longs;iusmet Archimedis &longs;criptis <expan abbr="cõfirmari">confirmari</expan> pote&longs;t. <lb/>in libro namque de conoidibus, & &longs;phæroidibus ante <expan abbr="quartã">quartam</expan> <lb/>propo&longs;itionem vbi Archimedes theorema proponit alibi de­<lb/>mon&longs;tratum, inquit, <emph type="italics"/>Hoc autem o&longs;ten &longs;um e&longs;t in conicis elementis.<emph.end type="italics"/> in <lb/>principio etiam libri de quadratura paraboles, cùm nonnulla <lb/>propo&longs;ui&longs;&longs;et; po&longs;t tertiam propo&longs;itionem &longs;cilicet, inquit <emph type="italics"/>De­<lb/>mon&longs;trata autem &longs;unthæc in elementis conicis.<emph.end type="italics"/> nonneigitur con&longs;tat <lb/>Archimedem <expan abbr="elem&etilde;ta">elementa</expan> conica &longs;crip&longs;i&longs;&longs;e? Obijciet verò aliquis, <lb/>non propterea con&longs;tare, h&ecedil;c elementa eonica, quorum me­<lb/>minit Archimedes, ip&longs;iusmet e&longs;&longs;e Archimedis; cùm non affir <lb/>met, hæcfui&longs;&longs;e ab ip&longs;o dem on &longs;trata. verùm illud in primis ma <lb/>nife&longs;tum e&longs;t, tempore Archimedis conica elementa extiti&longs;&longs;e. <lb/>vt nonnulli Euclidem quatuor conicorum libros edidi&longs;&longs;e <expan abbr="af-firmãt">af­<lb/>firmant</expan>; &longs;icut Pappusin &longs;eptimo <expan abbr="Mathematicarũ">Mathematicarum</expan> <expan abbr="collectionuũ">collectionuum</expan> <lb/>libro a&longs;&longs;erit. Sed ex modo loquendi Archimedis planè <expan abbr="cõ&longs;tat">con&longs;tat</expan> <lb/>hæc fui&longs;&longs;e ab ip&longs;o con&longs;cripta. Nam quando Archimedes ali­<lb/>qua &longs;upponitab alijs demon&longs;trata, <expan abbr="tũc">tunc</expan> addere con&longs;ueuit, illa <lb/>ab alijs demon&longs;trata e&longs;&longs;e; vtin vndecima propo&longs;itionedeco­<lb/>noidibus, & &longs;phæroidibus; cùm inquit. <emph type="italics"/>omnis coni ad conum pro­<lb/>portionem compo&longs;itam e&longs;&longs;e ex proportione ba&longs;ium, & proportione altitu­<lb/>dinum,<emph.end type="italics"/> quod quidem, quia ab alijs dem on&longs;tratum fuerat, &longs;ta­
<pb pagenum="122"/>tim inquit, <emph type="italics"/>demon&longs;tratum e&longs;t ab iis, qui ante nos fuerunt.<emph.end type="italics"/> &longs;imiliter <lb/>in libro de &longs;ph&ecedil;ra, & cylindro an te propo&longs;itionem decimam <lb/>&longs;eptimam, cùm nonnulla &longs;uppo&longs;uerit ab alijs demon &longs;trata in <lb/>quit. <emph type="italics"/>Hæc autem omnia à &longs;uperioribus &longs;unt demon&longs;trata.<emph.end type="italics"/> In &longs;ecunda <lb/>verò parte <expan abbr="quĩt&ecedil;">quint&ecedil;</expan> propo&longs;itionis hui^{9} &longs;ecudi libri cu inquit, <emph type="italics"/>De <lb/>mon&longs;tratum e&longs;t enim aliis in locis portiones &longs;e&longs;quitertias e&longs;&longs;e <expan abbr="triangulorũ">triangulorum</expan>.<emph.end type="italics"/><lb/>quod quia ip&longs;emet a&longs;&longs;ecutus e&longs;t in libro de quadratura para­<lb/>boles, idcircò non addit ab ip&longs;omethoc o&longs;ten&longs;um fui&longs;&longs;e. A­<lb/>liaquè huiu&longs;modi loca breuita tis &longs;tudio omitto o&longs;tendentia <lb/>ea, quæ Archimedes &longs;upponit tanquam demon&longs;trata, <expan abbr="quãdo">quando</expan> <lb/>non additab alijs o&longs;ten&longs;a e&longs;&longs;e, à &longs;e ip&longs;o demon&longs;trata fui&longs;&longs;e, vt <lb/>in demon&longs;tratione decimæ quart&ecedil; propo&longs;itionis primi libri, <lb/>nec non ex octaua huius &longs;ecundi libri dem on&longs;tratione; alij&longs;­<lb/>què locis per&longs;picuum e&longs;&longs;e pote&longs;t. Quare tùm ex præfntis Archi <lb/>medis locis, tùm Heraclij te&longs;tim onio manife&longs;tè elicipote&longs;t, <lb/>Archimedem elementa conica &longs;crip &longs;i&longs;&longs;e. Neque verò quicqua <lb/>nos turbare debet, quòd Apollo nius coni &longs;ectionibus nomina <lb/>impo&longs;uerit; &longs;i tamen ip&longs;e prim us fuit; cùm eas proprijs nomi­<lb/>nibus, vt potè parabolen, hyperbolen, & ellip&longs;im nuncupet; <lb/>& in quolibet cono omnes agnouerit &longs;ectiones. Nam quam­<lb/>uis v&longs;que ad Archimedis tempus hi termini nondum extite­<lb/>rint; & in &longs;ingulis conis pri&longs;ci illi vnicam <expan abbr="tãtùm">tantùm</expan> cognouerint <lb/>&longs;ectionem; Archimedes tamen vlteriùs progre&longs;&longs;us e&longs;t. etenim <lb/>hæc quoque <expan abbr="&longs;ectionũ">&longs;ectionum</expan> nomina ip&longs;i forta&longs;se minùs ignota fue­<lb/>runt: quandoquidem in demon&longs;tratione nonæ propo&longs;itio­<lb/>nis de conoidibus, & &longs;ph&ecedil;roidibus ellip&longs;im nominat. Pr&ecedil;te­<lb/>rea non &longs;olùm cognouit Archimedes conos &longs;ecari po&longs;&longs;e pla­<lb/>nis lateribus coni erectis, verùm etiam alijs modis: quod qui­<lb/>dem exemplo ellip&longs;is manife&longs;tari optimè pote&longs;t. Nam in o­<lb/>ctaua propo&longs;itione eiu&longs;dem libri ellip&longs;es latus coni ad angu­<lb/>los rectos minimè &longs;ecant. veluti quoque in nona propo&longs;itione <lb/><expan abbr="id&etilde;">idem</expan> &longs;&ecedil;pè <expan abbr="cõtingit">contingit</expan>. At verò in <expan abbr="eod&etilde;">eodem</expan> adhuc libro ante <expan abbr="primã">primam</expan> pro <lb/>po&longs;itionem inquit Archimedes. <emph type="italics"/>Si conus plano &longs;ecetur cum omnibus <lb/>eius lateribus coeunti, &longs;ectio vel erit circulus, vel acutianguli coni &longs;e­<lb/>ctio.<emph.end type="italics"/> Vnde per&longs;picuum e&longs;t non in vno duntaxat cono acutian <lb/>gulo, verùm in omnibus conis&longs;ectionem ellip&longs;is cognoui&longs;&longs;e. <lb/>Præterea ex hocloquendi modo liquet ip&longs;um &longs;ectionem quo
<pb pagenum="123"/>que noui&longs;&longs;e &longs;ubcontrariam; quæ cùm &longs;it ba&longs;i &longs;ubcontrai&longs;è po <lb/>&longs;ita, <expan abbr="oĩa">oina</expan> latera coni &longs;ecat; & <expan abbr="tñ">tnm</expan> <expan abbr="nō">non</expan> e&longs;t ellip&longs;is, &longs;ed circulus.
<arrow.to.target n="marg200"></arrow.to.target> qua­<lb/>propter &longs;i in omnibus conis ellip&longs;is nouit &longs;ectionem; cur in i­<lb/>p&longs;is, & parabolas, & hyperbolas minùs animaduertit? cùm <lb/>&longs;it manife&longs;tum ex dictis in cono obtu&longs;iangulo & <expan abbr="hyperbol&etilde;">hyperbolem</expan> <lb/>& ellip&longs;im; in rectangulo autem parabolem, ellip&longs;imquè co­<lb/>gnoui&longs;&longs;e? hòc certè non e&longs;t a&longs;&longs;erendum. Ex hoc enim per&longs;pi­<lb/>cuum e&longs;t Archimedem cognoui&longs;&longs;e conos &longs;ecari po&longs;&longs;e planis, <lb/>quæ non &longs;int &longs;emper ad coni latus erecta. dormita&longs;&longs;equè Eu­<lb/>tocium Geminum, & alios &longs;ecus hac in parte de Archimede <lb/>&longs;entientes. Ampliùs <expan abbr="nõ">non</expan> ne cognouit etiam Archimedes &longs;eca­<lb/>ri po&longs;&longs;e rectangulos conoides, itidemquè & <expan abbr="obtu&longs;iãgulos">obtu&longs;iangulos</expan> pla <lb/>nis, quæ neque &longs;int per axem ducta, neque axi æquidi&longs;tantia; <lb/>neque &longs;uper axem erecta. vt in duodecima, decimatertia, & <lb/>decima quarta propo&longs;itione eiu&longs;dem libri patet. quomodo i­<lb/>taque his quoque modis quemlibet conum &longs;ecari po&longs;&longs;e igno­<lb/>rauit? Non e&longs;t igitur ambigendum Archimedem cognoui&longs;­<lb/>&longs;e conos &longs;ecari po&longs;&longs;e planis ad latus coni differentem inclina­<lb/>tionem habentibus. Ex quibus per&longs;picuum e&longs;t, ip&longs;um in om­<lb/>nibus conis omnes ine&longs;&longs;e &longs;ectiones omnino animaduerti&longs;&longs;e. <lb/>At &longs;i concedamus etiam &longs;ua tempe&longs;tate nondum &longs;ectioni­<lb/>bus ip&longs;is propria fui&longs;&longs;e impo&longs;ita nomina; tam eam parabo­<lb/>lem, quæ erat rectanguli coni &longs;ectio; quàm quæ erat &longs;ectio <lb/>alterius coni, cùm &longs;it eadem &longs;ectio, eodem nomine nuncu­<lb/>pabat; nempè rectanguli coni &longs;ectionem. Et hoc, quia <lb/>priùs hæc &longs;ectio cognita &longs;uit in cono rectangulo (vnde &longs;i­<lb/>bi nomen vindicauit) quam in alio. quod idem dicen­<lb/>dum e&longs;t de alijs &longs;ectionibus. Vt manife&longs;tum e&longs;&longs;e pote&longs;t <lb/>exemplo &longs;ectionis acutianguli coni. Archimedes enim eo­<lb/>dem loco, anteprimam &longs;cilicet propo&longs;itionem de conoidi <lb/>bus, & &longs;ph&ecedil;roidibus inquit, <emph type="italics"/>Si cylindrus duobus planis æquidi­<lb/>stantibus &longs;ecetur; quæ cum omnibus ip&longs;ius lateribus coeant, &longs;ectio­<lb/>nes, uelerunt circuli; uel conorum acutiangulorum &longs;ectiones.<emph.end type="italics"/> vo­<lb/>catigitur Archimedes acutianguli coni &longs;ectionem, tam coni <lb/><expan abbr="&longs;ection&etilde;">&longs;ectionem</expan>, quàm <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan> cylindri. veluti <expan abbr="etiã">etiam</expan> in decimatertia, <lb/>& decimaquarta propo&longs;itione <expan abbr="eiu&longs;dē">eiu&longs;dem</expan> libri <expan abbr="acutiãguli">acutianguli</expan> coni &longs;e­<lb/>ctio ab ip&longs;o ea <expan abbr="nūcupatur">nuncupatur</expan> &longs;ectio, quæ <expan abbr="oīa">oina</expan> latera tam conoidis
<pb pagenum="124"/>rectanguli, quàm obtu&longs;ianguli ab&longs;cindit. dum modo non &longs;it <lb/>ad axem erecta. nullaquè alia de cau&longs;a hæ &longs;ectiones omnes i­<lb/>dem acutianguli coni &longs;ectionis nomen obtiuerunt; ni&longs;i quia <lb/>priùs hæc &longs;ectio à cono acutiangulo nomen accepit, quan do­<lb/>quidem in ip&longs;o forta&longs;se primùm cognita fuit, quaàm in alijs. <lb/>Ex dictis itaque manife&longs;tum e&longs;t, &longs;ententiam Heraclij veram <lb/>e&longs;&longs;e po&longs;&longs;e, & rationi valdè con&longs;entaneam; Archimedem &longs;cili <lb/>cet elementa conica &longs;crip&longs;i&longs;&longs;e; Apollonium què, cùm ea ab Ar <lb/>chimede nondum edita inueni&longs;&longs;et, &longs;icut propria &longs;ua edidi&longs;&longs;e. <lb/>Omitto interim multa ab Archimede in eius libris &longs;upponi, <lb/>quæ non ni&longs;i in conicis e&longs;&longs;e dcbebant, quæ quidem <expan abbr="hab&etilde;tur">habentur</expan> <lb/>&longs;olùm in conicis Apolloni. Negandum tamen non e&longs;t, vt <lb/>Eutocius quoque affirmat, ip&longs;um Apollonium multa auxi&longs;&longs;e, <lb/>multaquè ad conica &longs;pectantia adinueni&longs;&longs;e. vt ip&longs;emet Apol­<lb/>lonius in epi&longs;tola ad Eudemum fatetur. cùm tamen non &longs;it <lb/>&longs;emperfacilè inuentis addere. Sed de his hactenus. &longs;at &longs;it au­<lb/>tem noui&longs;&longs;e, Archimedem, <expan abbr="quãdo">quando</expan> in hoclibro nominat por <lb/>tionem recta linea, rectanguliquè coni &longs;ectione contentam, <lb/>eam &longs;ignificare fectionem, quæ parabole nuncupatur. </s></p>
<pb pagenum="125"/>
<p type="margin">
<s><margin.target id="marg200"></margin.target>5. <emph type="italics"/>primi co <lb/><expan abbr="nicorũ">nicorum</expan> A­<lb/>poll.<emph.end type="italics"/></s></p>
<p type="head">
<s>GVIDIVBALDI <lb/>E MARCHIONIBVS <lb/>MONTIS.</s></p>
<p type="head">
<s>IN SECVNDVM ARCHIMEDIS <lb/>ÆQVEPONDERANTIVM <lb/>LIBRVM.</s></p>
<p type="head">
<s>PARAPHRASIS <lb/>SCHOLIIS ILLVSTRATA.</s></p>
<figure></figure>
<p type="head">
<s>PROPOSITIO. I.</s></p>
<p type="main">
<s>Si duo &longs;pacia recta linea, & re <lb/>ctanguli coni &longs;ectione conten <lb/>ta, quæ ad datam rectam <expan abbr="lineã">lineam</expan> <lb/>applicare po&longs;&longs;umus, non ha <lb/>beantidem grauitatis <expan abbr="centrũ">centrum</expan>; <lb/>magnitudinis ex vtri&longs;que i­<lb/>p&longs;orum compo&longs;itæ centrum <lb/>grauitatis erit in recta linea, quæ ip&longs;orum centra <lb/>grauitatis coniungit; ita diuidens dictam rectam li <lb/>neam, vt ip&longs;ius portiones permutatim eandem ad <lb/>inuicem proportionem habeant, vt &longs;pacia. </s></p>
<pb pagenum="126"/>
<figure></figure>
<p type="main">
<s><emph type="italics"/>Sint duo &longs;pacia AB CD, qualia dicta &longs;unt. ip&longs;orum autem centra <lb/>grauitatis &longs;int puncta EF.<emph.end type="italics"/> iungaturquè EF, quæ diuidatur in <lb/>H; <emph type="italics"/>& quam proportionem habet AB ad CD, <expan abbr="eãdem">eandem</expan> habeat FH <lb/>ad HE. o&longs;tendendum e&longs;t magnitudmis ex utri&longs;què AB CD &longs;pa­<lb/>ciis compo&longs;itæ centrum grauitaias e&longs;&longs;e punctum H. &longs;it quidemip&longs;i EH <lb/>utraque ip&longs;arum FG FK æqualis; ip&longs;i autem FH, hocest GE<emph.end type="italics"/><lb/>(&longs;untenim EH GF æquales, à quibus dempta communi <lb/>GH remanent EG HF &ecedil;quales) <emph type="italics"/>&longs;it æqualis EL.<emph.end type="italics"/> & <expan abbr="quoniã">quoniam</expan> <lb/>FH e&longs;t æqualis LE, & FK ip&longs;i EH, <emph type="italics"/>erit & LH ip&longs;i KH <lb/>æqualis.<emph.end type="italics"/> Cùm autem &longs;it FH ad HE, vt AB ad CD; ip&longs;i <lb/>verò FH vtraque &longs;it æqualis LE EG. ip&longs;i autem HE vtra­<lb/>que æqualis GF FK, <emph type="italics"/>erit <expan abbr="etiã">etiam</expan> ut LG ad G<emph.end type="italics"/>k, <emph type="italics"/>ita AB ad CD.<emph.end type="italics"/><lb/>cùm &longs;it LG ad GK, vt FH ad HE; <emph type="italics"/>aupla enim est utraque<emph.end type="italics"/><lb/>EG GK <emph type="italics"/>utriu&longs;que<emph.end type="italics"/> FH HE. <emph type="italics"/>At uerò circa punctum<emph.end type="italics"/> E <emph type="italics"/>ip&longs;ius <lb/>AB,<emph.end type="italics"/> quod e&longs;t eius cen trum grauitatis, <emph type="italics"/>ex utraque parte lineæ LG, <lb/>ip&longs;i LG æquidistantes ducantur<emph.end type="italics"/> MO QN, quæ æqualiter ab <lb/>LG di&longs;tent, ductis &longs;cilicet MQ ON æquidi&longs;tantibus, &longs;int <lb/>LM LQ GO GN inter&longs;e æquales; <emph type="italics"/>ita ut &longs;pacium MN &longs;it <lb/>&longs;pacio AB æquale<emph.end type="italics"/>: quod quidem applicatum e&longs;t ad <expan abbr="lineã">lineam</expan> LG. <lb/>
<arrow.to.target n="marg201"></arrow.to.target> <emph type="italics"/>erit utique ip&longs;ius MN centrum grauitatis punctum E.<emph.end type="italics"/> cùm &longs;it <expan abbr="pũ-ctum">pun­<lb/>ctum</expan> E in medio lineæ LG, quæ bifariam diuidit latera <lb/>oppo&longs;ita MQ ON parallelogrammi MN. <emph type="italics"/>compleatur ita­<lb/>que &longs;pacium NX. habebit quidem MN. ad NX proportionem,<emph.end type="italics"/>
<pb pagenum="127"/><emph type="italics"/>quam<emph.end type="italics"/> habet QN ad NP, hoce&longs;t <emph type="italics"/>LG ad GK. habet autem & <lb/>AB ad CD proportionem ip&longs;ius LG ad G<emph.end type="italics"/>K. <emph type="italics"/>ut igitur AB ad<emph.end type="italics"/>
<arrow.to.target n="marg202"></arrow.to.target><lb/><emph type="italics"/>CD, &longs;ic est MN ad NX. & permutando<emph.end type="italics"/> vt AB ad MN, ita <lb/>CD ad NX. <emph type="italics"/>æquale autem est AB ip&longs;i MN, erit igitur & CD <lb/>ip&longs;i NX æquale. Centrum autem grauitatisip&longs;ius<emph.end type="italics"/> NX <emph type="italics"/>est <expan abbr="punotũ">punotum</expan> <lb/>F.<emph.end type="italics"/> propterea quod e&longs;t in medio lineæ GK, quæ
<arrow.to.target n="marg203"></arrow.to.target> parallelo­<lb/>grammi NX oppo&longs;ita latera ON XP bifariam &longs;ecat. <emph type="italics"/>& <lb/>quoniam æqualis e&longs;t LH ip&longs;i HK, totaquè LK appa&longs;ita latera<emph.end type="italics"/> MQ <lb/>XP <emph type="italics"/>bifariam diuidit, totius PM <expan abbr="centrũ">centrum</expan> grauitatis erit punctum Hr <lb/>Verùm ip&longs;um MP æquale est utri&longs;que MN NX,<emph.end type="italics"/> quorum, cùm <lb/>&longs;int centra grauitatis EF, æquepondera bunt &longs;pacia MN <lb/>NX ex di&longs;tantijs FH HE. &longs;i igitur loco parallelo gram mo­<lb/>rum MN NX ponatur AB in E, & CD in F, cùm &longs;it <lb/>AB ip&longs;i MN, & CD ip&longs;i NX æquale; &longs;pacia AB CD ex
<arrow.to.target n="marg204"></arrow.to.target><lb/>di&longs;tantijs FH HE æqueponderabunt. <emph type="italics"/>ac propterea magnitudi <lb/>nis ex utri&longs;que AB CD<emph.end type="italics"/> compo&longs;itæ <emph type="italics"/>centrum grauitatis <expan abbr="e&longs;tpunctũ">e&longs;tpunctum</expan> <lb/>H.<emph.end type="italics"/> quod quidem propo&longs;itum fuit. </s></p>
<p type="margin">
<s><margin.target id="marg201"></margin.target>2. <emph type="italics"/>cor.<emph.end type="italics"/> 9. <lb/><emph type="italics"/>primihui<emph.end type="italics"/>^{9}.</s></p>
<p type="margin">
<s><margin.target id="marg202"></margin.target>16.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg203"></margin.target>2.<emph type="italics"/>cor.<emph.end type="italics"/> 9. <lb/><emph type="italics"/>primihui<emph.end type="italics"/>^{9}.</s></p>
<p type="margin">
<s><margin.target id="marg204"></margin.target>8.<emph type="italics"/>po&longs;thui<emph.end type="italics"/>^{9}</s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Cùm &longs;it intentio Archimedis non nulla pertractare ad pa­<lb/>rabolen &longs;pectantia; primùm iacit fundamentum, parabolas <lb/>nempe ita &longs;e habere, vt permutatim di&longs;tantiæ, ex quibus <lb/>&longs;untcollocatæ, &longs;e habent. & <expan abbr="quãuis">quanuis</expan> vniuer&longs;im, atquè in om­<lb/>nibus mutuam hanc conuenientiam ex dictis ex primo libro <lb/>depræhendere liceat, hoc tamen loco peculiariter voluitad <lb/>huberiorem do ctrinam id ip&longs;um in parabolis demon&longs;trare.
<arrow.to.target n="marg205"></arrow.to.target><lb/>& quamuis in primo libro dixerit Archimedes magnitudi­<lb/>nes æqueponderare, quan do ita &longs;e habentinter&longs;e, ut di&longs;tan­<lb/>tiæ permutatim &longs;e habent; hocautem loco quærit <expan abbr="centrũ">centrum</expan> gra <lb/>uitatis magnitudinis ex parabolis compo&longs;itæ; non &longs;unt <expan abbr="tam&etilde;">tamen</expan> <lb/>propo&longs;itiones diuer&longs;æ. nam & in primo libro dum in demon <lb/>&longs;tratio ne quærit proportionem di&longs;tantiarum, o&longs;tendit, vbi <lb/>nam &longs;it centrum grauitatis magnitudinum. quare <expan abbr="quãnis">quannis</expan> pro <lb/>po&longs;itiones videantur diuer&longs;æ, non &longs;unt tamen diuer&longs;æ, ete­<lb/>nim vt po&longs;t tertiam primi libri propo&longs;itionem adnotauimus,
<pb pagenum="128"/>hæc planè&longs;e con&longs;equuntur, vt exempli gratia in figura pun­<lb/>ctum H centrum e&longs;t grauitatis magnitudinis ex vtri&longs;que <lb/>AB CD compo&longs;itæ. ergo AB, & CD ex di&longs;tantijs HEHF <lb/>æqueponderant. & è contra. hoc e&longs;t AB CD æqueponde­<lb/>rant ex di&longs;tantijs EH HF. ergo punctum H centrum e&longs;t <lb/>grauitatis magnitudinis ex vtri&longs;que AB CD compo&longs;rtæ; <expan abbr="cũ">cum</expan> <lb/>&longs;it EHF recta linea. Solent autem mathematici aliquan do <lb/>eandem propo&longs;itionem pluribusmedijs demon&longs;trare; idcirco <lb/>con&longs;iderandum e&longs;t, Archimedem in hac propo&longs;itione alio v­<lb/>ti medio ad o&longs;tendendum punctum H centrum e&longs;ie graui­<lb/>tatis, quo u&longs;us e&longs;t in &longs;exta propo&longs;itione primi libri. cùm in pri <lb/>mo libro per diui&longs;ionem magnitudinum, diui&longs;io nem què di <lb/>&longs;tantiarum vniuer&longs;aliter domon&longs;tret centrum grauitatis ma­<lb/>gnitudinum. hoc autem loco per parallelogramma MN <lb/>NX parabolis æqualia, & circa centra grauitatis EF con&longs;ti­<lb/>tuta, in uenit centrum grauitatis magnitudinis ex vtri&longs;que pa <lb/>
<arrow.to.target n="marg206"></arrow.to.target> rallelogrammis MN NX compo&longs;itæ. quod e&longs;t <expan abbr="quid&etilde;">quidem</expan> pun­<lb/>ctum H. medium nempè totius parallelogrammi MP. <lb/>quod idem punctum H centrum e&longs;t grauitatis vtriu&longs;que pa <lb/>raboles AB CD in EF collocatæ. </s></p>
<p type="margin">
<s><margin.target id="marg205"></margin.target>6.7.<emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg206"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 9.<emph type="italics"/>&<emph.end type="italics"/> 10 <lb/><emph type="italics"/>primihui<emph.end type="italics"/>^{9}.</s></p>
<p type="main">
<s>Ex his ob&longs;eruandum occurrit, hanc e&longs;&longs;e peculiarem metho <lb/>dum, qua po&longs;&longs;umus quorumlibet planorum æquepondera­<lb/>tionem o&longs;tendere; hoc e&longs;t plana ex di&longs;tantijs eandem permu <lb/>tatim proportionem habentibus, vt eadem met plana, æque­<lb/>ponderare; dum modo ip&longs;is æqualia parallelogramma con&longs;ti <lb/>tuere po&longs;&longs;imus. ac propterea &longs;upponit Archimedes, nos po&longs;&longs;e <lb/>applicare ad rectam lineam &longs;pacium æquale &longs;pacio recta li­<lb/>nea, rcctanguliquè coni &longs;ectione contento. quod <expan abbr="quid&etilde;">quidem</expan> &longs;pa­<lb/>cium &longs;upponit parallelogram mum exi&longs;tere, cùm pun­<lb/>ctum E centrum &longs;it grauitatis &longs;pacij MN, e&longs;t F <lb/>&longs;pacij NX. punctum verò H totius PM. quòd &longs;i MN <lb/>NX & MP non e&longs;&longs;ent parallelogramma, neque puncta EFH <lb/>eorum centra grauitatis exi&longs;terent. vt ex demon&longs;tranone pa­<lb/>tet. &longs;uppo&longs;uit tamen Archimedes nos po&longs;&longs;e applicare ad re­<lb/>ctam lineam parallelogrammum æquale &longs;pacio recta linea, <lb/>rectanguliquè coni&longs;ectione contento; quia duplici medio in
<pb pagenum="129"/>libro de quadratura paraboles, propo&longs;itione &longs;cilicet decima&longs;e <lb/>ptima, & vige&longs;imaquarta, docuit quamlibet portionem recta <lb/>linea, rectanguliquè coni &longs;ectione contentam &longs;e&longs;quitertiam <lb/>e&longs;&longs;e trianguli eandem ip&longs;i ba&longs;im habentis, & <expan abbr="altitudin&etilde;">altitudinem</expan> &ecedil;qua <lb/>lem. Ex qua propo&longs;itione facilè con&longs;tat nos parabol&ecedil; <expan abbr="&longs;paciū">&longs;pacium</expan> <lb/>ad rectam lineam applicare po&longs;&longs;e, vt propo&longs;itum fuit hoc <lb/>modo. </s></p>
<p type="head">
<s>PROBLEMA.</s></p>
<p type="main">
<s>Ad datam rectam lineam dat&ecedil; parabol&ecedil; &ecedil;quale parallelo­<lb/>grammum applicare, ita vt data linea oppo&longs;ita <expan abbr="parallelogrã-mi">parallelogran­<lb/>mi</expan> latera bi&longs;ariam diuidat. </s></p>
<figure></figure>
<p type="main">
<s>Data &longs;it parabole <lb/>ABC, &longs;itquè data recta <lb/>linea GK. oportet ad <lb/>GK <expan abbr="parallelogrãmum">parallelogrammum</expan> <lb/>applicare æquale por­<lb/>tioni ABC, ita vt GK <lb/>bifariam diuidat oppo <lb/>&longs;ita parallelogram mi <lb/>latera. Con&longs;tituatur &longs;u <lb/>per AC <expan abbr="triãgulũ">triangulum</expan> ABC, <lb/>qd ba&longs;im habeat AC, <lb/>eandemque portionis <lb/><expan abbr="altitudin&etilde;">altitudinem</expan>; quod <expan abbr="quid&etilde;">quidem</expan> <lb/>fiet, <expan abbr="inu&etilde;ta">inuenta</expan> diametro DB, quæ parabolen in B &longs;ecet, <expan abbr="iuncti&longs;&qacute;">iuncti&longs;que</expan>;
<arrow.to.target n="marg207"></arrow.to.target><lb/>AB BC. eritvtique parabole ABC trianguli ABC &longs;e&longs;quitertia. <lb/>Itaque diuidatur AC in tria &ecedil;qualia, quarum vna pars &longs;it CH.
<arrow.to.target n="marg208"></arrow.to.target><lb/>producaturquè AC. fiatquè CL ip&longs;i CH &ecedil;qualis<gap/> erit &longs;anè AL <lb/>ip&longs;ius AC &longs;e&longs;q uitertia. Et obid (iuncta BL) erit triangulum <lb/>ABL trianguli ABC &longs;e&longs;quitertium. &longs;unt quippè triangula ABL
<arrow.to.target n="marg209"></arrow.to.target><lb/>ABC inter&longs;e, vt ba&longs;es AL AC. ac per con&longs;equens triangulum <lb/>ABL patabol&ecedil; ABC exi&longs;tit &ecedil;quale. Applicetur itaque ad linea
<arrow.to.target n="marg210"></arrow.to.target><lb/>GK <expan abbr="parallelogrãmũ">parallelogrammum</expan> GS &ecedil;quale <expan abbr="triãgulo">triangulo</expan> ABL. erit GS parabo­
<pb pagenum="132"/>
<arrow.to.target n="fig65"></arrow.to.target><lb/>læ ABC &ecedil;quale. deinceps ducatur NP ip&longs;i GK <lb/>&ecedil;quidi&longs;tans, qu&ecedil; bifariam diuidat oppo&longs;ita latera GR <lb/>KS. producanturquè RG SK. fiantquè GO KX &ecedil;­<lb/>quales ip&longs;is GN KP. iungaturquè OX; erit nimi-­<lb/>rum parallelogram mum OP ip&longs;i GS &ecedil;quale. qua­<lb/>re parallelogram mum OP parabol&ecedil; ABC exi&longs;tit &ecedil;­<lb/>quale. Applicatum e&longs;t igitur ad GK parallelogram­<lb/>mum expo&longs;it&ecedil; parabol&ecedil; &ecedil;quale. lineaquè GK paralle­<lb/>logrammi OP bifariam diuidit oppo&longs;ita latera ON <lb/>XP. quod fieri oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg207"></margin.target>44. <emph type="italics"/><expan abbr="&longs;ecũdi">&longs;ecundi</expan> <lb/>conicorum <lb/>Apoll.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg208"></margin.target>17. 24. <emph type="italics"/>Ar <lb/>ch. dquad. <lb/>patab.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg209"></margin.target>1.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg210"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 44.<emph type="italics"/>pri­<lb/>mi.<emph.end type="italics"/></s></p>
<figure id="fig65"></figure>
<p type="main">
<s>Si in portione recta linea rectanguliquè coni <lb/>&longs;ectione contenta triangulum in&longs;cribatur, <expan abbr="eand&etilde;">eandem</expan> <lb/>ba&longs;im cum portione habens, & altitudinem æqua <lb/>lem: & rur&longs;us in reliquis portionibus triangula in­<lb/>&longs;cribantur, quæ ea&longs;dem ba&longs;es cum portionibus <lb/>habeant, & altitudinem æqualem; &longs;emper què in <lb/>re&longs;iduis portionibus triangula eodem modo <lb/>in&longs;cribantur: figura, quæ in portione oritur, <lb/>planè in&longs;cribi dicatur. Patet quidem lineas
<pb pagenum="131"/>huius figuræ in&longs;criptæ angulos, qui &longs;unt vertici <lb/>portionis proximi, eo&longs;què deinceps coniungen­<lb/>tes, ba&longs;i portionis æquidi&longs;tantes e&longs;&longs;e; bifariamquè <lb/>à diametro portionis diuidi; diametrum verò in <lb/>proportione diuidere numeris deinceps impari­<lb/>bus. vno deno minato ad verticem portionis. Hoc <lb/>autem ordinate o&longs;ten&longs;um e&longs;t. </s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Scopus Archimedis in hoc &longs;ecundo libio, vt initio primi <lb/>diximus, e&longs;t inuenire centrum grauitatis paraboles. & vt de­<lb/>ducatnosin hanc cognitionem, quadam vtitur figura rectili­<lb/>nea in parabole in&longs;cripta, qu&ecedil; plurimùm conducit, & e&longs;t <expan abbr="tã">tam</expan> <lb/>quam medium ad inueniendum hoc grauitatis centrum. his <lb/>igitur verbis docet, quo modo in parabole in &longs;criben da &longs;it h&ecedil;c <lb/>figura; in quibus multa quo que proponit tanquam &longs;it pro­<lb/>po&longs;itio quædam; in qua multa &longs;int o&longs;ten denda. quorum ta­<lb/>m&ecedil;n demon&longs;trationem omi&longs;it, ac tanquam ab eo alibi de­<lb/>mon&longs;tratam. Horum autem ex Apollonij Perg&ecedil;i conicis <lb/>demon&longs;trationem elicere quidem potui&longs;&longs;emus. at quoniam <lb/>Archimedes ip&longs;e non nulla ad hæ c&longs;pectantia alijs in locis de­<lb/>mon&longs;trauit ideo Archimedem per Archimedem declarare o­<lb/>portunum magis nobis vi&longs;um e&longs;t. </s></p>
<p type="main">
<s>Sit portio contenta recta linea, rectanguliquè coni &longs;ectio­<lb/>ne ABC, cuius diameter BD. Iunganturquè AB BC, diuida­<lb/>tur deinde AB bifariam in E, a quo ip&longs;i BD æquidi&longs;tans
<pb pagenum="132"/>ducatur EF, eritvti que punctum F vertex portionis AFB. <lb/>vt Archimedes demon&longs;trauit in libro de quadratura parabo­<lb/>les propo&longs;itione decimaoctaua. iunganturque AF FB. rur <lb/>fus bifariam diuidantur AF FB in punctis GH, à quibus <lb/>ip&longs;i BD ducantur æquidi&longs;tantes GI HK <gap/>b eandem cau­<lb/>&longs;am erit punctum I vertex portionis AIF. K verò portio­<lb/>nis FKB. connectanturquè AI IF FK KB. eademquè pror <lb/>fus ratione ad alteram partem in&longs;cribantur triangula CLB <lb/>
<arrow.to.target n="fig66"></arrow.to.target><lb/>CML, & LNB. Primùm <expan abbr="quid&etilde;">quidem</expan> triangulum ABC dicitur <lb/>planè in&longs;criptum, vt Archimedes ip&longs;e infra in demon&longs;tratio­<lb/>nibus quintæ, &longs;extæ, & octauæ propo&longs;itionis nominat. Dein <lb/>de figura AFBLC, figuraquè AIFKBNLMC dicuntur in <lb/>portione planè in&longs;criptæ. figuraquè AFBLC vna cum AC <lb/><expan abbr="pentagonũ">pentagonum</expan> in portione planè <expan abbr="in&longs;criptũ">in&longs;criptum</expan> dici <expan abbr="põt">pont</expan>. vt Archime <lb/>des in &longs;ecunda parte demon&longs;trationis quintæ propo&longs;itionis <lb/>huius libri nuncupat. ideòquè erit AIFKBNLMC nonago­<lb/>num in portione planè in&longs;criptum. & ita in alijs. <expan abbr="Connectã">Connectam</expan>
<pb pagenum="133"/>tur KN FL IM, quæ diametrum BD &longs;ecent in punctis <lb/>STV. o&longs;tendendum e&longs;t, lineas KN FL IM ba&longs;i AC &ecedil;qui <lb/>di&longs;tantes e&longs;&longs;e. deinde diametrum BD lineas KN FL IM <lb/>bifariam in punctis STV diuidere po&longs;tremo lineas KN F<gap/><lb/>IM ita diametrum BD di&longs;pe&longs;cere, vt po&longs;ito vno BS, linea ST <lb/>&longs;it tria, TV quinque; & VD &longs;eptem. Producantur FE KH <lb/>ad RX. quoniam enim FR e&longs;t æquid<gap/>tans BD, erit AE ad
<arrow.to.target n="marg211"></arrow.to.target><lb/>EB, vt AR ad RD; e&longs;tque AE ip&longs;i EB æqualis ergo AR i­<lb/>p&longs;i RD æqualis exi&longs;tit. eodem què modo o&longs;tendetur FX æ­<lb/>qualem e&longs;&longs;e XT. quandoquidem e&longs;t FX ad XT, vt FH ad <lb/>HB. &longs;imiliterquè ad alteram partem, exi&longs;tentibus LO NP i­<lb/>p&longs;i BD æquidi&longs;tantibus, erit DO ip&longs;i OC æqualis, & TP <lb/>ip&longs;i PL. quod quidem eodem pror&longs;us modo demon&longs;trabi­<lb/>tur. Quoniam autem AC bifariam à diametro diuiditur in <lb/>puncto D, erit DR ip&longs;i DO æqualis, cùm vnaquæque &longs;it <lb/>dimidia ip&longs;arum AD DC æqualium. e&longs;t igitur RD dimidia <lb/>ip&longs;ius AD, quæ dimidia e&longs;t ba&longs;is AC. quod idem euenit ip&longs;i <lb/>DO. quare BD &longs;e&longs;quitertia e&longs;t ip&longs;ius FR, & ip&longs;ius LO, ex de­<lb/>cimanona Archimedis de quadratura paraboles. ac propterea <lb/>eandem habet proportionem BD ad FR, quam ad LO. vnde
<arrow.to.target n="marg212"></arrow.to.target><lb/>&longs;equitur FR æqualem e&longs;&longs;e ip&longs;i LO. & obid FL ip&longs;i AC <expan abbr="æ-quidi&longs;tant&etilde;">æ­<lb/>quidi&longs;tantem</expan> e&longs;&longs;e. & FT ip&longs;i RD, & TL ip&longs;i DO &ecedil;qualem. <lb/>vnde FT ip&longs;i TL &ecedil;qualis exi&longs;tit. eadem quèratione pror&longs;us in <lb/>portione FBL o&longs;tendetur KN ip&longs;i FL, ac per con&longs;equens i­<lb/>p&longs;i AC &ecedil;quidi&longs;tantem e&longs;&longs;e. & KS ip&longs;i SN æqualem exi&longs;te­<lb/>re. Producatur IG ad Z, quæ ip&longs;am AB &longs;ecet in 9. linea ve­<lb/>rò LO &longs;ecet BC in <expan abbr="q;">que</expan> ductaquè MY ip&longs;i BD æquidi&longs;tans <lb/>ip&longs;am &longs;ecet BC in <foreign lang="greek">a</foreign>. & quoniam IZ e&longs;t æquidi&longs;tans FR, e­<lb/>rit AG ad GF, ut A9 ad 9E, & AZ ad ZR. & e&longs;t AG ip&longs;i
<arrow.to.target n="marg213"></arrow.to.target><lb/>GF æqualis, erit igitur A9 ip&longs;i 9E, & AZ ip&longs;i ZR æquaiis. <lb/>Eodemquè modo o&longs;tendetur C<foreign lang="greek">a</foreign> ip&longs;i <foreign lang="greek">a</foreign>Q, & CY ip&longs;i YO &ecedil;­<lb/>qualem e&longs;&longs;e. quo niam autem in portione AFB a dimidia ba&longs;i <lb/>ducta e&longs;t LF, à pun cto autem 9, hoc e&longs;t à dimidia dimidi&ecedil; ba <lb/>&longs;is AB (e&longs;t enim E9 dimidia ip&longs;ius AE, quæ dimidia e&longs;t ba&longs;is <lb/>AB) ducta e&longs;t 9I diametro æquidi&longs;tans, erit EF &longs;e&longs;quitertiai­<lb/>p&longs;ius I9 parique ratione o&longs;tendetur QL &longs;e&longs;quitereiam e&longs;&longs;e i­<lb/>p&longs;ius M<foreign lang="greek">a</foreign> quare vt FE ad I9, ita LQ ad M<foreign lang="greek">a</foreign>. ob&longs;imilitudinem
<pb pagenum="134"/>
<arrow.to.target n="marg214"></arrow.to.target> autem triangulorum ABD AER ita e&longs;t BD ad ER, vt DA <lb/>ad AR. eademqueiatione ita &longs;ehabet BD ad QO, vt DC <lb/>ad CO. Sed vt DA ad AR, ita e&longs;t DC ad CO, e&longs;t quip <lb/>pe DA ip&longs;ius AR dupla, veluti DC ip&longs;ius CO. quare i­<lb/>
<arrow.to.target n="marg215"></arrow.to.target> ta erit BD ad ER, vt BD ad QO. ac propterea ER ip&longs;i <lb/>
<arrow.to.target n="marg216"></arrow.to.target> QO &ecedil;qualis exi&longs;tit. o&longs;ten&longs;a verò e&longs;t RF &ecedil;qualis OL, reli­<lb/>quaigitur EF reliquæ QL e&longs;t æqualis, quia verò ita e&longs;t FE <lb/>
<arrow.to.target n="marg217"></arrow.to.target> ad I9, vt QL ad M<foreign lang="greek">a</foreign>, erit permutando FE ad QL, vt I9 <lb/>
<arrow.to.target n="fig67"></arrow.to.target><lb/>ad M<foreign lang="greek">a</foreign>. &longs;untquè FE QL &ecedil;quales, ergo I9 ip&longs;i M<foreign lang="greek">a</foreign> &ecedil;qua­<lb/>lis exi&longs;tit. quoniam autem ob triangu&longs;oium &longs;imilitudinem <lb/>AER A9Z, ita e&longs;t AR ad AZ, vt ER ad 9Z. ob &longs;imili­<lb/>tudinem vero triangulorum QOC <foreign lang="greek">a</foreign>YC ita e&longs;t CO ad CY, <lb/>vt QO ad <foreign lang="greek">a</foreign>Y: & e&longs;t RA ad AZ, vt OC ad CY, cùm <lb/>
<arrow.to.target n="marg218"></arrow.to.target> vtr&ecedil;que in dupla exi&longs;tant proportione; e<gap/>t ER ad 9Z, vt <lb/>QO ad <foreign lang="greek">a</foreign>Y. & permutando ER ad QO vt 9Z ad <foreign lang="greek">a</foreign>Y. e&longs;t <lb/>vero ER ip&longs;i QO, æqualis, ergo 9Z ip&longs;i <foreign lang="greek">a</foreign>Y &ecedil;qualis exi&longs;tit. at <lb/>vero o&longs;ten&longs;a e&longs;t I9 &ecedil;qualis M<foreign lang="greek">a</foreign>; to ta igitur IZ ip&longs;i MY e&longs;t &ecedil;-
<pb pagenum="135"/>æqualis, quæ cùm &longs;intip&longs;i BD æquidi&longs;tantes, erunt & inter&longs;e­<lb/>&longs;e parallelæ. quare IM ip&longs;i AC e&longs;t æquidi&longs;tans. Quoniam
<arrow.to.target n="marg219"></arrow.to.target> ita­<lb/>que AR e&longs;t æqualis CO, & horum dimidia, hoc e&longs;t RZ ip&longs;i <lb/>OY æqualis erit. atqui DR e&longs;t ip&longs;i DO æqualis; ergo DZ ip&longs;i <lb/>DY exi&longs;tit æqualis. ip&longs;i verò DZ e&longs;t æqualis IV, & ip&longs;i DY æ­<lb/>qualis VM. eruntigitur IV VM inter&longs;e equales. Iam itaque
<arrow.to.target n="marg220"></arrow.to.target><lb/>o&longs;ten&longs;um e&longs;t, lineas KN FL IM, qu&ecedil; coniunguntangulos fi <lb/>guræ in parabole planè in&longs;criptæ, ip&longs;i AC æquidi&longs;tantes e&longs;&longs;e. <lb/>Diametrum què BD ip&longs;as in punctis STV bifariam di&longs;pe&longs;cere. </s></p>
<p type="margin">
<s><margin.target id="marg211"></margin.target>2. <emph type="italics"/>lemma.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg212"></margin.target>9. <emph type="italics"/>quinti. <lb/>ex<emph.end type="italics"/> 33.34 <lb/><emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg213"></margin.target>2.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg214"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 4.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg215"></margin.target>11. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg216"></margin.target>9. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg217"></margin.target>16. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg218"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 11.<emph type="italics"/>quin <lb/>ti<emph.end type="italics"/> 16.<emph type="italics"/>qu<gap/>u<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg219"></margin.target>33.<emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg220"></margin.target>34.<emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<figure id="fig66"></figure>
<figure id="fig67"></figure>
<p type="main">
<s>Quoniam itaque in portione FBL à dimidia ba&longs;i ducta e&longs;t <lb/>TB, a dimidia verò dimidiæ ba&longs;is ducta e&longs;t XK, erit BT
<arrow.to.target n="marg221"></arrow.to.target> &longs;e&longs;­<lb/>quitertia ip&longs;ius KX, hoc e&longs;t ip&longs;ius ST. e&longs;t enim KT parallelo­<lb/>grammum, & ST ip&longs;i KX æqualis. Si igitur ponatur BT <lb/>quattuor, erit ST tria, & BS vnum. &longs;imiliter quoniam BD
<arrow.to.target n="marg222"></arrow.to.target><lb/>&longs;e&longs;quitertia e&longs;t ip&longs;ius FR, hoc e&longs;t ip&longs;ius TD, cùm &longs;it TD ip&longs;i <lb/>FR &ecedil;qualis. &longs;i ita que ponatur BD &longs;exdecim, erit vnaquæque <lb/>FR TD duodecim. & TB quattuor, vt po&longs;itum fuit. <expan abbr="Quoniã">Quoniam</expan> <lb/>autem (vt diximus) e&longs;t BD ad ER, vt DA ad AR, erit BD du­<lb/>pla ip&longs;ius RE. quare &longs;i BD e&longs;t &longs;exdecim, erit RE octo. & quo­<lb/>niam e&longs;t FR duodecim, erit EF quatuor. e&longs;t autem FE ip&longs;ius <lb/>I9 &longs;e&longs;quitertia, erit igitur I9 tria. & quoniam e&longs;t ER ad 9Z, vt <lb/>RA ad AZ, erit ER dupla ip&longs;ius 9Z. ac propterea erit 9Z quat <lb/>tuor, cum &longs;it ER octo, & e&longs;t 9I tria, tota ergo IZ, hoc e&longs;t DV, <lb/>&longs;eptem exi&longs;tet. &longs;ed quoniam e&longs;t DT duodecim, cuius pars <lb/>DV e&longs;t &longs;eptem, eritreliqua VT quinque. Po&longs;ito igitur BS v­<lb/>no, erit ST tria, TV quinque, & VD &longs;eptem. quod erat quo­<lb/>que demon&longs;trandum. Et hæc &longs;unt qu&ecedil; ab Archimede pro­<lb/>po&longs;ita fucrant. </s></p>
<p type="margin">
<s><margin.target id="marg221"></margin.target>19.<emph type="italics"/>Archi­<lb/>medis de <lb/>quad. pa­<lb/>rab.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg222"></margin.target>34. <emph type="italics"/>primi.<emph.end type="italics"/></s></p>
<p type="main">
<s>Ex his tamen nonnulla quoque colligemus ad ea, quæ &longs;e­<lb/>quuntur nece&longs;&longs;aria. ac primùm quidem con&longs;tat BD quadru­<lb/>plam e&longs;&longs;e ip&longs;ius BT, & ip&longs;ius FE. </s></p>
<pb pagenum="136"/>
<p type="main">
<s>O&longs;ten&longs;um e&longs;t enim BD &longs;exdecim e&longs;&longs;e, & BT quatuor, & FE <lb/>itidem quatuor exi&longs;tere. Ex demon&longs;tratione autem Archime <lb/>dis decimæ nonæ ptopo&longs;itionis de quadratura paraboles cla­<lb/>rè elicitur BD quadruplam e&longs;&longs;e ip&longs;ius BT. </s></p>
<p type="main">
<s>Ex quibus etiam &longs;equitur FE QL inter&longs;e æquales e&longs;&longs;e. am­<lb/>bo enim &longs;unt, vt quatuor. </s></p>
<figure></figure>
<p type="main">
<s>Præterea o&longs;tendendum e&longs;t triangulum AFB <expan abbr="triãgulo">triangulo</expan> BLC <lb/>&ecedil;quale e&longs;&longs;e, portionem què paraboles AFB portiom BLC &ecedil;qua <lb/>lem. Ampliùs triangulum AIF triangulo CML, & portio­<lb/>nem AIF portioni CML æqualem e&longs;&longs;e, & reliqua triangula <lb/>reliquis triangulis, acportiones portionibus &ecedil;quales e&longs;&longs;e. </s></p>
<p type="main">
<s>Ex vige&longs;ima prima propo&longs;itione Archimedis de quadratu­<lb/>ra paraboles triangulum ABC vniu&longs;cuiu&longs;que trianguli AFB <lb/>
<arrow.to.target n="marg223"></arrow.to.target> BLC e&longs;t <expan abbr="octuplũ">octuplum</expan>. ergo ad ambo <expan abbr="eand&etilde;">eandem</expan> <expan abbr="h&etilde;t">hent</expan> <expan abbr="proportion&etilde;">proportionem</expan>. qua <lb/>re triangula AFB BLC inter&longs;e &longs;unt &ecedil;qualia. At vero <expan abbr="quoniã">quoniam</expan>
<pb pagenum="137"/>portio AFB trianguli AFB e&longs;t &longs;e&longs;quitertia, quemadmodum
<arrow.to.target n="marg224"></arrow.to.target><lb/>portio BLC trianguli BLC, eritportio AFB ad triangulum <lb/>AFB, vt portio CLB ad triangulum CLB, & permutando <lb/>portio AFB ad portionem CLB, vt triangulum AFB ad
<arrow.to.target n="marg225"></arrow.to.target><lb/>ip&longs;um CLB <expan abbr="triãgula">triangula</expan> verò &longs;unt æqualia; ergo portiones AFB <lb/>CLB inter&longs;e &longs;unt æquales. Eademquè ratione <expan abbr="triangulũ">triangulum</expan> AFB <lb/>octuplum e&longs;t trianguli AIF, & triangulum CLB octuplum <lb/>ip&longs;ius CML. vnde triangula AIF CML &longs;unt æqualia. et ea­<lb/>rum quoque portiones AIF CML &longs;unt æquales, &longs;iquidem <lb/>&longs;unt triangulorum &longs;e&longs;quitertiæ. Et hoc modo reliqua trian­<lb/>gula FKB LNB, & portiones FKB LNB <expan abbr="o&longs;tend&etilde;tur">o&longs;tendentur</expan> æqua­<lb/>les. cùm &longs;it triangulum FBL dictorum triangulorum octu­<lb/>plum. quod oportebat quoque demon&longs;trate. </s></p>
<p type="margin">
<s><margin.target id="marg223"></margin.target>9. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg224"></margin.target>17.24. A<emph type="italics"/>r <lb/>chimedis <lb/>de quad. <lb/>parab.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg225"></margin.target>16. <emph type="italics"/>quimi<emph.end type="italics"/><lb/>21.<emph type="italics"/>Archi­<lb/>medis de <lb/>quad. pa­<lb/>rab.<emph.end type="italics"/></s></p>
<p type="main">
<s>His demon&longs;tratis &longs;equitur Archimedes qua&longs;i connectens &longs;e <lb/>quentem propo&longs;itionem cumijs, quæ &longs;uppo&longs;ita &longs;unt, inqui­<lb/>ens, <emph type="italics"/>&longs;i autem & in portione<emph.end type="italics"/> &c. </s></p>
<p type="head">
<s>PROPOSITIO. II.</s></p>
<p type="main">
<s>Si autem & in portione rectalinea, rectangu­<lb/>lique coni &longs;ectione contenta, figura rectilinea pla <lb/>ne in&longs;cribatur, in&longs;criptæ figuræ centrum grauita­<lb/>tis erit in diametro portionis. </s></p>
<pb pagenum="138"/>
<p type="main">
<s><emph type="italics"/>Sit portio ABC, qualis dicta e&longs;t, & in ip&longs;a planè in&longs;cribatur recti­<lb/>linea figura AEFGBHIKC. portionis verò diameter &longs;it BD. <expan abbr="o&longs;t&etilde;-">o&longs;ten-</expan><emph.end type="italics"/><lb/>
<arrow.to.target n="marg226"></arrow.to.target> <emph type="italics"/>dendum e&longs;t, rectilineæ figuræ centrum grauitatise&longs;&longs;e in linea BD.<emph.end type="italics"/> <expan abbr="iũ">ium</expan> <lb/>gantur GH FI EK. qu&ecedil; ip&longs;i AC, & inter&longs;e &ecedil;quidi&longs;tantes <lb/>erunt. h&ecedil; verò lineæ diametrum BD &longs;ecentin punctis NML <lb/>
<arrow.to.target n="fig68"></arrow.to.target><lb/><emph type="italics"/>Quoniam enim<emph.end type="italics"/> lineæ GH FI EK bifariam &longs;unt à diame­<lb/>tro BD diui&longs;æ in punctis NML, trapezium AEKC duas <lb/>
<arrow.to.target n="marg227"></arrow.to.target> habebit line as æ quidi&longs;tantes AC EK, quas bifariam diuidit <lb/>DL, quare <emph type="italics"/>trapezii AEKC centrum grauitatis est in LD. at<emph.end type="italics"/> ob <lb/>eandem cau&longs;am <emph type="italics"/>trapezii EFIK centrum est in ML; trapezii verò <lb/>FGHI centrum est in MN.<emph.end type="italics"/> lineæ enim LM MN bifariam <lb/>
<arrow.to.target n="marg228"></arrow.to.target> diuidunt parallela latera EK FI GH, <emph type="italics"/>&longs;ed & trianguli etiam <lb/>GBH centrum grauitatis e&longs;t in BN.<emph.end type="italics"/> quippè cùm BN ip&longs;am <lb/>GH bifariam diuidat. <emph type="italics"/>per&longs;picuum e&longs;t totius rectilineæ figuræ<emph.end type="italics"/><lb/>AEFGBHIKC <emph type="italics"/>centrum grauitatis e&longs;&longs;e in linea BD.<emph.end type="italics"/> quod de­<lb/>mon&longs;trare oportebat. </s></p>
<pb pagenum="139"/>
<p type="margin">
<s><margin.target id="marg226"></margin.target><emph type="italics"/><expan abbr="exdemõ">exdemom</expan> <lb/>stratis.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg227"></margin.target>15. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg228"></margin.target>13. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s></p>
<figure id="fig68"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Ecce qúo Archimedes incipit inue&longs;tigare centrum graui <lb/>tatis paraboles. nam ex hoc, quod o&longs;tendit centrum grauita­<lb/>tis figuræ in portione planè in&longs;criptæ e&longs;&longs;e in diametro por­<lb/>tionis, &longs;tatim colliget in quarta propo&longs;itione centrum graui­<lb/>tatis paraboles in diametro quoque ip&longs;ius portionis exi&longs;tere. <lb/>interponitautem Archimedes &longs;equentem propo&longs;itionem. <expan abbr="nã">nam</expan> <lb/>antequam inueniat centrum grauitatis paraboles, opus habet <lb/>prius o&longs;tendere centra grauitatis duarum, & vt ita dicam om <lb/>nium parabol<gap/>rum diametros in ea dem proportione &longs;ecare. <lb/>ad quod demon&longs;trandum, hanc <expan abbr="pa&longs;&longs;ion&etilde;">pa&longs;&longs;ionem</expan> figuris planè in&longs;cri­<lb/>ptis priùs accidere <expan abbr="o&longs;t&etilde;dit">o&longs;tendit</expan>. potui&longs;&longs;etquè Archimedes priùs quar <lb/>tam propo&longs;itionem o&longs;tendere, quam tertiam; &longs;equentem ve­<lb/>rò propo&longs;itionem immediatè po&longs;uit po&longs;t &longs;ecundam, ordo e­<lb/>nim &longs;ic po&longs;tulat. etenim ambæ deijs pertractant, quæ rectili­<lb/>neis figuris plane in&longs;criptis accidunt. Pr&ecedil;terea earum demon <lb/>&longs;trationes ferè circa eadem ver&longs;antur, cùm ijsdem rectis lineis <lb/>in portionibus eodem modo ductis vtantur; ob &longs;equentis ve­<lb/>rò propo&longs;itionis in telligentiam h&ecedil;c priùs o&longs;tendemus. </s></p>
<figure></figure>
<p type="head">
<s>LEMMA I.</s></p>
<p type="main">
<s>Eandem habeat proportionem AB ad CD, quam habet <lb/>GH ad KL. CD verò ad EF <expan abbr="eã">eam</expan>, <expan abbr="quã">quam</expan> habet kL ad MN. &longs;intquè
<pb pagenum="136"/>AB CD EF inter&longs;e &ecedil;quid&longs;tantes. &longs;imiliter GH KL MN <lb/>æquidi&longs;tantes, &longs;intautem ductæ BDF HLN rectæ lineæ; &longs;it­<lb/>què BD ad DF, vt HL ad LN. &longs;itquè maior AB quàm <lb/>CD, & CD, quàm EF. vnde erit quoquè GH maior KL, <lb/>& KL, quam MN. iuncti&longs;què AC CE, & GK KM. <lb/>Dico &longs;pacium ACDB ad &longs;pacium CEFD eandem habere <lb/>proportionem, quam &longs;pacium GKLH ad &longs;pacium KMNL. </s></p>
<figure></figure>
<p type="main">
<s>Producantur AC CE, quæ cum BF conueniantin OP. <lb/>productæquè GK KM cum HN conueniant in QR. <lb/>concurrentenim, quoniam CD KL &longs;unt minores ip&longs;is AB <lb/>
<arrow.to.target n="marg229"></arrow.to.target> GH, & EF MN minores ip&longs;is CD KL. Fiatquè vt AB <lb/>ad CD, ita CD ad V. & vt GH ad kL, ita KL ad X. <lb/>deinceps CD ad EF, ita EF ad Y. & vt KL ad MN, <lb/>ita MN ad Z. Quoniam igitur triangulum ABO &longs;imile <lb/>e&longs;t triangulo CDO, cùm &longs;it CD æquidi&longs;tansip&longs;i AB. ha <lb/>
<arrow.to.target n="marg230"></arrow.to.target> bebit triangulum ABO ad CDO, proportionem, quam ha <lb/>bet AB ad CD duplicatam. hoc e&longs;t quam hab et AB ad <lb/>V. Eodemquè modo o&longs;ten detur <expan abbr="triangulũ">triangulum</expan> GHQ ad KLQ <lb/>ita e&longs;&longs;e, vt GH ad X<gap/> quia verò AB CD V ita &longs;e <expan abbr="hab&etilde;t">habent</expan>, <lb/>
<arrow.to.target n="marg231"></arrow.to.target> vt GH kL X, erit ex æquali AB ad V, & GH ad X. <lb/>triangulum igitur ABO eandem habet proportionem ad
<pb pagenum="129"/>CDO, quam triangulum GHQ ad <expan abbr="KLq.">KLque</expan> quare diuiden­<lb/>do &longs;pacium ACDB ad triangulum CDO e&longs;t, vt &longs;pacium
<arrow.to.target n="marg232"></arrow.to.target><lb/>GKLH ad triangulum <expan abbr="kLq.">kLque</expan> Rur&longs;us quoniam ob triangu <lb/>lorum &longs;imilitudinem ABO CDO, ita e&longs;t AB ad CD, vt
<arrow.to.target n="marg233"></arrow.to.target><lb/>BO ad OD. &longs;imiliter ob &longs;imilitudinem <expan abbr="triangulorũ">triangulorum</expan> GHQ <lb/>KLQ ita e&longs;t GH ad kL, vt HQ ad QL. & e&longs;t AB ad CD, <lb/>vt GH ad KL, erit BO ad OD, vt HQ ad QL. &
<arrow.to.target n="marg234"></arrow.to.target> diui­<lb/>dendo BD ad DO, vt HL ad <expan abbr="Lq.">Lque</expan> deinde <expan abbr="conuert&etilde;do">conuertendo</expan> DO <lb/>ad DB, vt LQ ad LH. & e&longs;t BD ad DF, vt HL ad LN, erit
<arrow.to.target n="marg235"></arrow.to.target><lb/>ex &ecedil;quali DO ad DF, vt LQ ad LN. Quoniam autem &longs;imi <lb/>lium triangulorum CDP EFP latus CD ad latus EF ita &longs;e <lb/>habet, vt DP ad PF. &longs;imiliter exi&longs;tentibus &longs;imilibus triangu <lb/>lis KLR MNR ita e&longs;t KL ad MN, vt LR ad RN, & vt CD <lb/>ad EF, ita e&longs;t KL ad MN, erit DP ad PF, vt LR ad RN.
<arrow.to.target n="marg236"></arrow.to.target><lb/>& per conuer&longs;ionem rationis PD ad DF, vt RL ad LN. & <lb/>conuertendo DF ad DP, vt LN ad LR. diximus <expan abbr="aut&etilde;">autem</expan> OD <lb/>ad DF ita e&longs;&longs;e, vt QL ad LN, & e&longs;t DF ad DP, vt LN ad <lb/>LR. ergo ex &ecedil;quali erit OD ad DP, vt QL ad LR. At verò
<arrow.to.target n="marg237"></arrow.to.target><lb/>quoniam ita e&longs;t OD ad DP, vt triangulum OCD ad PCD, <lb/>& vt QL ad LR, ita e&longs;t triangulum QKL ad <expan abbr="triangulũ">triangulum</expan> RKL, <lb/>erit OCD ad PCD, vt QKL ad RKL. Quoniam <expan abbr="aut&etilde;">autem</expan> <expan abbr="triã">triam</expan> <lb/>gula CDP EFP &longs;unt &longs;imilia, triangulum CDP ad triangulum
<arrow.to.target n="marg238"></arrow.to.target><lb/>EFP proportionem habebit, quam CD ad EF duplicatam, <lb/>hoc e&longs;t quam habet CD ad Y, cùm &longs;int CD EF Y propor­<lb/>tionales. &longs;imiliter ob triangulorum KLR MNR &longs;imilitudi­<lb/>nem triangulum KLR ad MNR, ita erit vt KL ad Z, e&longs;t au­<lb/>tem CD ad Y, vt KL ad Z, erit igitur <expan abbr="triãgulum">triangulum</expan> CDP ad <lb/>EFP, vt KLR ad MNR, & diuiden do <expan abbr="&longs;paciũ">&longs;pacium</expan> CEFD ad trian
<arrow.to.target n="marg239"></arrow.to.target><lb/>gulum EFP, vt &longs;pacium KMNL ad triangulum MNR. & <expan abbr="cõ">com</expan>
<arrow.to.target n="marg240"></arrow.to.target><lb/>uertendo triangulum EFP ad &longs;pacium CEFD, vt <expan abbr="triangulũ">triangulum</expan> <lb/>MNR ad &longs;pacium KMNL. Itaque quoniam o&longs;ten&longs;um e&longs;t i­<lb/>ta e&longs;&longs;e &longs;pacium ACDB ad triangulum CDO, vt &longs;pacium <lb/>GKLH ad triangulum <expan abbr="KLq.">KLque</expan> & vt <expan abbr="triangulũ">triangulum</expan> CDO ad trian <lb/>gulum CDP, ita triangulum KLQ ad <expan abbr="triangulũ">triangulum</expan> KLR, dein <lb/>de, vt triangulum CDP ad triangulum EFP, ita <expan abbr="triãgulum">triangulum</expan> <lb/>KLR ad triangulum MNR; deniquè vt triangulum EFP ad <lb/>&longs;pacium CEFD, ita triangulum MNR ad &longs;pacium kMNL,
<pb pagenum="142"/>
<arrow.to.target n="marg241"></arrow.to.target> erit ex æquali à primo ad vltimum &longs;pacium ACDB ad <expan abbr="&longs;paciũ">&longs;pacium</expan> <lb/>CEFD, vt &longs;pacium GKLH ad &longs;pacium KMNL. quod <expan abbr="demõ">demom</expan> <lb/>&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg229"></margin.target>11. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg230"></margin.target>9. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg231"></margin.target>22 <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg232"></margin.target>17. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg233"></margin.target><emph type="italics"/>e&longs;t<emph.end type="italics"/> 4.<emph type="italics"/>&longs;exti<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg234"></margin.target>17.<emph type="italics"/>quinti. <lb/>cor.<emph.end type="italics"/>4. <emph type="italics"/><expan abbr="quī">quim</expan> <lb/>ti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg235"></margin.target>22. <emph type="italics"/>quinti<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg236"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 11. <emph type="italics"/><expan abbr="quĩ">quim</expan> <lb/>ti. <lb/>cor.<emph.end type="italics"/> 19. <lb/><emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg237"></margin.target>22. <emph type="italics"/>quinti <lb/>ex<emph.end type="italics"/> 1.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg238"></margin.target>19. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg239"></margin.target><emph type="italics"/>ex quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg240"></margin.target><emph type="italics"/>cor.<emph.end type="italics"/>4. <emph type="italics"/><expan abbr="quī">quim</expan> <lb/>ti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg241"></margin.target>22. <emph type="italics"/>quinti<emph.end type="italics"/></s></p>
<p type="head">
<s>LEMMA II.</s></p>
<p type="main">
<s><expan abbr="Æquidi&longs;tãtes">Æquidi&longs;tantes</expan> verò line&ecedil; AB CD ita &longs;e habeant, vt æquidi­<lb/>&longs;tantes EF GH, &longs;itquè maior AB, quàm CD, & EF, quam <lb/>GH. & &longs;uper CD GH &longs;int triangula CDP GHR, <expan abbr="&longs;int&qacute;">&longs;intque</expan>; BDP <lb/>FHR rectæ lineæ, & vt BD ad DP, ita &longs;it FH ad HR. <expan abbr="iunctis&qacute;">iunctisque</expan>; <lb/>AC EG. Dico &longs;pacium ACDB ad <expan abbr="triangulũ">triangulum</expan> CDP ita e&longs;&longs;e, vt <lb/>&longs;pacium EG HF ad triangulum GHR. </s></p>
<figure></figure>
<p type="main">
<s>Eadem enim pror&longs;us ratione productis AC EG, quæ cum <lb/>BP FR conueniantin OQ, o&longs;tendetur &longs;pacium AD ad trian <lb/>gulum CDO ita e&longs;&longs;e, vt &longs;pacium EH ad triangulum <expan abbr="GHq.">GHque</expan> & <lb/>e&longs;&longs;e OD ad DB, ut QH ad HF. & quoniam e&longs;t BD ad DP, vt <lb/>
<arrow.to.target n="marg242"></arrow.to.target> FH ad HR, erit ex &ecedil;quali OD ad DP, vt QH ad HR. & vt OD <lb/>ad DP, ita e&longs;t triangulum CDO ad triangulum CDP, & vt <lb/>QH ad HR, ita triangulum GHQ ad GHR. cùm itaque &longs;it <lb/>AD ad CDO, vt EH ad GHQ, & vt CDO ad CDP, ita <lb/>
<arrow.to.target n="marg243"></arrow.to.target> GHQ ad GHR. ex æquali erit &longs;pacium AD ad triangulum <lb/>CDP, vt &longs;pacium EH ad triangulum GHR. quod demon&longs;tra <lb/>re oportebat. </s></p>
<pb pagenum="143"/>
<p type="margin">
<s><margin.target id="marg242"></margin.target>22 <emph type="italics"/>quinti.<emph.end type="italics"/><lb/><gap/>. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg243"></margin.target>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="head">
<s>LEMMA. III.</s></p>
<figure></figure>
<p type="main">
<s>Sit A ad CD, vt E ad FG, diuidan <lb/><expan abbr="tur&qacute;">turque</expan>; CD FG in <expan abbr="ead&etilde;">eadem</expan> proportione in HK, <lb/>ita vt &longs;it CH ad HD, vt FK ad KG. <lb/>Dico A ad DH ita e&longs;&longs;e, vt E ad KG. <lb/>A verò ad CH, vt E ad Fk. </s></p>
<p type="main">
<s>Quoniam enim ita e&longs;t CH ad HD, vt FK ad kG; e­<lb/>rit componendo CD ad DH, vt FG ad GK. e&longs;t autem A
<arrow.to.target n="marg244"></arrow.to.target><lb/>ad CD, vt E ad FG; CD verò e&longs;t ad DH, vt FG ad G<emph type="italics"/>K<emph.end type="italics"/>; er <lb/>go ex æquali A erit ad DH, vt E ad GK. Deinde
<arrow.to.target n="marg245"></arrow.to.target> quo­<lb/>niam e&longs;t GH ad HD, vt FK ad kG; erit conuertendo
<arrow.to.target n="marg246"></arrow.to.target><lb/>DH ad HC, vt GK ad KF. rur&longs;us igitur ex æquali A e­<lb/>rit ad CH, vt E ad FK. quod o&longs;tendere oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg244"></margin.target>18.<emph type="italics"/>qumti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg245"></margin.target>22 <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg246"></margin.target><emph type="italics"/>cor.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="quĩ-ti">quin­<lb/>ti</expan>.<emph.end type="italics"/></s></p>
<p type="head">
<s>PROPOSITIO. III.</s></p>
<p type="main">
<s>Siin <expan abbr="vtraq;">vtraque</expan> <expan abbr="duarũ">duarum</expan> <expan abbr="&longs;imiliũ">&longs;imilium</expan> <expan abbr="portionũ">portionum</expan> recta linea re <lb/>ctanguliquè coni &longs;ectione contentarum rectili­<lb/>neæ figuræ planè in&longs;cribantur; figuræ verò in&longs;cri­<lb/>ptæ latera inter&longs;e multitudine æqualia habeant; <lb/>rectilinearum centra grauitatum portionum dia­<lb/>metros &longs;imiliter &longs;ecabunt. </s></p>
<pb pagenum="144"/>
<figure></figure>
<p type="main">
<s><emph type="italics"/>Sint duæ portiones ABC XOP, in ip&longs;i&longs;què planè in &longs;cribantur recti­<lb/>lineæ figuræ<emph.end type="italics"/> AEFGBHIKC XSYQOZVTP; <emph type="italics"/>quæ omnia latera <lb/>inter&longs;e numero æqualia habeanta, Diametri verò portionum &longs;int BD<emph.end type="italics"/>
<pb pagenum="145"/><emph type="italics"/>OR. <expan abbr="iungãtur&qacute;">iunganturque</expan>; E<emph.end type="italics"/>k <emph type="italics"/>FI GH.<emph.end type="italics"/> quæ inter&longs;e, & ip&longs;i AC <expan abbr="çquidi&longs;tãtes">çquidi&longs;tantes</expan>
<arrow.to.target n="marg247"></arrow.to.target><lb/>erunt; bifariam què à diametro BD in punctis LMN diui&longs;æ e­<lb/>runt. Iungantur &longs;imiliter <emph type="italics"/>& ST YV QZ<emph.end type="italics"/>, quas bifariam dia­<lb/>meter OR in punctis 9<foreign lang="greek">ab</foreign> diuidet. eruntquè ductæ lineæ ip&longs;i <lb/>XP, & inter&longs;e æquidi&longs;tantes. <emph type="italics"/>Quoniam igitur BD diuiditur à lineis <lb/>æquidi&longs;tantibus<emph.end type="italics"/> GH FI EK <emph type="italics"/>in proportionibus numeris deinceps impa­<lb/>ribus;<emph.end type="italics"/> po&longs;ito enim vno BN, e&longs;t quidem NM tria, ML quinque, <lb/>& LD &longs;eptem. &longs;ed <emph type="italics"/>& RO &longs;imiliter<emph.end type="italics"/> à lineis QZ YV ST in pro­<lb/>portionibus diuiditur numeris deinceps imparibus, <expan abbr="ead&etilde;">eadem</expan>. n. <lb/>ratione &longs;i ponatur O<foreign lang="greek">b</foreign> vnum, erit <foreign lang="greek">ba</foreign> tria, <foreign lang="greek">a</foreign>9 <expan abbr="quin&qacute;">quinque</expan>;, & 9R <lb/>&longs;eptem. <emph type="italics"/>& portiones ip&longs;orum<emph.end type="italics"/> diametrorum BD OR <emph type="italics"/>&longs;unt numero æ <lb/>quales.<emph.end type="italics"/> quot.n &longs;unt BN NM ML LD, tot &longs;unt O<foreign lang="greek">b ba a</foreign> 9 9R. <emph type="italics"/>pa <lb/>tet diametrorum portiones in eadem e&longs;&longs;e proportione<emph.end type="italics"/>, vt quem <expan abbr="admodũ">admodum</expan> <lb/>e&longs;t BN ad NM, & NM ad ML, & ML ad LD, ita e&longs;&longs;e O<foreign lang="greek">b</foreign> ad <lb/><foreign lang="greek">ba</foreign>, & <foreign lang="greek">ba</foreign> ad <foreign lang="greek">a</foreign>9, & <foreign lang="greek">a</foreign>9 ad 9R. Atverò quoniam ita e&longs;t DB ad BL, <lb/>vt RO ad O9; (&longs;unt.n.ut &longs;exdecim ad nouem) & ut DB ad BL,
<arrow.to.target n="marg248"></arrow.to.target><lb/>ita e&longs;t quadratum ex AD ad <expan abbr="quadratũ">quadratum</expan> ex EL; & vt RO ad O9, <lb/>ita e&longs;t <expan abbr="quadratũ">quadratum</expan> ex XR ad quadratum ex S<emph type="italics"/>9<emph.end type="italics"/>; erit <expan abbr="quadratũ">quadratum</expan> ex <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/>ergo ut AD ad EL, ita XR ad S9. & horum dupla <expan abbr="n&etilde;pè">nempè</expan> AC ad <lb/>EK, vt XP ad ST: <expan abbr="eadem&qacute;">eademque</expan>; pror&longs;us <expan abbr="rõne">ronne</expan>, quoniam ita e&longs;t LB
<arrow.to.target n="marg249"></arrow.to.target><lb/>ad BM, vt 9O ad O<foreign lang="greek">a</foreign> (&longs;unt.n.ut nouem ad quatuor) o&longs;tendetur <lb/>EL ad FM ita e&longs;&longs;eut S9 ad Y<foreign lang="greek">a</foreign>, & horum dupla, &longs;cilicet EK ad FI <lb/>ita e&longs;&longs;e, ut ST ad YV. <expan abbr="Cùm&qacute;">Cùmque</expan>; &longs;it MB ad BN, vt <foreign lang="greek">a</foreign>O ad O<foreign lang="greek">b</foreign>, ut &longs;ci <lb/>licet quatuor ad vnum; &longs;imiliter o&longs;tendetur FM ad GN ita e&longs;&longs;e <lb/>vt Y<foreign lang="greek">a</foreign> ad Q<foreign lang="greek">b</foreign>; FI uerò ad GH, vt YV ad QZ. vnde colligitur <expan abbr="nõ">non</expan> <lb/>&longs;olùm portiones diametrorum (ut dixim us) in eadem e&longs;&longs;e pro­<lb/>portione, &longs;ed <emph type="italics"/>& parallelas<emph.end type="italics"/> AC EK FI GH, & XP ST YV QZ <emph type="italics"/>in <lb/><expan abbr="eadē">eadem</expan> e&longs;&longs;e proportione. & T rapeziorum ip&longs;ius quidem AE<emph.end type="italics"/>k<emph type="italics"/>C, & ip&longs;ius<emph.end type="italics"/>
<arrow.to.target n="marg250"></arrow.to.target><lb/><emph type="italics"/>XSTP centra grauitatum e&longs;&longs;e in lineis LD 9R &longs;imiliter po&longs;ita, cùm <lb/>eandem habeant proportionem AC EK, quam XP ST.<emph.end type="italics"/> lineæquè <lb/>LD 9R bifariam diuidant &longs;uas æquidi&longs;tantes AC EK. <lb/>& XP ST. etenim &longs;i ponatur trapezij AK centrum graui <lb/>tatis <foreign lang="greek">g</foreign>, ip&longs;ius vcrò XT centrum grauitatis <foreign lang="greek">d</foreign>, erit L<foreign lang="greek">g</foreign> ad <foreign lang="greek">g</foreign>D, <lb/>vt dupla ip&longs;ius AC cum EK ad duplam ip&longs;ius EK
<arrow.to.target n="marg251"></arrow.to.target><lb/>cum AC. & 9<foreign lang="greek">d</foreign> ad <foreign lang="greek">d</foreign>R erit, vt dupla ip&longs;ius XP cum <lb/>ST ad duplam ST cum XP. quoniam autem ita e&longs;t AC ad EK,
<pb pagenum="146"/>
<arrow.to.target n="fig69"></arrow.to.target>
<pb pagenum="147"/>vt XP ad ST, & an tecedentium dupla, hoc e&longs;t dupla i­<lb/>p&longs;ius AC ad EK erit, vt dupla ip&longs;ius XP ad ST. <lb/>& componendo dupla ip&longs;ius AC cum EK, vt dupla
<arrow.to.target n="marg252"></arrow.to.target> i­<lb/>p&longs;ius XP cum ST ad ST. At verò EK ad duplam <lb/>ip&longs;ius EK, ita e&longs;t, vt ST ad duplam ip&longs;ius ST, &longs;ed EK
<arrow.to.target n="marg253"></arrow.to.target><lb/>ad AC e&longs;t, vt ST ad XP, erit EK ad vtra&longs;que con&longs;e­<lb/>quentes &longs;im ul &longs;umptas, hoc e&longs;t ad duplam ip&longs;ius EK cum <lb/>AC, vt ST ad &longs;uas con&longs;equentes, nempe ad duplam ip&longs;ius <lb/>ST cum XP. Itaque quoniam ita e&longs;t dupla ip&longs;ius AC <lb/><expan abbr="cũ">cum</expan> EK ad Ek, vt dupla ip&longs;ius XP cum ST ad ST, & e&longs;t EK <lb/>ad duplam ip&longs;ius EK cum AC, vt ST ad duplam ip&longs;ius <lb/>ST cum XP. erit ex &ecedil;quali dupla ip&longs;ius AC cum EK ad du
<arrow.to.target n="marg254"></arrow.to.target><lb/>plam ip&longs;ius EK cum AC, vt dupla ip&longs;ius XP cum ST ad <lb/>duplam ip&longs;ius ST cum XP. ac propterea ita e&longs;t L<foreign lang="greek">g</foreign> ad <foreign lang="greek">g</foreign>D, <lb/>vt 9<foreign lang="greek">d</foreign> ad <foreign lang="greek">d</foreign>R, & ob id centra <foreign lang="greek">gd</foreign> erunt in lineis LD 9R &longs;i­<lb/>militer po&longs;ita. <emph type="italics"/>Rur&longs;us<emph.end type="italics"/> eodem modo (ne eadem &longs;æpiùs repetan <lb/>tur) <emph type="italics"/>Trapeziorum EFI<emph.end type="italics"/>k <emph type="italics"/>S<emph.end type="italics"/><foreign lang="greek">*g</foreign><emph type="italics"/>VT centragrauitatum<emph.end type="italics"/>, quæ &longs;int <foreign lang="greek">ez</foreign>, <emph type="italics"/>&longs;i <lb/>militer<emph.end type="italics"/> hoc e&longs;t in eadem proportione <emph type="italics"/>diuident lineas LM<emph.end type="italics"/> 9<foreign lang="greek">a</foreign>, i­<lb/>ta vt &longs;it L<foreign lang="greek">e</foreign> ad <foreign lang="greek">e</foreign>M, vt 9<foreign lang="greek">z</foreign> ad <foreign lang="greek">za</foreign>. <emph type="italics"/>& in trapezits FH<emph.end type="italics"/> <foreign lang="greek">*g</foreign><emph type="italics"/>Z centra <lb/>grauitatum<emph.end type="italics"/> <foreign lang="greek">*hk</foreign> <emph type="italics"/>&longs;imiliter diuident MN<emph.end type="italics"/> <foreign lang="greek">ab</foreign>, ita ut M<foreign lang="greek">*h</foreign> ad <foreign lang="greek">*h</foreign>N &longs;it, vt <lb/><foreign lang="greek">ak</foreign> ad <foreign lang="greek">kb</foreign> <emph type="italics"/>&longs;ed & triangulorum GBH QOZ centra grauitatum<emph.end type="italics"/> <foreign lang="greek">lm</foreign><lb/><emph type="italics"/>in lineis B N<emph.end type="italics"/> O<foreign lang="greek">b</foreign> <emph type="italics"/>erunt &longs;imiliter po&longs;ita<emph.end type="italics"/>, &longs;iquidem B<foreign lang="greek">l</foreign> ad <foreign lang="greek">l</foreign>N e&longs;t, vt
<arrow.to.target n="marg255"></arrow.to.target><lb/>O<foreign lang="greek">m</foreign> ad <foreign lang="greek">mb</foreign>; quippè cùm in dupla &longs;int proportione. <emph type="italics"/>eandem au­<lb/>tem habent proportionem Trapezia, & triangula:<emph.end type="italics"/> Nam cùm <lb/>&longs;it AD ad EL, vt XR ad S9, & ut EL ad FM, ita S9 ad Y; <lb/>e&longs;tquè DL ad LM, ut R9 ad 9<foreign lang="greek">a</foreign>, cùm &longs;int, vt &longs;eptem ad quin <lb/>que; erit &longs;pacium AL ad &longs;pacium EM, vt &longs;pacium X9 ad
<arrow.to.target n="marg256"></arrow.to.target> &longs;pa­<lb/>cium S. &longs;imiliterquè o&longs;tendetur DK ad LI ita e&longs;&longs;e, vt RT <lb/>ad 9V. quare totum trapezium AK ad EI e&longs;t, vt XT ad SV. <lb/>pariquè ratione o&longs;tendeturita e&longs;&longs;e trapezium EI ad FH, vt <lb/>SV ad YZ. quia verò ita e&longs;t FM ad GN, vt Y<foreign lang="greek">a</foreign> ad Q<foreign lang="greek">d</foreign>, <lb/>e&longs;t autem MN ad NB, vt <foreign lang="greek">ab</foreign> ad <foreign lang="greek">b</foreign>O, &longs;unt quippè ut tria ad <lb/>vnum, erit &longs;pacium FN ad triangulum GBN, vt &longs;pacium
<arrow.to.target n="marg257"></arrow.to.target><lb/>Y<foreign lang="greek">b</foreign> ad triangulum Q<foreign lang="greek">b</foreign>O. codemquè modo o&longs;tendetur ita <lb/>e&longs;&longs;e &longs;pacium IN ad triangulum BNH, vt &longs;pacium V<foreign lang="greek">b</foreign> ad <lb/>triangulum O<foreign lang="greek">b</foreign>Z. Ex quibus &longs;equitur ita e&longs;&longs;e <expan abbr="trapeziũ">trapezium</expan> FH <lb/>ad triangulum BGH, vt trapezium YZ ad <expan abbr="triangulũ">triangulum</expan> OQZ.
<pb pagenum="148"/>
<arrow.to.target n="fig70"></arrow.to.target>
<pb pagenum="149"/>&longs;i itaque diuidatur <foreign lang="greek">ge</foreign> in <foreign lang="greek">n</foreign>, ita ut &longs;it <foreign lang="greek">en</foreign> ad <foreign lang="greek">ng</foreign>, vt <expan abbr="trapeziũ">trapezium</expan> AK <lb/>ad EI. erit punctum <foreign lang="greek">n</foreign> centrum grauitatis figur&ecedil; AEFIKC.
<arrow.to.target n="marg258"></arrow.to.target><lb/>&longs;imiliquè modo diuidatur <foreign lang="greek">dz</foreign> in <foreign lang="greek"><10></foreign>, ita vt &longs;it <foreign lang="greek">z<10></foreign> ad <foreign lang="greek"><10>d</foreign>, vt trape <lb/>zium XT ad SV; erit punctum <foreign lang="greek"><10></foreign> grauitatis centrum figuræ <lb/>XSYVTP. quia verò ita e&longs;t AK ad EI, vt XT ad SV, erit <foreign lang="greek">en</foreign><lb/>ad <foreign lang="greek">ng</foreign>, vt <foreign lang="greek">z<10></foreign> ad <foreign lang="greek"><10>d</foreign>. Diuidatur <expan abbr="aũt">aunt</expan> deinceps <foreign lang="greek">l*h</foreign> in <foreign lang="greek">s</foreign>, <expan abbr="&longs;it&qacute;">&longs;itque</expan>; <foreign lang="greek">ls</foreign> ad <foreign lang="greek">s*h</foreign>, vt <lb/>FH ad triangulum BGH, erit punctum <foreign lang="greek">s</foreign> centrum grauitatis <lb/>figuræ FGBHI. eademquè ratione diuidatur <foreign lang="greek">mk</foreign> in <foreign lang="greek">t</foreign>, &longs;itquè <lb/><foreign lang="greek">mt</foreign> ad <foreign lang="greek">tk</foreign>, vt YZ ad triangulum OQZ; erit punctum <foreign lang="greek">t</foreign> cen­<lb/>trum grauitatis figuræ YQOZV. &longs;ed e&longs;t FH ad BGD, vt YZ <lb/>ad OQZ, eritigitur <foreign lang="greek">ls</foreign> ad <foreign lang="greek">sh</foreign>, vt <foreign lang="greek">mt</foreign> ad <foreign lang="greek">tk</foreign>. Quoniam autem <lb/>ita e&longs;t Ak ad EI, vt XT ad SV, erit componendo AEFIKC
<arrow.to.target n="marg259"></arrow.to.target><lb/>ad EI, vt figura XSYVTP ad SV; & e&longs;t EI ad FH, vt SV ad
<arrow.to.target n="marg260"></arrow.to.target><lb/>YZ. ergo ex æquali figura AEFIKC erit ad FH, vt figura <lb/>XSYVTP ad YZ. e&longs;t autem FH ad BGH, vt YZ ad OQZ. e­<lb/>ritigitur figura AEFIKC ad &longs;uas con&longs;equentes, ad figuram
<arrow.to.target n="marg261"></arrow.to.target><lb/>&longs;cilicet FGBHI, vt figura XSYVTP ad &longs;uas con&longs;equentes, hoc <lb/>e&longs;t ad figuram YQOZV. Diuidatur itaque <foreign lang="greek">sn</foreign> in <foreign lang="greek">x</foreign>, ita ut <foreign lang="greek">sx</foreign><lb/>ad <foreign lang="greek">x</foreign> &longs;it, vt &longs;igura AEFIKC ad figuram FGBHI, erit punctum
<arrow.to.target n="marg262"></arrow.to.target><lb/><foreign lang="greek">x</foreign> <expan abbr="centrũ">centrum</expan> grauitatis totius figur&ecedil; AEFGBHIKC. &longs;imiliter di­<lb/>uidatur <foreign lang="greek">t<10></foreign> in <foreign lang="greek">c</foreign>, &longs;itque <foreign lang="greek">tc</foreign> ad <foreign lang="greek">c<10></foreign>, ut figura XSYVTP ad figu­<lb/>ram YQOZV, erit punctum <foreign lang="greek">c</foreign> centrum grauitatis totius fi­<lb/>guræ XSYQOZVTP. quia verò ita e&longs;t figura AEFIKC ad fi <lb/>guram FGBHI, vt figura XSYVTP ad figuram YQOZV. e­<lb/>rit <foreign lang="greek">sx</foreign> ad <foreign lang="greek">xn</foreign>, vt <foreign lang="greek">tc</foreign> ad <foreign lang="greek">c<10></foreign>. Itaque quoniam BD ad DL e&longs;t, vt <foreign lang="greek">sn</foreign><lb/>ad R9, cùm &longs;in^{4} ut&longs;exdecim ad &longs;eptem. & e&longs;t L<foreign lang="greek">g</foreign> ad <foreign lang="greek">g</foreign>D, vt 9<foreign lang="greek">d</foreign><lb/>ad <foreign lang="greek">d</foreign>R, erit BD ad L<foreign lang="greek">g</foreign>, vt <foreign lang="greek">sn</foreign> ad 9<foreign lang="greek">d</foreign>. & vt BD ad <foreign lang="greek">g</foreign>D, ita OR ad
<arrow.to.target n="marg263"></arrow.to.target><lb/><foreign lang="greek">d</foreign>R. rur&longs;us quoniam BD ad LM e&longs;t, vt OR ad 9<foreign lang="greek">a</foreign>, nempe vt &longs;ex <lb/>decim ad quinque; & e&longs;t L<foreign lang="greek">e</foreign> ad <foreign lang="greek">e</foreign>M, ut 9<foreign lang="greek">z</foreign> ad <foreign lang="greek">za</foreign>, erit BD ad <foreign lang="greek">e</foreign>L, <lb/>vt OR ad 9<foreign lang="greek">z</foreign>. e&longs;t verò BD ad L<foreign lang="greek">g</foreign>, vt OR ad 9<foreign lang="greek">d</foreign>; erit igitur BD ad <lb/>vtram que &longs;imul <foreign lang="greek">e</foreign>L L<foreign lang="greek">g</foreign>, hoc e&longs;t ad <foreign lang="greek">eg</foreign>, vt OR ad <foreign lang="greek">zd</foreign>. &longs;ed <expan abbr="quoniã">quoniam</expan>
<arrow.to.target n="marg264"></arrow.to.target><lb/>e&longs;t <foreign lang="greek">gn</foreign> ad <foreign lang="greek">ne</foreign>, vt <foreign lang="greek">d<10></foreign> ad <foreign lang="greek"><10>z</foreign>, erit BD ad <foreign lang="greek">gn</foreign>, vt OR ad <foreign lang="greek">d<10></foreign>. e&longs;t <expan abbr="aut&etilde;">autem</expan> BD <lb/>ad D<foreign lang="greek">g</foreign>, vt OR ad R<foreign lang="greek">d</foreign>, vt dictum e&longs;t, ergo BD ad D<foreign lang="greek">n</foreign> e&longs;t, vt OR <lb/>ad R<foreign lang="greek"><10></foreign>. &longs;imiliterquè <expan abbr="o&longs;t&etilde;detur">o&longs;tendetur</expan> BD ad BA ita e&longs;&longs;e, vt OR ad O<foreign lang="greek">t</foreign>. <lb/>Cùm itaque &longs;it BD ad DR, & ad B<foreign lang="greek">s</foreign>, ut OR ad R<foreign lang="greek"><10></foreign>, & ad O<foreign lang="greek">t</foreign>; e­<lb/>rit BD ad DR B<foreign lang="greek">s</foreign> &longs;imul, vt OR ad R<foreign lang="greek"><10></foreign> O<foreign lang="greek">t</foreign> &longs;imul, & permutan­<lb/>do tota BD ad totam OR, vt ablata D<foreign lang="greek">n</foreign>B<foreign lang="greek">s</foreign> ad ablatam R<foreign lang="greek"><10>ot</foreign>.
<pb pagenum="150"/>
<arrow.to.target n="marg265"></arrow.to.target> ergo & reliqua <foreign lang="greek">sn</foreign> ad reliquam <foreign lang="greek">t<10></foreign> e&longs;t, ut tota BD ad <expan abbr="totã">totam</expan> OR. <lb/>rur&longs;u&longs;què permutando <foreign lang="greek">sn</foreign> ad BD ut <foreign lang="greek">t<10></foreign> ad OR, <expan abbr="conuertendo&qacute;">conuertendoque</expan>; <lb/>BD ad <foreign lang="greek">sn</foreign> e&longs;t, ut OR ad <foreign lang="greek">t<10></foreign>, Quia verò ita e&longs;t <foreign lang="greek">sx</foreign> ad <foreign lang="greek">xn</foreign>, ut <foreign lang="greek">tc</foreign> ad <foreign lang="greek">c<10></foreign>; <lb/>
<arrow.to.target n="marg266"></arrow.to.target> erit BD ad <foreign lang="greek">sx</foreign>, vt OR ad <foreign lang="greek">tc</foreign> atverò BD ad b<foreign lang="greek">s</foreign> e&longs;t, vt OR ad O<foreign lang="greek">t</foreign>. <lb/>eritigitur BD ad B<foreign lang="greek">x</foreign>, ut O<foreign lang="greek">g</foreign> ad O<foreign lang="greek">c</foreign>. ac propterea diuidendo D<foreign lang="greek">x</foreign><lb/>ita &longs;e habet ad <foreign lang="greek">x</foreign>B, vt R<foreign lang="greek">c</foreign> ad <foreign lang="greek">c</foreign>O. <emph type="italics"/>Quare manifestum est totius recti­<lb/>lineæ figuræ in portione ABC in&longs;criptæ centrum grauitatis<emph.end type="italics"/> <foreign lang="greek">x</foreign> <emph type="italics"/>in eadem <lb/>proportione diuidere BD, veluti centrum grauitatis<emph.end type="italics"/> <foreign lang="greek">c</foreign> <emph type="italics"/>figuræ rectilineæ <lb/>in portione XOP<emph.end type="italics"/> in&longs;criptæ <emph type="italics"/>ip&longs;am OR<emph.end type="italics"/> diametrum. <emph type="italics"/>quod demonstra­<lb/>re oportebat.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg247"></margin.target><emph type="italics"/>ex iis qu&ecedil; <lb/>po&longs;t <gap/> pri­<lb/>mi huius <lb/>demon&longs;tra <lb/>ta &longs;unt.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg248"></margin.target>3. A<emph type="italics"/>rchi. <lb/>de quad. <lb/>parab. &<emph.end type="italics"/><lb/>20, <emph type="italics"/>primi <lb/>conicorum <lb/>Apoll.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg249"></margin.target>22. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg250"></margin.target>15. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg251"></margin.target>15. <emph type="italics"/>primi <lb/>buius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg252"></margin.target>18. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg253"></margin.target>2. <emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> an <lb/>te<emph.end type="italics"/> 13. <emph type="italics"/>pri <lb/>mi huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg254"></margin.target>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg255"></margin.target><emph type="italics"/><expan abbr="ãte">ante</expan><emph.end type="italics"/> 13.<emph type="italics"/>pri <lb/>mi huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg256"></margin.target>1.<emph type="italics"/>lemma.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg257"></margin.target>2.<emph type="italics"/>lemma.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg258"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 6. <emph type="italics"/>pri­<lb/>mi huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg259"></margin.target>18. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg260"></margin.target>22.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg261"></margin.target><emph type="italics"/>cor.<emph.end type="italics"/> 2. <emph type="italics"/>lem <lb/>ma m<emph.end type="italics"/> 13. <lb/><emph type="italics"/>primi hui<emph.end type="italics"/>^{9}</s></p>
<p type="margin">
<s><margin.target id="marg262"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 6. <emph type="italics"/>pri <lb/>mi huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg263"></margin.target>3. <emph type="italics"/>lemma.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg264"></margin.target>2. <emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> an <lb/>te<emph.end type="italics"/> 13. <emph type="italics"/>pri <lb/>mi huius.<emph.end type="italics"/><lb/>3. <emph type="italics"/>lcmma.<emph.end type="italics"/><lb/>2. <emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> an <lb/>te<emph.end type="italics"/> 13. <emph type="italics"/>pri­<lb/>mi huius<emph.end type="italics"/><lb/>16.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg265"></margin.target>19.<emph type="italics"/>quinti. <lb/>co.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="quīti">quinti</expan>.<emph.end type="italics"/><lb/>3.<emph type="italics"/>lemma.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg266"></margin.target>2. <emph type="italics"/>lemma <lb/>ante<emph.end type="italics"/> 13. <lb/><emph type="italics"/>primi hui<emph.end type="italics"/>^{9} <lb/>18. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<figure id="fig69"></figure>
<figure id="fig70"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Hinc colligere licet parabolas omnes inter&longs;e &longs;imiles e&longs;&longs;e. Re <lb/>fert enim Eutocius hoc in loco, Apollonium perg&ecedil;um in &longs;ex <lb/>to Conicorum libro. (qui nondum in lucem prodijt) &longs;imiles <lb/>coni &longs;ectiones dixi&longs;&longs;e eas e&longs;&longs;e, quando in vnaquaque &longs;ectione <lb/>line&ecedil; <expan abbr="ducũtur">ducuntur</expan> ba&longs;i <expan abbr="æquidi&longs;tãtes">æquidi&longs;tantes</expan> numero pares; hoc e&longs;t totin v­<lb/>na, quotin alia; vt in &longs;uperioribus figuris ductæ fuerunt, in v­<lb/>na quidem EK FI GH ip&longs;i AC æquidi&longs;tan tes; & in altera ST <lb/>YV QZ ip&longs;i PX æquidi&longs;tantes; qu&ecedil; quidem efficiant, vt dia­<lb/>metri in eadem proportione diui&longs;æ proueniant; vt&longs;unt BN <lb/>NM ML LD; & O<foreign lang="greek">b ba a</foreign>9 9R. Deinde <expan abbr="æquidi&longs;tãtes">æquidi&longs;tantes</expan> AC EK <lb/>FI GH in eadem &longs;int proportione ip&longs;arum XP ST YV QZ. <lb/>& quoniam hæ conditiones in omnibus po&longs;&longs;unt accidere pa <lb/>rabolis; vt ex ijs, quæ demon&longs;trata &longs;unt, manife&longs;tum e&longs;t; id­<lb/>circo parabolæ omnes &longs;unt &longs;imiles. Neque verò <expan abbr="exi&longs;timandũ">exi&longs;timandum</expan> <lb/>e&longs;t, quoniam parabolæ &longs;unt &longs;imiles, figur as quoque planè <lb/>in&longs;criptas, vt AEFGBHIKC & XSYQOZVTP &longs;imiles e&longs;&longs;e in <lb/>ter &longs;e, ea præ&longs;ertim &longs;imilitudine, qua &longs;unt figuræ rectilineæ; <lb/>vt&longs;cilicet anguli &longs;int æquales, & circum &ecedil;quales angulos late­<lb/>ra proportionalia. in parabolis <expan abbr="nõ">non</expan> attenditur h&ecedil;c &longs;imilitudo. <lb/>&longs;atenim e&longs;t, vt præfatæ ad&longs;int conditiones; ex quibus &longs;equi­<lb/>tur (vt o&longs;tendimus) trapezia AK EI FH, triangulum què <lb/>BGH in eadem e&longs;&longs;e proportione trapeziorum XT SV YZ, ac
<pb pagenum="151"/>trianguli OQZ. ac propterea quando Archimedesin propo <lb/>&longs;itione inquit <emph type="italics"/>&longs;i in vtraque &longs;imilium portionum rectalmea, rectangu­<lb/>liquè coni &longs;ectione contentarum,<emph.end type="italics"/> non propterda exi&longs;timandum e&longs;t <lb/>reperiri po&longs;&longs;e aliquas parabolas recta linea terminatas no e&longs;&longs;e <lb/>&longs;imiles inter&longs;e; ea nimirumiam explicata &longs;imilitu dine. &longs;unte­<lb/>nim Archimedis verba hoc modo intelligenda, nempè, &longs;i in <lb/>vtraque portionum recta linea rectanguliquè coni &longs;ectione <lb/>contentarum, quæ omnes &longs;unt &longs;imiles, & c. veluti &longs;i dicere­<lb/>mus. In &longs;imilibus &longs;emicirculis anguli omnes &longs;untrecti. non <lb/>e&longs;t in telligendum nonnullos &longs;emicirculos inter&longs;e di&longs;&longs;imiles <lb/>exi&longs;tere po&longs;&longs;e. &longs;ed hoc modo; in &longs;emicirculis, qui omnes &longs;unt <lb/>&longs;imiles, anguli&longs;unt recti. Et hoc modo &longs;emperintelligere o­<lb/>portet, quando in &longs;equentibus Archimedes parabolas &longs;imiles <lb/>nominat. Nam & Archimedes cognouit omnes parabolas <lb/>inter&longs;e &longs;imiles e&longs;&longs;e; vt ip&longs;e in demon&longs;tratione octauæ propo&longs;i <lb/>tionis huius &longs;upponere videtur. Oportebatenim aliquam in <lb/>parabolis demon&longs;trare &longs;imilitudinem, vt dem on&longs;trari po&longs;&longs;et <lb/>centrum grauitatis in omnibus parabolis e&longs;&longs;e in certo, ac de­<lb/>terminato &longs;itu ip&longs;ius figuræ. in figuris enim, quæ aliquam in­<lb/>ter&longs;e non habent &longs;imilitudinem, in ip&longs;is centrum grauitatis <lb/>determinari minimè po&longs;&longs;e videtur. Dicet autem forta&longs;&longs;e ali­<lb/>quis, determinatur tamen centrum grauitatis in omnibus <expan abbr="triã">triam</expan> <lb/>gulis, quæ quidem in ter&longs;e non &longs;unt&longs;imilia. Cui re&longs;ponden­<lb/>dum; triangula omnia inter&longs;e &longs;imilia non e&longs;&longs;e &longs;imilitudine <lb/>rectilinearum figurarum, nempè vt anguli &longs;intæquales, & cir­<lb/>cum æqualesangulos latera proportionalia. quòd tamen nul­<lb/>lam inter&longs;e&longs;e habeant conuenientiam, omnino negatur. <expan abbr="nã">nam</expan> <lb/>triangula omnia &longs;imul quodam modo illam habent conue­<lb/>nientiam, & &longs;imilitudinem; quæ parabolis accidit. </s></p>
<p type="main">
<s>In triangulis enim ABC DEF duct&ecedil; &longs;int AG DH ab angu­<lb/>lis ad dimidias ba&longs;es. &longs;intquè diui&longs;a triangulorum latera in ea <lb/>dem proportione, in punctis kL, OP. & vt AK KL LB, ita &longs;it <lb/>AM MN NC, & DQ QR RF. ducti&longs;què KM LN OQ PR,
<arrow.to.target n="marg267"></arrow.to.target><lb/>quæ lineas AG DH &longs;ecentin punctis ST VX; primùm <expan abbr="quid&etilde;">quidem</expan> <lb/>erunt KM LN OQ PR ba&longs;ibus BC EF æquidi&longs;tantes; quas <lb/>lineæ AG DH in punctis ST VX bifariam diuident, cùm &longs;it
<pb pagenum="152"/>BG ad GC, vt LT ad TN, & KS ad SM. & ut EH ad HF ita <lb/>PX ad XR, & OV ad <expan abbr="Vq.">Vque</expan> Deinde erunt AG DH à lineis KM <lb/>LN OQ PR in eadem proportione diui&longs;æ; &longs;iquidem ita e&longs;t <lb/>AS ST TG, ut DV VX XH. cùm &longs;int, ut expo&longs;itæ propor­<lb/>tiones AK KL LB, & DO OP PE. Præterea erit &longs;pacium, <lb/>BN ad LM, vt ER ad PQ, & LM ad triangulum AK M, <lb/>
<arrow.to.target n="fig71"></arrow.to.target><lb/>vt PQ ad triangulum <expan abbr="DOq.">DOque</expan> Nam quoniam triangulu AEC <lb/>&longs;imile e&longs;t triangulo ALN, oblatus LN ip&longs;i BC æquidi&longs;tans; <lb/>erit ABC ad ALN, ut AB ad AL duplicata. eodemquè modo <lb/>erit DEF ad DPR, vt DE ad DP duplicata; eandem aut<gap/>m, <lb/>habet proportionem AB ad AL, quam DE ad DP: quadoqui <lb/>dem latera AB DE in eadem &longs;unt proportione diui&longs;a; eritigi­<lb/>tur triangulum ABC ad ALN, vt triangulum DEF ad DPR. <lb/>&longs;imiliterquè o&longs;tendetur ALN ad AkM ita e&longs;&longs;e, ut DPR ad <lb/><expan abbr="DOq.">DOque</expan> Quoniam autem ABC e&longs;t ad ALN, ut DEF ad DPR, <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/>ALN ad AKM e&longs;t, vt DPR ad <expan abbr="DOq;">DOque</expan> erit per conuer&longs;io­<lb/>nem rationis ALN ad LM, vt DPR ad <expan abbr="Pq.">Pque</expan> qua­<lb/>
<arrow.to.target n="marg269"></arrow.to.target> re ex &ecedil;quali BN e&longs;t ad LM, ut ER ad <expan abbr="Pq.">Pque</expan> Cùm au<gap/>em &longs;it <lb/>ALN ad AKM, ut DPR ad <expan abbr="DOq;">DOque</expan> erit diuidendo LM ad <lb/>AKM, vt PQ ad <expan abbr="DOq.">DOque</expan> Quocirca erit &longs;pacium BN ad <lb/>LM, vt ER ad PQ, & LM ad triangulum AKM, <lb/>vt PQ ad triangulum <expan abbr="DOq.">DOque</expan> Ex quibus per&longs;picuum <lb/>e&longs;t omnia triangula aliquam inter&longs;e habere &longs;imilitudinem, <lb/>ex qua po&longs;&longs;ibile fuit determinare in omnibus &longs;itum, vb<gap/>epe-
<pb pagenum="153"/>ritur centrum graurtatis. Quòd &longs;i figur&ecedil; nullam conuenien­<lb/>tiam, nullamquè &longs;imilitudinem inter&longs;e habuerint; ut in qua <lb/>drilateris, pentagonis, & reliquis figuris, quæ inter&longs;e neque <lb/>latera neque angulos &ecedil;quales <expan abbr="habeãt">habeant</expan>; & propterea nullam in­<lb/>ter&longs;e conuenientiam, & &longs;imilitudinem habere po&longs;&longs;unt; im­<lb/>po&longs;&longs;ibile quidem e&longs;&longs;et in ip&longs;is determinare &longs;itum <expan abbr="c&etilde;tri">centri</expan> grauita <lb/>tis; ita vt omnibus quadrilateris, ac omnibus pentagonis quo <lb/>modo cunque factis, & ita c&ecedil;teris figuris de&longs;eruire po&longs;&longs;it. Cum <lb/>exempli gratia in pentagonis modò in vno, modò in alio &longs;i­<lb/>tu centrum reperiatur; prout &longs;unt diuer&longs;&ecedil; figuræ. Po&longs;&longs;umus <lb/>quidem in vnaquaque figura reperire punctum po&longs;itione, <lb/>quod &longs;it quidem centrum grauitatis illius determinatæ figu­<lb/>r&ecedil;t. vtin fine primilibri o&longs;ten dimus. e&longs;&longs;et tamen impo&longs;&longs;ibile <lb/>in omnibus proprium certum, ac determinatum &longs;itum repe­<lb/>rire; vt &longs;cilicet &longs;it in tali linea, taliquè modo diui&longs;a, vtomnib^{9} <lb/>pentagonis, & hexagonis, cæteri&longs;què huiu&longs;modi de&longs;eruire <lb/>po&longs;&longs;it. vt determinatur in triangulis, & vt determinari pote&longs;t <lb/>in quadrilateris; quæ vel &longs;int parallelogramma, vel duo <expan abbr="&longs;alt&etilde;">&longs;altem</expan> <lb/>latera &longs;int æquidi&longs;tantia. cùm in his conuenientia, quàm <lb/>triangulis accidere o&longs;tendimus, reperiatur; quandoquidem <lb/>&longs;unt <expan abbr="triãgulorum">triangulorum</expan> portiones. &longs;imiliter in parallelogrammis fa <lb/>cilè erit o&longs;tendere aliquam inter&longs;e &longs;imilitudinem exi&longs;tere. <expan abbr="p&etilde;-tagona">pen­<lb/>tagona</expan> verò hexagona, & cæteræ figuræ, quæ angulos æqua­<lb/>les, & æqualia latera habent; iam con&longs;tat &longs;imilia e&longs;&longs;e inter &longs;e. <lb/>præterea circuliomnes &longs;unt &longs;imiles. Ellip&longs;es quoque inter&longs;e <lb/>aliquam habent &longs;imilitudinem, in quibus de&longs;cribitur figura, <lb/>planè in&longs;cripta. vt per&longs;picuum e&longs;t in libro Federici Comman <lb/>dini de centro grauitatis &longs;olidorum. ac propterea in his, & in <lb/>alijs, quibusinter&longs;e aliqua &longs;imililudo reperiri pote&longs;t, centrum <lb/>quoque grauitatis determinari poterit. </s></p>
<p type="margin">
<s><margin.target id="marg267"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 2.<emph type="italics"/>&longs;exti <lb/>ex lèmate <lb/><expan abbr="ĩ">im</expan> <expan abbr="&longs;ecũdã">&longs;ecundam</expan> d <lb/><expan abbr="mõ&longs;tratio-ne">mon&longs;tratio­<lb/>ne</expan><gap/>. pri­<lb/>mi huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg268"></margin.target>17. <emph type="italics"/>quinti. <lb/>coro.<emph.end type="italics"/> 19. <lb/><emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg269"></margin.target>22. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<figure id="fig71"></figure>
<p type="head">
<s>LEMMA.</s></p>
<p type="main">
<s>Sint quatuor magnitudines ABCD. &longs;itquè A maior B; <lb/>&C maior D. Dico A ad D maiorem habere proportio­<lb/>nem, quàm habet B ad C. </s></p>
<pb pagenum="154"/>
<p type="main">
<s>Hoc à nobis o&longs;ten&longs;um fuitinitio tractatus devecte in no­<lb/>&longs;tris mechanicishoc pacto. </s></p>
<figure></figure>
<p type="main">
<s>
<arrow.to.target n="marg270"></arrow.to.target> Quoniam enim A ad C maiorem habet pro<gap/><lb/>portionem, quam B ad C; & A ad D maiorem <lb/>quoque habet proportionem, quàm habetad C; <lb/>A igitur ad D maiorem habebit, quàm B ad C. <lb/>quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg270"></margin.target>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="head">
<s>PROPOSITIO. IIII.</s></p>
<p type="main">
<s>Omnis portionis recta linea, rectanguliquè co <lb/>ni &longs;ectione contentæ, centrum grauitatis e&longs;tin dia <lb/>metro portionis. </s></p>
<figure></figure>
<p type="main">
<s><emph type="italics"/>Sit portio, vt dicta e&longs;t, ABC; cuius diameter &longs;it BD. demon­<lb/>&longs;trandum est dictæ portionis centrum grauitatis e&longs;&longs;e in linea BD. &longs;i.n. <lb/>non, &longs;it punctum E. & ab ip&longs;o ducatur ip&longs;i BD aquidistans EF; at­<lb/>que in portione in&longs;cribatur triangulum ABC eandem ba&longs;im<emph.end type="italics"/> AC <lb/><emph type="italics"/>habens, & altitudinem<emph.end type="italics"/> portioni <emph type="italics"/>æqualem. & quam proportionem <lb/>habet CF ad FD, eandem habeat triangulum ABC ad &longs;pacium<emph.end type="italics"/>
<pb pagenum="155"/>k. <emph type="italics"/>in portione autem planè in&longs;cribatur figura rectilinea<emph.end type="italics"/> AGBNC, <emph type="italics"/>ita <lb/>vt relictæ portiones<emph.end type="italics"/> AOG GPB BQN NRC &longs;imul <emph type="italics"/>&longs;int minores<emph.end type="italics"/>
<arrow.to.target n="marg271"></arrow.to.target><lb/><emph type="italics"/>ip&longs;o K. in&longs;criptæ quidem rectilineæ figuræ centrum grauitatis est in linea <lb/>B D. &longs;it punctum H. connectaturquè HE, & producatur; &<emph.end type="italics"/> à pun <lb/>cto C <emph type="italics"/>ip&longs;i B D ducatur æquidistans CL.<emph.end type="italics"/> Quoniam autem por <lb/>tiones AOG GPB BQN NRC &longs;imul &longs;unt ip&longs;o K mino­<lb/>res; maiorem habebit proportionem triangulum ABC ad
<arrow.to.target n="marg272"></arrow.to.target> di­<lb/>ctas portiones, quàm ad K; in&longs;cripta verò figura AGBNC ma <lb/>ior e&longs;t triangulo ABC, K verò maius e&longs;t reliquis portionibus. <lb/><emph type="italics"/>Mani&longs;e&longs;tum est<emph.end type="italics"/> igitur <emph type="italics"/>figuram rectilineam<emph.end type="italics"/> ACBNC <emph type="italics"/>in portione in-<emph.end type="italics"/>
<arrow.to.target n="marg273"></arrow.to.target><lb/><emph type="italics"/>&longs;criptam <expan abbr="maior&etilde;">maiorem</expan> habere proportionem adreliquas portiones<emph.end type="italics"/> AOG GPB <lb/>BQN, NRC, <emph type="italics"/>quàm triangulum ABC ad K. &longs;ed vt triangulum <lb/>ABC ad K, ita est CF ad FD; figura igitur in&longs;cripta ad reliquas por­<lb/>tiones maiorem habebit proportionem, quam CF ad FD; hoc e&longs;t LE ad <lb/>EH.<emph.end type="italics"/> Cùm &longs;int LH CD à lineis æquidi&longs;tantibus LC EF
<arrow.to.target n="marg274"></arrow.to.target><lb/>HD drui&longs;æ. quare cùm figura in&longs;cripta ad reliquas portio­<lb/>nes maiotem habeat proportionem, quàm LE ad EH; linea, <lb/>quæ ad EH eandem habeat <expan abbr="proportion&etilde;">proportionem</expan>, quàm figura in&longs;cri­<lb/>pta ad reliquas portiones, maior erit, <expan abbr="quã">quam</expan> LE. <emph type="italics"/>Habeat igitur ME<emph.end type="italics"/>
<arrow.to.target n="marg275"></arrow.to.target><lb/><emph type="italics"/>ad EH <expan abbr="proportion&etilde;">proportionem</expan> eam, <expan abbr="quã">quam</expan> figura in&longs;cripta ad portiones. Quoniam igi­<lb/>tur punctum E centrum e&longs;t grauitatis totius portionis, figuræ <expan abbr="aut&etilde;">autem</expan> in ip&longs;a <lb/>in&longs;criptæ<emph.end type="italics"/> centrum grauitatis <emph type="italics"/>est punctum H: constat reliquæ magni­<lb/>tudinis ex circumrelictis portionibus compo&longs;itæ centrum grauitatis e&longs;&longs;e in <lb/>linea HE producta; ita vt a&longs;&longs;umpta aliqua recta linea<emph.end type="italics"/> ME <emph type="italics"/>eam proportio <lb/>nem habeat ad EH, quam figura in&longs;cripta ad circumrelictas portiones. <lb/>Quare magnitudinis ex circumrelictis portionibus compo&longs;itæ centrum gra <lb/>uitatis e&longs;t punctum M. quod est ab&longs;urdum. Ducta enim linea<emph.end type="italics"/> ST <emph type="italics"/>per <lb/>punctum M ip&longs;i BD æquidi&longs;tante, in ea omnes circumrelictæ portiones <lb/>centra grauitatis habebunt.<emph.end type="italics"/> hoc e&longs;t magnitudinis ex portioni­<lb/>bus BPG-BQN compo&longs;itæ centrum grauitatis e&longs;&longs;et in parte <lb/>MS. centrum verò grauitatis portionum AOG CRN e&longs;&longs;etin <lb/>parte MX; ita ut M omnium dictarum portionum e&longs;&longs;et gra­<lb/>uitatis centrum. quæ &longs;untquidem inconuenientia. quippè <lb/>quæ etiam eodem modo &longs;equentur, &longs;i ST ip&longs;i BD <expan abbr="æquidi&longs;tãs">æquidi&longs;tans</expan> <lb/>non e&longs;&longs;et. <emph type="italics"/>Patet igitur centrum grauitatis<emph.end type="italics"/> portionis ABC <emph type="italics"/>e&longs;&longs;e in <lb/>linea BD.<emph.end type="italics"/> quod demon&longs;trare oportebat. </s></p>
<pb pagenum="156"/>
<p type="margin">
<s><margin.target id="marg271"></margin.target>2. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg272"></margin.target>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg273"></margin.target><emph type="italics"/>lemma.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg274"></margin.target><emph type="italics"/>1: <expan abbr="tem-ĩ">tem-im</expan><emph.end type="italics"/> 13. <lb/><emph type="italics"/>primi hui<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg275"></margin.target>8. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>In hac demon&longs;tratione ob&longs;eruandum e&longs;t; quòd <expan abbr="quãdo">quando</expan> Ar­<lb/>chimedes inquit, <emph type="italics"/>in portione autem planè in&longs;cribatur figura<emph.end type="italics"/> &c. in­<lb/>telligendum e&longs;t, in&longs;cribatur primò pentagonum AGBNC <lb/>in portione planè in&longs;criptum; quod quidem relinquet por­<lb/>tiones AOG GPB BQN NRC, quæ &longs;imul uel erunt minores <lb/>&longs;pacio K, vel minùs. &longs;i non, rur&longs;us planè adhuc in&longs;cribatur <lb/>in portione ABC nonagonum; deinde alia figura; idquè &longs;em­<lb/>per fiat, donec circumrelictæ portiones &longs;imul &longs;int &longs;pacio K <lb/>minores. quod quidem fieri po&longs;&longs;e ex prima decimi Euclidis <lb/>
<arrow.to.target n="marg276"></arrow.to.target> patet. Aufertur enim &longs;emper maius, <expan abbr="quã">quam</expan> dimidium. Cùm quæ <lb/>libet portio paraboles trianguli plane in ip&longs;a in&longs;eripti &longs;it &longs;e&longs;­<lb/>quitertia. Vnde triangulum ABC maius e&longs;t, quàm <expan abbr="dimidiũ">dimidium</expan> <lb/>portionis ABC. triangulum què AGB maius, quàm <expan abbr="dimidiũ">dimidium</expan> <lb/>portionis AGB. &longs;imiliter triangulum BNC portionis BNC & <lb/>ita in alijs. Quæ quidem omnia &longs;untquoque manife&longs;ta ex vi <lb/>ge&longs;ima propo&longs;itione, eiu&longs;què demon&longs;tratione de quadratura <lb/>paraboles Archimedis. </s></p>
<p type="margin">
<s><margin.target id="marg276"></margin.target>17. <emph type="italics"/>Archi. <lb/>de quad. <lb/>parab.<emph.end type="italics"/></s></p>
<p type="main">
<s>Demon&longs;trato centro grauitatis cuiu&longs;libet paraboles in eius <lb/>diametro exi&longs;tere; o&longs;tendet Archimedes, (vt diximus) in pa­<lb/>rabolis grauitatum centra in eadem proportione diametros <lb/>di&longs;pe&longs;cere. antequam autem hoc demon&longs;tret, duas pr&ecedil;mittit <lb/>&longs;equentes propo&longs;itiones ad demon&longs;trationem nece&longs;&longs;arias. </s></p>
<p type="head">
<s>PROPOSITIO. V.</s></p>
<p type="main">
<s>Si in portione recta linea, rectanguliquè coni <lb/>&longs;ectione contenta rectilinea figura planè in&longs;criba <lb/>tur, totius portionis <expan abbr="centrũ">centrum</expan> grauitatis <expan abbr="propĩquius">propinquius</expan> <lb/>e&longs;t vertici portionis, <expan abbr="quã">quam</expan> <expan abbr="centrũ">centrum</expan> figuræ in&longs;criptæ. </s></p>
<pb pagenum="157"/>
<p type="main">
<s><emph type="italics"/>Sit portio ABC, qualis dictaest, ip&longs;ius verò diameter &longs;it BD. <lb/>primùmquè in ip&longs;a planè in&longs;eribatur triangulum ABC. & diuidatur<emph.end type="italics"/>
<arrow.to.target n="marg277"></arrow.to.target><lb/><emph type="italics"/>BD in E, ita vt dupla &longs;it BE ip&longs;ius ED. erit vtiquè trtanguli ABC <lb/>centrum grauitatis punctum E. Diuidatur itàque bi&longs;ariam vtraque <lb/>AB BC in punctis FG. & <gap/>punctis FG ip&longs;i BD ducantur æquidi­<lb/>&longs;tantes FK GL. erit &longs;anè portionis A<emph.end type="italics"/>k<emph type="italics"/>B centrum grauitatis in linea<emph.end type="italics"/>
<arrow.to.target n="marg278"></arrow.to.target><lb/><emph type="italics"/>F<emph.end type="italics"/>k. <emph type="italics"/>portionis verò BLC centrum grauit atis erit in linea GL. &longs;int ita­<lb/>que puncta HI. connectanturquè HI FG.<emph.end type="italics"/> quæ BD &longs;ecent in QN. <lb/>
<arrow.to.target n="fig72"></arrow.to.target><lb/>erit vtique punctum Q vertici B propinquius, quàm N. quia
<arrow.to.target n="marg279"></arrow.to.target><lb/>verò e&longs;t BF ad FA, vt BG ad GC, erit FG <expan abbr="æquidi&longs;tãsip&longs;i">æquidi&longs;tansip&longs;i</expan> AC, <lb/>eritquè FN ad NG, vt AD ad DC. e&longs;t verò AD ip&longs;i DC æqua­<lb/>lis, ergo FN NG inter&longs;e &longs;unt æquales. quoniam autem FN <lb/>e&longs;t ip&longs;i AD æquidi&longs;tans, erit AF ad FB, vt DN ad NB. e&longs;t au
<arrow.to.target n="marg280"></arrow.to.target><lb/>tem AF dimidia ip&longs;ius AB; cùm &longs;int AF FB &ecedil;quales ergo & <lb/>DN dimidia e&longs;t ip&longs;ius DB. at verò quoniam DE terria e&longs;t <lb/>pars ip&longs;ius DB, &longs;iquidem e&longs;t BE ip&longs;ius ED dupla, erit pun­<lb/>ctum N propinquius vertici B portionis, quàm pun­<lb/>ctum E. <emph type="italics"/>Et quoniam parallelogrammum est HFGI. & æqualis est <lb/>FN ip&longs;i NG, erit QH ip&longs;i QI æqualis. ac propterea magnitudinis ex <lb/>vtri&longs;que A<emph.end type="italics"/>k<emph type="italics"/>B BLC portionibus compo&longs;itæ centrum grauitatis e&longs;t in<emph.end type="italics"/>
<arrow.to.target n="marg281"></arrow.to.target><lb/><emph type="italics"/>medio lineæ HI, cùm portiones<emph.end type="italics"/> AKB BLC <emph type="italics"/>&longs;int æquales. erit &longs;cilicet <lb/>punctum <expan abbr="q.">que</expan> Quoniam autem trianguli ABC centrum grauitatis e&longs;t <lb/>punctum E, magnitudinis verò ex vtri&longs;què A<emph.end type="italics"/>k<emph type="italics"/>B BLC compo&longs;isæ<emph.end type="italics"/>
<pb pagenum="158"/><emph type="italics"/>e&longs;t punctum <expan abbr="q.">que</expan> con&longs;tat totius portionis ABC centrum grauitatis e&longs;&longs;e<emph.end type="italics"/><lb/>
<arrow.to.target n="marg282"></arrow.to.target> <emph type="italics"/>in linea QE. hoc est inter puncta QE. Quare totius portionis <expan abbr="cētrum">centrum</expan> <lb/>grauitatis propinquius e&longs;t vertici portionis, quam<emph.end type="italics"/> centrum grauitatis <lb/><emph type="italics"/>trianguli planè in&longs;cripti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg277"></margin.target><emph type="italics"/>ante pri­<lb/>mi huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg278"></margin.target>4. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg279"></margin.target>2. <emph type="italics"/>&longs;exti­<lb/>lemma ta <lb/>aliter<emph.end type="italics"/> 13. <lb/><emph type="italics"/>primi hui^{9}<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg280"></margin.target>2. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg281"></margin.target>4. <emph type="italics"/>primi <lb/>buius. <lb/>ex its quæ <lb/>ante<emph.end type="italics"/> 2. <emph type="italics"/>hu <lb/>ius demon <lb/>&longs;trata &longs;unt. <lb/>ex<emph.end type="italics"/> 8. <emph type="italics"/>pri­<lb/>mi huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg282"></margin.target>*</s></p>
<figure id="fig72"></figure>
<figure></figure>
<p type="main">
<s><emph type="italics"/>Rur&longs;us in portione pent agonum rectilineum AKBLC planè in&longs;cri­<lb/>batur. &longs;itquè totius portionis diameter BD, vtriu&longs;que autem portionis<emph.end type="italics"/><lb/>AKB. BLC <emph type="italics"/>diameter &longs;it vtraque KF LG. & quoniam in portione <lb/>AKB planè in&longs;cripta est figura rectilinea<emph.end type="italics"/> trilatera AKB, <emph type="italics"/>totius por <lb/>tionis<emph.end type="italics"/> AKB <emph type="italics"/>centrum grauitatis est propinquius vertici<emph.end type="italics"/> K, <emph type="italics"/>quam <lb/>centrum rectilineæ figuræ<emph.end type="italics"/> AKB. <emph type="italics"/>&longs;it itaque portionis A<emph.end type="italics"/>k<emph type="italics"/>B centrum <lb/>grauitatis punctum H; trianguli verò punctum 1. Rur&longs;us autem &longs;it por <lb/>tionis BLC centrum grauitatis punctum M. trianguli verò<emph.end type="italics"/> BLC <emph type="italics"/>pun <lb/>ctum N. iunganturquè HM JN<emph.end type="italics"/>; quæ BD &longs;ecentin punctis <lb/>QT. erit vtique punctum Q vertici B propinquius, <expan abbr="quã">quam</expan> <lb/>T. & quoniam (&longs;i ducta e&longs;&longs;et FG) lineæ HM IN FG ab æ <lb/>
<arrow.to.target n="marg283"></arrow.to.target> quidi&longs;tantibus lineis KF BD LG in eadem <expan abbr="diuidũtur">diuiduntur</expan> pro­<lb/>portione. FG verò, vt o&longs;ten&longs;um e&longs;t, bifariam à linea BD di­<lb/>uideretur; ergo & lineæ HM IN bifariam diui&longs;&ecedil; <expan abbr="proucni&etilde;t">proucnient</expan>. <lb/><emph type="italics"/>æqualis est igitur HQ ip&longs;i QM; & IT ip&longs;i TN. &longs;ed triangulo <lb/>AKB æquale est triangulum BLC; portio vero A<emph.end type="italics"/>k<emph type="italics"/>B portioni <lb/>BLC e&longs;t æqualis. Demonstratum e&longs;t enim alis in loçis portiones<emph.end type="italics"/>
<pb pagenum="159"/><emph type="italics"/>&longs;e&longs;quitertias e&longs;&longs;e triangulorum, erit igitur magnitudinis ex vtri&longs;que por-<emph.end type="italics"/>
<arrow.to.target n="marg284"></arrow.to.target><lb/><emph type="italics"/>tionibus A<emph.end type="italics"/>k<emph type="italics"/>B BLC compo&longs;itæ centrum grauitatis punctum <expan abbr="q.">que</expan> magni­<lb/>tudinis verò ex vtri&longs;que triangulis AKB BLC compo&longs;itæ punctum <lb/>T. Rur&longs;us itaque quoniam trianguli ABC centrum grauitatis e&longs;t <expan abbr="punctū">punctum</expan> <lb/>E, magnitudinis verò ex vtri&longs;que A<emph.end type="italics"/>k<emph type="italics"/>B BLC portionibus punctum <lb/><expan abbr="q.">que</expan> manifestum e&longs;t totius portionis A<emph.end type="italics"/>B<emph type="italics"/>C centrum grauitatis e&longs;&longs;e in linea <lb/>QE ita diui&longs;a<emph.end type="italics"/> in O puncto, <emph type="italics"/>vt quam proportionem habet trian­<lb/>gulum ABC ad vtra&longs;que portiones A<emph.end type="italics"/>k<emph type="italics"/>B BLC, eandem habeat por<emph.end type="italics"/>
<arrow.to.target n="marg285"></arrow.to.target><lb/><emph type="italics"/>tio ip&longs;ius terminum habens punctum Q,<emph.end type="italics"/> hoc e&longs;t OQ <emph type="italics"/>ad portionem <lb/>minorem<emph.end type="italics"/> OE. <emph type="italics"/>pentagoni autem AKBLC,<emph.end type="italics"/> hoc e&longs;t magnitudinis <lb/>ex triangulo ABC, trianguli&longs;què AKB BLC compo&longs;itæ <lb/><emph type="italics"/>centrum grauitatis e&longs;t in linea ET &longs;ic diui&longs;a<emph.end type="italics"/> in S, <emph type="italics"/>vt quam habet <lb/>proportionem triangulum ABC ad triangula AKB BLC, eande ha­<lb/>beat portio ip&longs;ius ad T terminata,<emph.end type="italics"/> hoc e&longs;t ST <emph type="italics"/>ad reliquam<emph.end type="italics"/> SE. <lb/><emph type="italics"/>Quoniam igitur maiorem habet proportionem triangulum ABC ad <expan abbr="triã">triam</expan><emph.end type="italics"/>
<arrow.to.target n="marg286"></arrow.to.target><lb/><emph type="italics"/>gula KAB LBC, quam ad portiones<emph.end type="italics"/> AKB BLC; minora enim <lb/>&longs;unt triangula portionibus. habebit TS ad SE <expan abbr="mior&etilde;">miorem</expan> pro­<lb/>portio nem, quam QO ad OE ac propterea erit <expan abbr="punctũ">punctum</expan> S <lb/>propinquiusip&longs;i E, quàm O. Nam &longs;i punctum S primùm <lb/>e&longs;&longs;et in eodem puncto O, tunc TO ad OE, non quidem <lb/>maiorem, &longs;ed minorem haberet proportionem, quàm QO
<arrow.to.target n="marg287"></arrow.to.target><lb/>ad OE, cùm &longs;it TO minor QO. &longs;imiliter ob eadem cau <lb/>&longs;am &longs;i punctum S e&longs;&longs;et inter OT, minorem haberet
<arrow.to.target n="marg288"></arrow.to.target> pro­<lb/>portionem TS ad SE, quàm QS ad SE, quare & ad huc <lb/>maiorem haberet proportionem QO ad OE, quàm TS <lb/>ad SE. nece&longs;&longs;e e&longs;t igitur punctum S e&longs;&longs;e inter puncta OE. <lb/>Itaquè cùm punctum O &longs;it <expan abbr="centrũ">centrum</expan> grauitatis portionis ABC, <lb/>punctum verò S centrum &longs;it grauitatis rectilineæ figuræ <lb/>AK BLC; <emph type="italics"/>constat portionis ABC centrum grauitatis propinquius <lb/>e&longs;&longs;e vertici B, quàm centrum rectilineæ figuræ in&longs;criptæ. Et in om­<lb/>nibus rectilineis figuris in portionibus planè in&longs;criptis eadem e&longs;t ratio.<emph.end type="italics"/><lb/>quod demon&longs;trare oportebat. </s></p>
<pb pagenum="160"/>
<p type="margin">
<s><margin.target id="marg283"></margin.target><emph type="italics"/>prima lem <lb/>ma in<emph.end type="italics"/> 13. <lb/><emph type="italics"/>primi bui^{9}.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg284"></margin.target>4. <emph type="italics"/>primi <lb/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg285"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 8. <emph type="italics"/>pri­<lb/>mi huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg286"></margin.target>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg287"></margin.target>8. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg288"></margin.target>8.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>
<arrow.to.target n="marg289"></arrow.to.target> In fine primæ demon&longs;trationis in vltima conclu&longs;ione <expan abbr="quã-do">quan­<lb/>do</expan> inquit Archimedes. <emph type="italics"/>Quare totius portionis centrum propinquius <lb/>e&longs;t vertici portionis, <expan abbr="quã">quam</expan> trianguli planè in &longs;cripti<emph.end type="italics"/> Gra cus codexita &longs;e <lb/>habet <foreign lang="greek">w)\s t) e)\ihka e)ggute<10>on ta=s tou= tma/matos ko<10>ufa=s to\ ke/nt<10>on tou= o)\lou <lb/>tma/matos, h)/ tou= e)gg<10>afome/nou t<10>igw/nou gnw<10>i/mws</foreign>. verbaquè <foreign lang="greek">e)/ih ka</foreign> malè in <lb/>terpo&longs;ita &longs;unt, nullumquè cum alijs rectum &longs;en&longs;um habent, <lb/>quare horum loco ponerem <foreign lang="greek">e)si/</foreign>, vt&longs;en&longs;us &longs;it, <foreign lang="greek">w)/s te)/ggu/te<10>on e)si ta=s <lb/>tou= tma/matos</foreign>, &c. </s></p>
<p type="margin">
<s><margin.target id="marg289"></margin.target>*</s></p>
<figure></figure>
<p type="main">
<s>Ob&longs;eruandum autem occurrit in demon&longs;trationibus, ab <lb/>Archimede allatis; quòd in prima demon&longs;tratione &longs;upponit <lb/>Archimedes, HFGI e&longs;&longs;e parallelogrammum. quòd vt &longs;it pa­<lb/>rallelogrammum, nece&longs;&longs;e e&longs;t &longs;upponere centra grauitatis HI <lb/>&longs;ecare lineas KF LG in partes inuicem proportionales. quod <lb/>tamen &longs;upponi po&longs;&longs;e minimè videtur. Et &longs;i quis ex quinto <lb/>po&longs;tulato obijceret, centragrauitatis in æqualibus, &longs;imilibu&longs;­<lb/>què figuris e&longs;&longs;e æqualiter po&longs;ita; admitti quidem pote&longs;t; quo-
<pb pagenum="161"/>niam figuræ, ipforum què centra inter&longs;e coaptari po&longs;&longs;unt. vt <lb/>omnibus figuris rectilineis &ecedil;qualibus, & &longs;imilib^{9} accidere po­<lb/>te&longs;t. Hoc tamé contingere po&longs;&longs;e in parabolis, vt AKB BLC, vi <lb/>detur in <expan abbr="cõueniés">conueniés</expan>. <expan abbr="Nã">Nam</expan> quamuis AKB BLC &longs;int æquales, & &longs;int <lb/><expan abbr="etiã">etiam</expan> &longs;imiles; non &longs;unt tamen &longs;imiles ea &longs;i militudine, vt &longs;untre <lb/>ctilineæ figuræ; vtantea diximus. Quod etiam <expan abbr="per&longs;picuũ">per&longs;picuum</expan> fit ex <lb/>hoc, quia non &longs;emper coaptari porei&longs;t portio AKB <expan abbr="cũ">cum</expan> portio­<lb/>ne BLC. <expan abbr="nõ">non</expan>. n.&longs;emper recta linea BC erit æqualisip&longs;i BA; <expan abbr="ne&qacute;">neque</expan>; <lb/>&longs;ectionis linea BLC &longs;ectionis line&ecedil; BKA &ecedil;qualis exi&longs;tet. <expan abbr="Cũ">Cum</expan> <expan abbr="nõ">non</expan> <lb/>&longs;emper AC, & quæ &longs;untip&longs;i AC æquidi&longs;tates ad rectos &longs;int an <lb/>gulos diametro BD. &longs;i.n. &ecedil;quidi&longs;tantes line&ecedil; diametro fuerint <lb/>perpendiculares, tunc AB BC inter&longs;e &ecedil;quales e&longs;&longs;ent; <expan abbr="portio&qacute;">portioque</expan>; <lb/>AKB <expan abbr="cũ">cum</expan> portione BLC coaptari po&longs;&longs;et: &longs;ecùs autem minimè. <lb/>Quare centra grauiratis HI lineas KFLG in eadem proportio <lb/>ne &longs;ecare minimè&longs;upponi po&longs;&longs;e videtur; tùm exijs, quæ dicta <lb/>&longs;unt; tú quia hoc o&longs;tendet Archimedes in &longs;eptima propo&longs;itio <lb/>ne. quòd &longs;i adhuc non e&longs;t <expan abbr="demõ&longs;tratú">demon&longs;tratú</expan>, <expan abbr="nõ">non</expan> pote&longs;t <expan abbr="quo&qacute;">quoque</expan>; &longs;uppo <lb/>ni; præ&longs;ertim cùm &longs;it demon&longs;trabile. ac propterea <expan abbr="demõ&longs;tra-tio">demon&longs;tra­<lb/>tio</expan> nullam videturvim haberead <expan abbr="o&longs;tendendũ">o&longs;tendendum</expan>, quod propo&longs;i­<lb/>tú fuit. Huic <expan abbr="tam&etilde;">tamen</expan> occurri po&longs;&longs;evidetur <expan abbr="cũ">cum</expan> Eutocio in exphca <lb/>tione huiusloci dicendo, hoc &longs;upponere Archimedé, quia por <lb/>tiones AKBBLC &longs;unt&ecedil;quales, quarú diametri KFLG &longs;unt &ecedil;­<lb/>quales, & <expan abbr="&ecedil;quidi&longs;tãtes">&ecedil;quidi&longs;tantes</expan>, quæ &longs;imiliter diuiduntur à punctis HI; <lb/>vnde erit kG ad HF, vt LI ad IG. ex quibus colligit HF ip&longs;i IG <lb/><expan abbr="æqual&etilde;">æqualem</expan> e&longs;&longs;e; ac propterea HG <expan abbr="parallelogrãmũ">parallelogrammum</expan> exiltere. Quæ <expan abbr="tñ">tnm</expan> <lb/>re&longs;pon&longs;io <expan abbr="nõ">non</expan> e&longs;t Eutocio digna. cùm ex dictis <expan abbr="nõ">non</expan> &longs;it omninò <lb/>demon&longs;tratiua, vtres mathematic&ecedil; <expan abbr="requirũt">requirunt</expan>; quapropter omit <lb/>tenda e&longs;t.hac.n.ratione&longs;upponitur centra HI lineas KFLG in <lb/>eadem proportione &longs;ecare.quod nullo modo &longs;upponi pote&longs;t. <lb/>Quare dici poterit, & forta&longs;le rectiùs, quòd vis demon&longs;tratio­<lb/>nis videtur in hoc e&longs;&longs;e con&longs;tituta, vt &longs;upponatur puncta HI <expan abbr="v-bicun&qacute;">v­<lb/>bicunque</expan>; e&longs;&longs;e po&longs;&longs;e in lineis KFLG; ita vt &longs;iue ducta HI fuerit, <lb/>&longs;iue etiam non fuerit ip&longs;i FG æquidi&longs;tans, demon&longs;tratio <expan abbr="tam&etilde;">tamen</expan> <lb/>&longs;uam &longs;emper habebit vim, <expan abbr="id&etilde;&qacute;">idenque</expan>; concludet. Nam ex <expan abbr="præced&etilde;">præcedem</expan>. <lb/>ti patet centra grauitatis portionum AKB BLC e&longs;&longs;e in lineis <lb/>KF LG; hoce&longs;t inter puncta KF, & LG. <expan abbr="&longs;upponãturita&qacute;">&longs;upponanturitaque</expan>; <expan abbr="c&etilde;-tra">cen­<lb/>tra</expan> grauitatis <expan abbr="portionũ">portionum</expan> AKB BLC e&longs;&longs;e puncta HI <expan abbr="vbicũ&qacute;">vbicunque</expan>; po­
<pb pagenum="162"/>&longs;ita, <expan abbr="dũmodo">dummodo</expan> &longs;int in lineis KF LG, veluti Archimedes ip&longs;e in <lb/>d mon&longs;tratione &longs;upponit. <expan abbr="Ducatur&qacute;">Ducaturque</expan>; HI; quæ vel ip&longs;i FG æ­<lb/>quidi&longs;tans erit, vel minùs: &longs;i e&longs;t æquidi&longs;tans, <expan abbr="parallelogrãmũ">parallelogrammum</expan> <lb/>e&longs;t HFGI, & vera e&longs;t demon&longs;tratio Archimedis. &longs;i verò <expan abbr="nõ">non</expan> e&longs;t <lb/><expan abbr="æquidi&longs;tãs">æquidi&longs;tans</expan>, nihilominus veri&longs;&longs;ima e&longs;t eadem <expan abbr="demõ&longs;tratio">demon&longs;tratio</expan>. <expan abbr="Nã">Nam</expan> <lb/>&longs;i HI ip&longs;i FG <expan abbr="nõ">non</expan> e&longs;t <expan abbr="&ecedil;quidi&longs;tãs">&ecedil;quidi&longs;tans</expan>, patet in primis <expan abbr="pũctũ">punctum</expan> Qpropin <lb/>quius e&longs;&longs;e vertici B portionis ABC, <expan abbr="quã">quam</expan> <expan abbr="punctũ">punctum</expan> N, ac per con­<lb/>&longs;equens, <expan abbr="quã">quam</expan> punctum E centrum grauitatis trianguli ABC. <lb/>Etquoniam lineæ HI FG à lineis diuiduntur KF BN LG &ecedil; <lb/>
<arrow.to.target n="fig73"></arrow.to.target><lb/>
<arrow.to.target n="marg290"></arrow.to.target> quidi&longs;tantibus, erit HQ ad QI, vt FN ad NG. e&longs;t autem FN i­<lb/>pGNG &ecedil;qualis, ergo HQ ip&longs;i QI &ecedil;qualis quoque erit. itaque <lb/>quoniam portiones AKBBLC &longs;unt æquales, erit magnitudi­<lb/>nis ex vtri&longs;que AKB BLC portionibus compo&longs;it&ecedil; <expan abbr="centrũ">centrum</expan> gra­<lb/>uitatis in medio line&ecedil; HI. ergo eritpunctum <expan abbr="q.">que</expan> quo cognito <lb/>eadem demon&longs;tratio Archimedis o&longs;tendet centrum grauita­<lb/>tis portionis ABC e&longs;&longs;e inter puncta <expan abbr="Eq.">Eque</expan> Nam ex verbis ip&longs;ius, <lb/>cùm ait, <emph type="italics"/>Quoniam autem trianguli ABC centrum grauitatis est <lb/>punctum E magnitudinis verò ex vtri&longs;que AkB BLC compo&longs;icæ <lb/>est punctum <expan abbr="q;">que</expan> constat totius portionis ABC centrum grauitatis <lb/>e&longs;&longs;e in in linea QE. hoc est inter puncta QE. Quare totius portionis <lb/>centrum grauitatis propinquius e&longs;t vertici portionis, quàm trian­<lb/>guli planè in&longs;cripti.<emph.end type="italics"/> <expan abbr="manife&longs;tũ">manife&longs;tum</expan> e&longs;t igitur centrum grauitatis por <lb/>tionis ABC, &longs;iuè &longs;it HI ip&longs;i FG æquidi&longs;tans, &longs;iue non æ. <lb/>quidi&longs;tans, propinquius e&longs;&longs;e vertici B portionis, quàm <expan abbr="c&etilde;trum">centrum</expan>
<pb pagenum="163"/>grauitatis trianguli ABC<gap/> Quare cuca <gap/>erba demon&longs;tratio­<lb/>nis, cùm inquit Archimedes, <emph type="italics"/>& quoniam parallelogrammum est <lb/>HFGJ, & æqualisest FN ip&longs;i NG.<emph.end type="italics"/> &c. immitando &longs;ecun­<lb/>dam Archimedis demon&longs;trationem huius propo&longs;itionis, vel <lb/>delenda &longs;untverba, <emph type="italics"/>parallelogrammum e&longs;t HFGI, &<emph.end type="italics"/> tamquam <lb/>ab aliquo ad dita; ita vt verba &longs;int hoc modo vniuer&longs;alia, <emph type="italics"/>& <lb/>quoniam æqualis e&longs;t FN ip&longs;i NG,<emph.end type="italics"/> & quæ &longs;equuntur. vel &longs;at for­<lb/>ta&longs;&longs;e Archimedi vi&longs;um e&longs;t. &longs;e o&longs;tendi&longs;&longs;e hoc contingere exi­<lb/>&longs;tente HI ip&longs;i FG æquidi&longs;tante. quòd &longs;i etiam non fuerit HI <lb/>æquidi&longs;tans FG, idem &longs;equi tanquam notum omi&longs;it. cùm per <lb/>facilis &longs;it demon&longs;tratio, vt dictum e&longs;t. Archimede&longs;què res val <lb/>dè notas &longs;&ecedil;pè prætermittere&longs;olet. </s></p>
<p type="margin">
<s><margin.target id="marg290"></margin.target>1.<emph type="italics"/><expan abbr="l&etilde;waĩ">lenwaim</expan><emph.end type="italics"/> 15 <lb/><emph type="italics"/>primu hu­<lb/>ius.<emph.end type="italics"/></s></p>
<figure id="fig73"></figure>
<p type="main">
<s>Hocidem etiam con&longs;iderari pote&longs;t in &longs;ecunda demon&longs;tra <lb/>tione quamuis verba hanc difficultatem non habeant. <expan abbr="nã">nam</expan> ea­<lb/>dem &longs;equltur demon&longs;tratio, &longs;iuè&longs;it HM lineæ IN &ecedil;quidi&longs;tás, <lb/>vel non æquidi&longs;tans, vt ex verbis Archimedis per&longs;picuum e&longs;t.
<arrow.to.target n="marg291"></arrow.to.target><lb/>etenim manife&longs;tum e&longs;t centra grauitatis portionum AKB <lb/>BLC e&longs;&longs;einlineis KF LG. &longs;imiliter centra grauitatis
<arrow.to.target n="marg292"></arrow.to.target> trian­<lb/>gulorum AKB BLC in ijsdem e&longs;&longs;e lineis KF LG. vtin <expan abbr="pũ-ctis">pun­<lb/>ctis</expan> IN; quæ nece&longs;&longs;ariò diuidunt KF LG in partes propor­<lb/>tionales, vnde FI GN euadunt æquales. & quoniam por­<lb/>tionum centra HM &longs;unt propinquiora verticibus KL, quam <lb/>triangulorum centra IN; ideo nece&longs;&longs;e e&longs;t <expan abbr="pũcta">puncta</expan> HM in lineis <lb/>KI LN exi&longs;tere. quare &longs;int puncta HM vbicúque in lineis KI <lb/>LN con&longs;tituta; <expan abbr="ducta&qacute;">ductaque</expan>; HM, quæ &longs;iuè &longs;it ip&longs;i IN &ecedil;quidi&longs;tans, <lb/>&longs;iuenon æquidi&longs;tans, &longs;em per erit <expan abbr="pũctum">punctum</expan> Qpropinquius ver <lb/>tici B, quam T. eodem què modo erit punctum Q <expan abbr="mediũ">medium</expan> li­<lb/>neæ HM <expan abbr="centrũ">centrum</expan> grauitatis magnitudinis ex portionib^{9} AKB <lb/>BLC compo&longs;itæ. &longs;iquidem portiones &longs;unt &ecedil;quales. qu&ecedil; <expan abbr="quid&etilde;">quidem</expan> <lb/>omnia ex ip&longs;amet demon&longs;tratione &longs;unt manife&longs;ta. &longs;untquè <lb/>hæc <expan abbr="ead&etilde;">eadem</expan> <expan abbr="ob&longs;eruãda">ob&longs;eruanda</expan> in duabus <expan abbr="&longs;equ&etilde;tibus">&longs;equentibus</expan> <expan abbr="demõ&longs;trationib^{9}">demon&longs;trationib^{9}</expan>. </s></p>
<p type="margin">
<s><margin.target id="marg291"></margin.target>4. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg292"></margin.target><emph type="italics"/>ante<emph.end type="italics"/> 15. <lb/><emph type="italics"/>primi hu­<lb/>ius.<emph.end type="italics"/></s></p>
<p type="head">
<s>PROPOSITIO. VI.</s></p>
<p type="main">
<s>Data portione rectalinea, rectanguliquè coni <lb/>&longs;ectione <expan abbr="cõtenta">contenta</expan>, in portione figurarectilinea pla <lb/>ne in&longs;cribi pote&longs;t; ita vt linea inter centrum graui­
<pb pagenum="164"/>tatis portionis, & figuræ rectilineæ in&longs;criptæ, mi­<lb/>nor &longs;it propo&longs;ita recta linea. </s></p>
<figure></figure>
<p type="main">
<s><emph type="italics"/>Data &longs;it portio ABC, qualis dicta est. cuius centrum grauitatis &longs;it <lb/>punctum H. & in ip&longs;a planè in&longs;cribatur triangulum ABC. &longs;itquè pro <lb/>po&longs;ita recta linea F. & quam proportionem habet BH ad F, eandem <lb/>habeat triangulum ABC ad &longs;pacium<emph.end type="italics"/> k. <emph type="italics"/>inportione autem ABC pla<emph.end type="italics"/><lb/>
<arrow.to.target n="marg293"></arrow.to.target> <emph type="italics"/>nè in&longs;cribatur figura rectilinea AGB LC, ita vt circumrelict æ portio <lb/>nes<emph.end type="italics"/> ANG GOB BPL LQC &longs;imul &longs;umpt&ecedil; <emph type="italics"/>&longs;int minoresip&longs;o<emph.end type="italics"/> k<emph type="italics"/>: <lb/>ip&longs;iu&longs;què figuræ in&longs;criptæ centrum grauitatis &longs;it punctum E. Dico li­<lb/>neam HE minorem e&longs;&longs;e ip&longs;a F. N am&longs;i non, vel æqualis est, vel <lb/>maior. Quoniam autem<emph.end type="italics"/> maior e&longs;t figura rectilinea AGBLC, <lb/>quàm triangulum ABC, maius verò e&longs;t &longs;pacium K portio­<lb/>nibus ANG GOB BPL LQC &longs;imul &longs;umptis, ideo <emph type="italics"/>rectili-<emph.end type="italics"/><lb/>
<arrow.to.target n="marg294"></arrow.to.target> <emph type="italics"/>nea figura AGBLC ad circumrelictas portiones maiorem habet pro­<lb/>portionem, quàm triangulum ABC ad K. hoc est HB ad F. at ue <lb/>rò BH nonhabet minorem proportionem ad F, quàm habet ad HE. <lb/>cùmnon &longs;it minor HE ip&longs;a F.<emph.end type="italics"/> &longs;i enim ponatur HE ip&longs;i F
<pb pagenum="165"/>æqualis, eandem habebit proportionem BH ad HE, <expan abbr="quã">quam</expan>
<arrow.to.target n="marg295"></arrow.to.target><lb/>ad F. quæ e&longs;t proportio trianguli ABC ad. K. vnde figu­<lb/>ra rectilinea AGBLC ad circumrelictas portiones maiorem, <lb/>habebit proportionem, quàm BH ad HE. &longs;i verò ponatur <lb/>HE maior, quàm F, habebit BH ad F, hoc e&longs;t <expan abbr="triangulũ">triangulum</expan>
<arrow.to.target n="marg296"></arrow.to.target><lb/>ABC ad K maiorem proportionem, quàm BH ad HE. <lb/><emph type="italics"/>multo igitur maiorem habet proportionem figura rectilinea AGBLC ad <lb/>circumrelictas portiones, quàm BH ad HE. Quare &longs;i fiat ut rectili­<lb/>linea figura AGBLC ad circumrelictas portiones, &longs;ic alia quædam li­<lb/>nea ad HE. erit maior, quàm BH. &longs;itquè HM. Cùm enim portio­<lb/>nis ABC centrum grauitatis &longs;it H. figuræ verò rectilineæ AGBLC <lb/>punctum E. producta EH, a&longs;&longs;umptaquè aliqua recta linea proportione <lb/>babente ad EH, quam rectilineum AGBLC ad circumtelictas por­<lb/>tiones; maior erit quàm HB. habeat igitur<emph.end type="italics"/> (vt dictum e&longs;t) <emph type="italics"/>MH ad <lb/>HE<emph.end type="italics"/> proportionem eam, quam habet figura AGBLC ad reli
<arrow.to.target n="marg297"></arrow.to.target><lb/>quas portiones, <emph type="italics"/>ergopunctum M centrum est grauit atis magnitudi­<lb/>nis ex circumrelictis portionibus compo&longs;itæ. quod e&longs;&longs;e non pote&longs;t. Ducta <lb/>enimrecta linea<emph.end type="italics"/> RS <emph type="italics"/>per M ip&longs;i AC æquidistante, inip&longs;a &longs;unt centra <lb/>grauitatis vnicuiquè portioni re&longs;pondentia<emph.end type="italics"/>; ita &longs;cilicet vt centrum <lb/>magnitudinis ex portionibus ANG GOB compo&longs;itæ &longs;it in <lb/>linea RS. &longs;ed in parte MR. in parteverò MS &longs;it grauitatis <lb/>centrum magnitudinis ex reliquis portionibus BPL LQC <lb/>compo&longs;itæ; ita vt punctum M magnitudinis ex omnibus <lb/>portionibus compo&longs;itæ centrum grauitatisexi&longs;tat. quæ <expan abbr="tam&etilde;">tamen</expan> <lb/>e&longs;&longs;e non po&longs;&longs;unt. quod idem accideret, &longs;i etiam RS ip&longs;i AC <lb/>æquidi&longs;tans non e&longs;&longs;et. <emph type="italics"/>Patetigitur HE minorem e&longs;&longs;e, quam F.<emph.end type="italics"/><lb/>cùm neque maior, neque &ecedil;qualis e&longs;&longs;e po&longs;&longs;it. <emph type="italics"/>quod quidem de­<lb/>mon&longs;trare oportebat.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg293"></margin.target>A</s></p>
<p type="margin">
<s><margin.target id="marg294"></margin.target><emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> in<emph.end type="italics"/> 4. <lb/><emph type="italics"/><expan abbr="&longs;ecũdi">&longs;ecundi</expan> hui<emph.end type="italics"/>^{9}</s></p>
<p type="margin">
<s><margin.target id="marg295"></margin.target>7. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg296"></margin.target>8.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg297"></margin.target>8.<emph type="italics"/>primihu <lb/>ius.<emph.end type="italics"/></s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>In hac quoque demon&longs;tratione ob&longs;eruandum e&longs;t, quod
<arrow.to.target n="marg298"></arrow.to.target><lb/>po&longs;t quartam huius adnotauimus; nimirum &longs;i pentagonum <lb/>AGBLC in portione planèin&longs;criptum relinqueret portiones <lb/>ANG GOB BPL LQC, quæ &longs;imul maiores, vel etiam æ-
<pb pagenum="166"/>quales e&longs;&longs;ent &longs;pacio K. Rur&longs;us planè adhuc in&longs; cribatur in <lb/>portione ABC nonagonum, deinde altera figura, idquè &longs;em <lb/>per fiat, donec circumrelict&ecedil; portiones &longs;imul &longs;int &longs;pacio K <lb/>minores. quod quidem fieri po&longs;&longs;e ibidem o&longs;tendimus: </s></p>
<p type="margin">
<s><margin.target id="marg298"></margin.target>A</s></p>
<p type="head">
<s>PROPOSITIO. VII.</s></p>
<p type="main">
<s>Duabus portionibus &longs;imilibus recta linea, re­<lb/>ctanguliquè coni &longs;ectione contentis, centra gra­<lb/>uitatum diametros in eadem proportione di&longs;pe­<lb/>&longs;cunt. </s></p>
<figure></figure>
<p type="main">
<s><emph type="italics"/>Sint duæ portiones, quales dictæ &longs;unt ABC EFG. quarum diame­<lb/>tri BD FH. &longs;itquè portionis ABC centrum grauitatis punctum K. <lb/>ip&longs;ius verò EFG punctum L. Demonstrandum est, puncta<emph.end type="italics"/> k<emph type="italics"/>L in <lb/>eadem proportione diametros diuidere,<emph.end type="italics"/> ita vt BK ad KD &longs;it, vt FL
<pb pagenum="167"/>ad LH. <emph type="italics"/>&longs;i autemnon.<emph.end type="italics"/> &longs;i fieri pote&longs;t, <emph type="italics"/>&longs;it BK ad<emph.end type="italics"/> k<emph type="italics"/>D, vt FM ad <lb/>MH. & in portione EFG rectilineum planè in&longs;cribatur, ita vt linea <lb/>inter centrum<emph.end type="italics"/> grauitatis <emph type="italics"/>portionis, &<emph.end type="italics"/> centrum grauitatis <emph type="italics"/>figuræ<emph.end type="italics"/>
<arrow.to.target n="marg299"></arrow.to.target><lb/><emph type="italics"/>in&longs;criptæ minor &longs;it, quàm LM. &longs;itquè figuræ in&longs;criptæ centrum graui­<lb/>tatis punctum X.<emph.end type="italics"/> eritvtiquè punctum L propinquius vertici
<arrow.to.target n="marg300"></arrow.to.target><lb/>F, quàm punctum X. & quoniam LX minor e&longs;t, quàm <lb/>LM, erit quoque punctum X vertici F propinquius, quàm <lb/>M. <emph type="italics"/>Jn portione autem ABC in&longs;cribatur figura rectilinea &longs;imilis figu <lb/>ræin portione EFG in&longs;criptæ. hoc est &longs;imiliter planè,<emph.end type="italics"/> (ita nempè vt <lb/>figur&ecedil; latera multitudine &ecedil;qualia habeant) <emph type="italics"/>cuius centrum graui­<lb/>tatis<emph.end type="italics"/> &longs;it punctum N. & quoniam figuræ in porrionibus pla­<lb/>nèin&longs;cript&ecedil; habentlatera multitudine æqualia, ip&longs;arum cen­<lb/>tra grauitatis diametros BD FH in eadem proportione di&longs;pe­<lb/>&longs;cent. quare erit BN ad ND, vt FX ad XH. po&longs;itum <expan abbr="aut&etilde;">autem</expan>
<arrow.to.target n="marg301"></arrow.to.target><lb/>fuitita e&longs;&longs;e FM ad MH, vt BK ad KD. &longs;i itaque <expan abbr="punctũ">punctum</expan> <lb/>X propinquius e&longs;t ip&longs;i F, quàm M; erit & punctum N i­<lb/>p&longs;i B propinquius, quàm K. e&longs;tverò punctum K <expan abbr="centrũ">centrum</expan> <lb/>grauitatis portionis ABC, punctum verò N centrum figuræ <lb/>in&longs;cripte; ergo centrum grauitatis figur&ecedil; in&longs;criptæ <emph type="italics"/>propinquius <lb/>erit vertici portionis,<emph.end type="italics"/> quam centrum ip&longs;ius portionis. <emph type="italics"/>quod fieri <expan abbr="nõ">non</expan> <lb/>potest. Manife&longs;tum est igitur eandem habere proportionem BK ad KD. <lb/>quam FL ad LH.<emph.end type="italics"/> quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg299"></margin.target>6. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg300"></margin.target>5. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg301"></margin.target>3. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Pr&ecedil;&longs;ens demon&longs;tratio ea tantùm ratione e&longs;&longs;icax e&longs;&longs;e vide­<lb/>tur, quatenus &longs;upponitur punctum L vertici F propinqui^{9} <lb/>e&longs;&longs;e, quàm M. ex hoc enim &longs;equitur punctum X e&longs;&longs;e ip&longs;i F <lb/>propinquius, quàm M. vnde euenitab&longs;urdum, nempè, <expan abbr="pũ">pum</expan> <lb/>ctum N e&longs;&longs;evertici B propinquius, quàm K. Quòd &longs;i &longs;up <lb/>po&longs;itum fuerit Bk ad KD ita e&longs;&longs;e, vt FP ad PH; fuerit <lb/>autem P inter LF; tunc centrum grauitatis figur&ecedil; in EFG
<pb pagenum="168"/>planè in&longs;criptæ e&longs;&longs;etinter puncta PH; vnde centrum ctiam <lb/>figur&ecedil; in ABC &longs;imiliter planè in&longs;cript&ecedil; inter KD eueniret; <lb/>e&longs;&longs;etquè centrum grauitatis portionis ABC vertici B propin­<lb/>quius, quam centrum figuræ planè in&longs;criptæ. ideoquè <expan abbr="nullũ">nullum</expan> <lb/>accideret ab&longs;urdum. Quare &longs;i &longs;uppo&longs;itum fuerit FP ad PH <lb/>e&longs;&longs;e, vt BK ad KD, tunc (vt eadem demon&longs;tratio rei propo <lb/>&longs;itæ in&longs;eruire po&longs;&longs;et) diuidenda e&longs;&longs;et diameter BD in <expan abbr="q;">que</expan> i­<lb/>ta vt BQ ad QD &longs;it, vt FL ad LH. & quoniam maio­<lb/>
<arrow.to.target n="marg302"></arrow.to.target> rem habet proportionem FL ad LH, quàm FP ad PH; &longs;iqui­<lb/>dem maior e&longs;t FL, quàm FP, & PH maior, quàm LH. Vtverò <lb/>FL ad LH, ita e&longs;t BQ ad QD; & vt FP ad PH. ita BK ad KD; <lb/>maiorem quoque habebit proportionem BQ ad QD, quàm <lb/>
<arrow.to.target n="marg303"></arrow.to.target> BK ad KD. & componendo BD ad DQ maiorem, quàm ea <lb/>
<arrow.to.target n="marg304"></arrow.to.target> dem BD ad Dk. Quare maior e&longs;t DK, quàm <expan abbr="Dq.">Dque</expan> & ob id <lb/>punctum K propinquius erit vertici B, quàm <expan abbr="q.">que</expan> Deinde <lb/>planè in&longs;cribenda e&longs;&longs;et figura in portione ABC, ita vt linea <lb/>inter centrum figuræ in&longs;criptæ, & centrum portionis minor <lb/>e&longs;&longs;et, quàm <expan abbr="Kq;">Kque</expan> & reliqua quæ &longs;equuntur, ita tamen, vt qu&ecedil; <lb/>facta &longs;untin EFG, fiant in ABC; & quæ in ABC, <expan abbr="fiãt">fiant</expan> in EFG. <lb/>o&longs;tendeturquè centrum figur&ecedil; in&longs;cript&ecedil; in portione EFG pro <lb/>pinquius e&longs;&longs;e vertici F, quàm centrum grauitatis ip&longs;ius portio <lb/>nis EFG. quod quidem fieri non pote&longs;t. Ex quibus perlpi­<lb/>cuum fit demon&longs;trationem e&longs;&longs;e vniuer&longs;alem. & hanc <expan abbr="demõ">demom</expan> <lb/>&longs;trationis partem Archimedem omi&longs;i&longs;&longs;e, vt notam. Etvt an­<lb/>tea admonuimus, quòd centra grauitatis diametros in eadem <lb/>proportione diuidunt, omnibus parabolis competere intelli­<lb/>gendum e&longs;t. &longs;iquidem omnes &longs;unt&longs;imiles. quo demon&longs;trato, <lb/>in &longs;equenti, quo in loco, & in qua diametri parte reperitur <expan abbr="c&etilde;">cem</expan> <lb/>trum grauitatis paraboles demon&longs;trat, quòd vt res per&longs;picua <lb/>reddatur; hæc priùs demon&longs;trabimus. </s></p>
<p type="margin">
<s><margin.target id="marg302"></margin.target><emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> in<emph.end type="italics"/> 4. <lb/><emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg303"></margin.target>28.<emph type="italics"/>quinti. <lb/>addi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg304"></margin.target>10.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="head">
<s>LEMMA. I.</s></p>
<p type="main">
<s>Si magnitudo magnitudinis fuerit quadrupla, minorverò <lb/>magnitudo alterius magnitudinis &longs;it tripla, maior magnitu­<lb/>do vtrarum què &longs;imul magnitudinum tripla erit. </s></p>
<pb pagenum="169"/>
<p type="main">
<s>Quadrupla &longs;it magnitudo A magnitudinis BC. <lb/>
<arrow.to.target n="fig74"></arrow.to.target><lb/>&longs;it verò BC alterius magnitudinis CD tripla. Di <lb/>co magnitudinem A vtrarumquè &longs;imul BC CD, <lb/>hoc e&longs;t BD triplam e&longs;se. Quoniam enim BC tri­<lb/>pla e&longs;t ip&longs;ius CD, erit componendo BC cum CD, <lb/>hoc e&longs;t BD ip&longs;ius CD quadrupla. &longs;ed magnitudo <lb/>quoque A quadrupla e&longs;t ip&longs;ius BC, eandem igitur <lb/>habetproportionem A ad BC, vt BD ad CD. & <lb/>permutando A ad BD, vt BC ad CD. & e&longs;t
<arrow.to.target n="marg305"></arrow.to.target> qui­<lb/>dem BC tripla ip&longs;ius CD, ergo A ip&longs;ius BD tri­<lb/>pla exi&longs;tit. quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg305"></margin.target>16.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<figure id="fig74"></figure>
<p type="head">
<s>LEMMA. II.</s></p>
<p type="main">
<s>Si magnitudo magnitudinis fuerit &longs;e&longs;quitertia, erit magni­<lb/>tudo minor ip&longs;ius exce&longs;&longs;us tripla. </s></p>
<p type="main">
<s>Sit magnitudo AB magnitudinis C &longs;e&longs;quiter <lb/>
<arrow.to.target n="fig75"></arrow.to.target><lb/>tia; exce&longs;&longs;us verò, quo AB &longs;uperat C, &longs;it BD. Dico <lb/><expan abbr="magnitudin&etilde;">magnitudinem</expan> C ip&longs;ius BD triplam e&longs;&longs;e. quod qui <lb/>dem ex &longs;e patet. Nam quoniam BD e&longs;t exce&longs;­<lb/>&longs;us, quo AB &longs;uperat C. magnitudo autem AB i­<lb/>p&longs;am C &longs;uperat tertia ip&longs;ius C parte, cum &longs;it AB <lb/>ip&longs;ius C &longs;e&longs;quitertia. erit igitur BD tertia pars i­<lb/>&longs;ius C. quare magnitudo C ip&longs;ius BD tripla <lb/>exi&longs;tit. quod o&longs;tendere oportebat. </s></p>
<figure id="fig75"></figure>
<p type="head">
<s>LEMMA III.</s></p>
<p type="main">
<s>Sit magnitudo AB ip&longs;ius BC &longs;extupla. &longs;it verò AD tripla <lb/>ip&longs;ius AC. Dico BD ip&longs;ius BA &longs;e&longs;quialteram e&longs;se. </s></p>
<pb pagenum="170"/>
<p type="main">
<s><expan abbr="Quoniã">Quoniam</expan>. n. AD multiplex e&longs;t ip&longs;ius AG, erit AC pars ip&longs;i^{9} <lb/>AD. ac propterea ip&longs;am AD metictur. rur&longs;us quoniam AB, <lb/>hoc e&longs;t AC vnà cum CB &longs;extupla e&longs;t ip&longs;ius BC, erit <expan abbr="diuid&etilde;do">diuidendo</expan> <lb/>AC ip&longs;ius CB quintupla. vndè CB ip&longs;am AC, ac propterea <expan abbr="etiã">etiam</expan> <lb/>ip&longs;am AB metietur. Vtautem AC ad AD, ita fiat <lb/>
<arrow.to.target n="fig76"></arrow.to.target><lb/>CB ad aliam <expan abbr="magnitudin&etilde;">magnitudinem</expan> G; eritvti <expan abbr="&qacute;">que</expan>; CB ip&longs;ius <lb/>G pars tertia, cùm &longs;it AC ip&longs;ius AD pars quoque <lb/>tertia. Itaque quoniam CB ad G e&longs;t, vt AC ad AD, <lb/>
<arrow.to.target n="marg306"></arrow.to.target> erit perm utando CB ad CA, vt G ad AD. BC verò <lb/>ip&longs;am CA metitur, eiu&longs;què e&longs;t pars quinta; ergo <lb/>Gip&longs;am quoque AD metietur, eritquè ip&longs;ius pars <lb/>quinta. Quoniam autem BC ip&longs;am BA metitur, <lb/>eademquè BC ip&longs;am quoque G metitur, erit BC <lb/>ip&longs;arum AB G communis men&longs;ura. quia verò AB <lb/>&longs;extupla e&longs;t ip&longs;ius CB, G verò e&longs;t eiu&longs;dem CB tri­<lb/>pla, erit AB ad G, ut &longs;extupla ad triplam. hoc e&longs;t <lb/>&longs;e habebunt in dupla proportione. quapropter <lb/>AB dupla e&longs;t ip&longs;ius G; ac per con&longs;equens Gip&longs;am <lb/>AB metitur. Quoniam igitur G totam AD metitur, & <lb/>ablatam AB quoque metitur; metietur G reliquam BD. G <lb/>igitur ip&longs;arum AB BD communis exi&longs;tit men&longs;ura. & <expan abbr="quoniã">quoniam</expan> <lb/>AB dupla e&longs;t ip&longs;ius G, tota verò AD eiu&longs;dem G quintupla <lb/>exi&longs;tit, erit reliqua BD tripla ip&longs;ius G. Ex quibus&longs;equitur <lb/>DB ad BA ita &longs;e habere, vt tripla ad duplam. Quare DB <lb/>ip&longs;ius BA &longs;e&longs;quialtera exi&longs;tit. quod o&longs;tendere oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg306"></margin.target>16,<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<figure id="fig76"></figure>
<p type="head">
<s>PROPOSITIO. VIII.</s></p>
<p type="main">
<s>Omnis portionis recta linea, rectanguliquè co <lb/>ni &longs;ectione contentæ centrum grauitatis diame­<lb/>trum portionis ita diuidit, vt pars ip&longs;ius ad verti­<lb/>cem portionis reliquæ ad ba&longs;im &longs;it &longs;e&longs;quialtera. </s></p>
<pb pagenum="171"/>
<p type="main">
<s><emph type="italics"/>Sit portio ABC, qualis dicta est. ip&longs;ius verò diameter &longs;it BD. cen­<lb/>trum autem grauitatis &longs;it punctum H. o&longs;tendendum e&longs;t BH ip&longs;ius HD <lb/>&longs;e&longs;quialteram e&longs;&longs;e. Planè in&longs;cribatur in portione ABC triangulum ABC. <lb/>cuius centrum grauitatis &longs;it punctum E. bi&longs;ariamquè diuidatur vtra­<lb/>què AB BC in punctis FG. & ip&longs;i BD æquidi&longs;tantes ducantur F<emph.end type="italics"/>k <lb/><emph type="italics"/>GL. erunt vtique<emph.end type="italics"/> FK GL <emph type="italics"/>diametri portionum A<emph.end type="italics"/>k<emph type="italics"/>B BLC. &longs;it ita­<lb/>que portionis A<emph.end type="italics"/>k<emph type="italics"/>B centrum grauitatis M; portionis verò BLC pun­<lb/>ctum N. connectanturque FG MN<emph.end type="italics"/> k<emph type="italics"/>L<emph.end type="italics"/>, quæ diametrum BD &longs;e­<lb/>
<arrow.to.target n="fig77"></arrow.to.target><lb/>cent in punctis OQS. Quoniam igitur puncta MN in <expan abbr="ead&etilde;">eadem</expan> <lb/>proportione diuidunt KF LG, erit KM ad MF, vt LN ad
<arrow.to.target n="marg307"></arrow.to.target><lb/>NG. & componendo KF ad FM, vt LG ad GN. &
<arrow.to.target n="marg308"></arrow.to.target> per­<lb/>mutando KF ad LG, vt FM ad GN. &longs;untquè KF LG <lb/>æquales; erit FM ip&longs;i GN &ecedil;qualis; & reliqua Mk reliquæ
<arrow.to.target n="marg309"></arrow.to.target><lb/>LN æqualis. & quoniam FM GN, & Mk NL &longs;unt
<arrow.to.target n="marg310"></arrow.to.target> &ecedil;qui­<lb/>di&longs;tantes, erunt FG MN KL inter&longs;e &ecedil;quales, &
<arrow.to.target n="marg311"></arrow.to.target> <expan abbr="æquidi&longs;tã-tes">æquidi&longs;tan­<lb/>tes</expan>. & e&longs;t BD æquidi&longs;tans KF, eritigitur SQ ip&longs;i KM æ­<lb/>qualis. quia verò KF BD LG &longs;unt æquidi&longs;tantes, erit MQ ad
<arrow.to.target n="marg312"></arrow.to.target><lb/>QN, vt FO ad OG. Cùm autem &longs;it BF ad FA, vt BG ad GC,
<pb pagenum="172"/>
<arrow.to.target n="marg313"></arrow.to.target> crit FG ip&longs;i AC &ecedil;quidi&longs;tans. & vt AD ad DC, ita FO ad <lb/>OG. &longs;untautem AD DC æquales, ergo FO OG, ac per con­<lb/>&longs;equens MQ QN inter&longs;e &longs;unt æquales. itaque quoniam por <lb/>
<arrow.to.target n="marg314"></arrow.to.target> tiones AKB BLC &longs;unt æquales, <emph type="italics"/>magnitudinis ex vtri&longs;que portio­<lb/>nibus<emph.end type="italics"/> AKB BLC <emph type="italics"/>compo&longs;itæ centrum grauitatis erit<emph.end type="italics"/> in medio li­<lb/>
<arrow.to.target n="marg315"></arrow.to.target> neç MN; hoc e&longs;t erit <emph type="italics"/>punctum <expan abbr="q.">que</expan> & quoniam BH ad HD est,<emph.end type="italics"/><lb/>
<arrow.to.target n="marg316"></arrow.to.target> <emph type="italics"/>vt KM ad MF<emph.end type="italics"/> (centra enim grauitatum portionum in ea­<lb/>
<arrow.to.target n="fig78"></arrow.to.target><lb/>dem proportione diametros &longs;ecare nece&longs;&longs;e e&longs;t) <emph type="italics"/>& componendo<emph.end type="italics"/><lb/>
<arrow.to.target n="marg317"></arrow.to.target> BD ad DH, vt KF ad FM. <emph type="italics"/>permutandoquè vt BD ad KF,<emph.end type="italics"/><lb/>
<arrow.to.target n="marg318"></arrow.to.target> <emph type="italics"/>ita HD ad MF. at verò BD quadrupla est ip&longs;ius KF. Hoc enim<emph.end type="italics"/><lb/>
<arrow.to.target n="marg319"></arrow.to.target> <emph type="italics"/>in fine demon&longs;tratum est, vbi est &longs;ignum hoc, H. quadrupla igitur est<emph.end type="italics"/><lb/>
<arrow.to.target n="marg320"></arrow.to.target> <emph type="italics"/>& DH ip&longs;ius MF. Quare & reliqua BH reliquæ<emph.end type="italics"/> k<emph type="italics"/>M, hoc est i­<lb/>p&longs;ius SQ, quadrupla existit.<emph.end type="italics"/> exi&longs;tente autem tota BH, quæ <expan abbr="cõ">com</expan>. <lb/>po&longs;ita e&longs;t ex BS QH, & SQ, quadrupla ip&longs;ius <expan abbr="Sq;">Sque</expan> dempta <lb/>SQ ab ip&longs;is BS QH SQ, <emph type="italics"/>reliqua igitur ex vtri&longs;que BS QH<emph.end type="italics"/><lb/>con&longs;tans <emph type="italics"/>tripla est ip&longs;ius <expan abbr="Sq.">Sque</expan> &longs;it BS tripla ip&longs;ius SX.<emph.end type="italics"/> & <expan abbr="quoniã">quoniam</expan> <lb/>tota HQ cum SB ad totam QS e&longs;t, vt ablata BS ad ab­<lb/>
<arrow.to.target n="marg321"></arrow.to.target> latam SX; &longs;unt quidem tripl&ecedil;; erit reliqua HQ ad <expan abbr="reliquã">reliquam</expan> <lb/>
<arrow.to.target n="marg322"></arrow.to.target> QX in eadem proportione. <emph type="italics"/>ergo & QH ip&longs;ius XQ e&longs;t tripla. <lb/>Et quoniam quadrupla est BD ip&longs;ius BS. hoc enim demon&longs;tratum<emph.end type="italics"/><lb/>
<arrow.to.target n="marg323"></arrow.to.target> <emph type="italics"/>e&longs;t. ip&longs;a verò BS ip&longs;ius SX e&longs;t tripla<emph.end type="italics"/>; erit BD ip&longs;ius BX tripla.
<pb pagenum="173"/>ac propterea <emph type="italics"/>erit XB ip&longs;ius BD pars tertia. Verùm ED ip&longs;ius <lb/>DB parstertia existit. Cùm centrum grauitatis trianguli ABC &longs;it <lb/>p<gap/>nctum E.<emph.end type="italics"/> quod ita diuidit BD, vt BE ip&longs;ius ED &longs;itdupla.
<arrow.to.target n="marg324"></arrow.to.target><lb/>At verò quoniam totius lineæ BD (quæ compo&longs;ita e&longs;t ex DE <lb/>EX XB) tertia pars e&longs;t ip&longs;a DE. & tertia quoque ip&longs;a BX; <lb/><emph type="italics"/>reliqua igitur XE tertia est pars ip&longs;ius BD. & quoniam totius por­<lb/>tionis centrum grauitatis est punctum H; magnitudinis verò ex v­<lb/>tr<gap/>que portionibus A<emph.end type="italics"/>k<emph type="italics"/>B BLC compo&longs;itæ centrum grauitatis est pun <lb/>ctum <expan abbr="q;">que</expan> trianguli verò ABC est punctum E; erit triangulum ABC <lb/>ad circumrelictas portiones<emph.end type="italics"/> AKB BLC, <emph type="italics"/>vt QH ad HE, <expan abbr="triplũ">triplum</expan><emph.end type="italics"/>
<arrow.to.target n="marg325"></arrow.to.target><lb/><emph type="italics"/>autem e&longs;t triangulum ABC portionum. Cùm totaportio<emph.end type="italics"/> ABC <emph type="italics"/>&longs;e&longs;qui­<lb/>tertia &longs;it trianguli ABC<emph.end type="italics"/>, exce&longs;&longs;us verò, quo portio ABC
<arrow.to.target n="marg326"></arrow.to.target> &longs;upe­<lb/>rat triangulum ABC, &longs;int portiones AKB BLC &longs;imul &longs;um <lb/>ptæ. <emph type="italics"/>tripla igitur est QH ip&longs;ius HE. osten&longs;a verò e&longs;t etiam QH <lb/>tripla ip&longs;ius QX.<emph.end type="italics"/> quare erit QX ip&longs;i HE æqualis. &
<arrow.to.target n="marg327"></arrow.to.target> quo­<lb/>niam HQ e&longs;t tripla ip&longs;ius QX, erit HQ cum QX, hoc <lb/>e&longs;t tota BX quadrupla ip&longs;ius QX, hoc e&longs;t ip&longs;ius HE. &longs;i­<lb/>militer quoniam XH quadrupla e&longs;t ip&longs;ius HE; <emph type="italics"/>quintupla i­<lb/>gitur e&longs;t<emph.end type="italics"/> XH cum HE, tota &longs;cilicet <emph type="italics"/>XE ip&longs;ius EH; hoc est <lb/>DE ip&longs;ius EH. inuicem enim &longs;unt æquales<emph.end type="italics"/> EX ED, vt o&longs;ten­<lb/>&longs;um e&longs;t. Cùm itaque &longs;it DE ip&longs;ius EH quintupla; erit DE <lb/>cum EH &longs;extupla ip&longs;ius EH. <emph type="italics"/>Quare &longs;extupla est<emph.end type="italics"/> tota <emph type="italics"/>DH <lb/>ip&longs;ius HE. & e&longs;t BD ip&longs;ius DE tripla; &longs;equialtera igitur e&longs;t BH<emph.end type="italics"/>
<arrow.to.target n="marg328"></arrow.to.target><lb/><emph type="italics"/>ip&longs;ius HD.<emph.end type="italics"/> Quare centrum grauitatis H ita diuidit diame­<lb/>trum BD, vtpars BH ad HD &longs;e&longs;quialtera exi&longs;tit. quod de <lb/>mon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg307"></margin.target>7. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg308"></margin.target>18.16 <emph type="italics"/><expan abbr="quĩti">quinti</expan><emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg309"></margin.target><emph type="italics"/>po&longs;t <expan abbr="primã">primam</expan> <lb/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg310"></margin.target>33. <emph type="italics"/>primi<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg311"></margin.target>34, <emph type="italics"/>primi<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg312"></margin.target>1. <emph type="italics"/>lemma <lb/>in<emph.end type="italics"/> 13. <emph type="italics"/>pri <lb/>mi huius<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg313"></margin.target><emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> in ali <lb/>ter<emph.end type="italics"/> 13 <emph type="italics"/>pri <lb/>mi huius<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg314"></margin.target><emph type="italics"/>po&longs;t <expan abbr="primã">primam</expan> <lb/>huius<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg315"></margin.target>4. <emph type="italics"/>primi hu <lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg316"></margin.target>7. <emph type="italics"/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg317"></margin.target>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg318"></margin.target>16.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg319"></margin.target>A</s></p>
<p type="margin">
<s><margin.target id="marg320"></margin.target>19 <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg321"></margin.target>19.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg322"></margin.target>B</s></p>
<p type="margin">
<s><margin.target id="marg323"></margin.target>1.<emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> hui^{9}<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg324"></margin.target><emph type="italics"/>ante<emph.end type="italics"/> 1;.<emph type="italics"/>pri <lb/>mi huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg325"></margin.target>8.<emph type="italics"/>primi hu <lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg326"></margin.target>2.<emph type="italics"/>lemma <lb/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg327"></margin.target>9.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg328"></margin.target>3.<emph type="italics"/>lemma <lb/>huius.<emph.end type="italics"/></s></p>
<figure id="fig77"></figure>
<figure id="fig78"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Ea verba in demon&longs;tratione po&longs;ita nempè <emph type="italics"/>Hoc enim in fine<emph.end type="italics"/>
<arrow.to.target n="marg329"></arrow.to.target><lb/><emph type="italics"/>demon&longs;tratum e&longs;t, vbi est &longs;ignum hoc, H.<emph.end type="italics"/> ita credo e&longs;&longs;e intell igen­<lb/>da, quòd &longs;cilicet Archimedes alicubi, & in fine, &longs;iue huius, &longs;i­<lb/>ue alicuius alterius demon&longs;trationis, demon&longs;trauerit lineain
<pb pagenum="174"/>BD quadruplam e&longs;&longs;e ip&longs;ius KF. & vbi hoc demon&longs;tratum <lb/>erat, ibi quoque pro &longs;igno po&longs;ita fuerit littera H. quod qui­<lb/>dem o&longs;ten&longs;um e&longs;t à nobis paulò ante &longs;ecundam huius propo&longs;i <lb/>tionem; vbi etiam appo&longs;uim us pro &longs;igno hanc literam H. </s></p>
<p type="margin">
<s><margin.target id="marg329"></margin.target>A</s></p>
<p type="main">
<s>
<arrow.to.target n="marg330"></arrow.to.target> Rur&longs;um in demon&longs;tratione paulò infra Archimedes dixit, <lb/><emph type="italics"/>Hoc enim demonstratum e&longs;t<emph.end type="italics"/>, &longs;cilicet BD ip&longs;ius BS quadruplam <lb/>e&longs;&longs;e. &longs;upponit autem hoc tanquam demon&longs;tratum po&longs;t pri­<lb/>mam <expan abbr="propo&longs;ition&etilde;">propo&longs;itionem</expan> huius, vbi tota BD e&longs;t &longs;exdccim, & BS qua <lb/>tuor, vt eodem in loco o&longs;ten&longs;um fuità nobis. Vel ad ea re­<lb/>&longs;pexit Archimedes, quæ ab ip&longs;o in decimanona propo&longs;itione <lb/>de quadratura paraboles demon&longs;tra ta fuerunt. vbi circa <expan abbr="fin&etilde;">finem</expan> <lb/>demon&longs;trationis o&longs;tendit BD quadruplam e&longs;&longs;e ip&longs;ius BS. </s></p>
<p type="margin">
<s><margin.target id="marg330"></margin.target>B</s></p>
<p type="main">
<s>Inuento itaque centro grauitatis paraboles, vult Archime­<lb/>des in ue&longs;tigare centrum grauita tis fru&longs;ti à parabole ab&longs;ci&longs;&longs;i. <lb/>quemadmodum in primo libro po&longs;t inuentionem centri gra <lb/>uitatis trianguli, adin uenit etiam centrum grauitatis trapezij. <lb/>quod e&longs;t tan quam fru&longs;tum à triangulo ab&longs;ci&longs;sum. quare duo <lb/>adhuc theoremata proponit, in quorum po&longs;tremo, vbi &longs;it <expan abbr="c&etilde;">cem</expan> <lb/>trum grauitatis fru&longs;ti demon&longs;trat. in &longs;equenri verò quædam <lb/>demon&longs;trat nece&longs;&longs;aria, vt huiu&longs;modi centrum determinare <lb/>po&longs;&longs;it. Quoniam autem &longs;equens theorema arduum, difficile­<lb/>què &longs;e&longs;e offert; non nulla priùs quibu&longs;dam lemmatibus o&longs;ten <lb/>demus, ne &longs;i in demon&longs;tratione ea in&longs;ererentur, longa nimis <lb/>euaderet, ac tædio&longs;a demon&longs;tratio. quæ quidem &longs;umma in di <lb/>get attentione. quamquàm in hoc theoremate explicando ad <lb/>vitandam ob&longs;curitatem copio&longs;um &longs;ermonem adhibendum <lb/>curauimus; ne breuitati &longs;tudentes ob&longs;curiores e&longs;&longs;emus. </s></p>
<p type="head">
<s>LEMMA. I.</s></p>
<p type="main">
<s>Si qua tuor magnitudines in continua fuerint proportione, <lb/>& earum exce&longs;&longs;us in eadem erunt proportione <expan abbr="magnitudinũ">magnitudinum</expan>. </s></p>
<pb pagenum="175"/>
<p type="main">
<s>Sint quatuor magnitudines AF BH CL D in continua <lb/>proportione; vt &longs;cilicet &longs;it AF ad BH, vt BH ad CL; & CL <lb/>ad D. exce&longs;&longs;us verò, quo AF &longs;uperat BH, &longs;it EF. & exce&longs;&longs;us, quo <lb/>BH &longs;uperat CL, &longs;it GH. exce&longs;&longs;us denique, quo CL &longs;uperat <lb/>D, &longs;it KL. eruntutique AE BH inter &longs;e &ecedil;quales, itidemquè <lb/>
<arrow.to.target n="fig79"></arrow.to.target><lb/>BG CL æquales. Dico EF GH KL in eadem e&longs;&longs;e proportio <lb/>ne, vt &longs;unt magnitudines AF BH CL, & vt BH CL D. Quo­<lb/>niam enim tota AF ad totam BH e&longs;t, vt BH ad CL; hoc e&longs;t
<arrow.to.target n="marg331"></arrow.to.target><lb/>vt ablata EA ad ablatam GB. erit reliqua EF ad reliquam GH; <lb/>vt AF ad BH. Pariquè ratione o&longs;tendetur GH ad kL ita e&longs;­<lb/>&longs;e, vt BH ad CL. ergo exce&longs;&longs;us EF GH KL in eadem &longs;unt <lb/>proportione, vt magnitudines AF BH CL. quæ cùm &longs;int, vt <lb/>magnitudines BH CL D, &longs;iquidem omnes in continua &longs;unt <lb/>proportione; exce&longs;&longs;us igitur EF GH KL in eadem quoque <lb/>&longs;unt proportione, vt magnitudines BH CL D. quæ quidem <lb/>demon&longs;trare oportebat. </s></p>
<pb pagenum="176"/>
<p type="margin">
<s><margin.target id="marg331"></margin.target>19.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<figure id="fig79"></figure>
<p type="head">
<s>LEMMA. II.</s></p>
<p type="main">
<s>Si tres fuerint magnitudines, & aliæ ip&longs;is numero æquales, <lb/>& in eadem proportione, in primis magnitudinibus prima; <lb/>& &longs;ecunda ad tertiam erunt, vt in &longs;ecundis magnitudinibus <lb/>prima & &longs;ecunda ad tertiam. </s></p>
<figure></figure>
<p type="main">
<s>Sint tres magnitudines ABC, & aliæ tres DEF in <expan abbr="ead&etilde;">eadem</expan> pro­<lb/>portione. Dico AB &longs;imul ad C ita e&longs;&longs;e, vt DE &longs;imul ad F. <lb/>
<arrow.to.target n="marg332"></arrow.to.target> Quoniam enim A ad B e&longs;t, ut D ad E, erit <expan abbr="compon&etilde;do">componendo</expan> AB <lb/>
<arrow.to.target n="marg333"></arrow.to.target> ad B, ut DE ad E. &longs;ed vt B ad C, ita e&longs;t E ad F. ergo ex &ecedil;quali <lb/>AB &longs;imul ad C e&longs;t, vt DE &longs;imul ad F. quod demon&longs;trare opor <lb/>tebat. </s></p>
<p type="margin">
<s><margin.target id="marg332"></margin.target>18,<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg333"></margin.target>22.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="head">
<s>LEMMA. III.</s></p>
<p type="main">
<s>Si fuerit AB ad AC, vt DE ad DF. Dico exce&longs;&longs;um BC ad <lb/>
<arrow.to.target n="marg334"></arrow.to.target> CA ita e&longs;&longs;e, vt exce&longs;&longs;us EF ad FD. </s></p>
<p type="margin">
<s><margin.target id="marg334"></margin.target><emph type="italics"/>cor.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="quĩti">quinti</expan><emph.end type="italics"/></s></p>
<p type="main">
<s>Quoniam enim e&longs;t AB ad AC, vt DE ad DF, erit con-
<pb pagenum="177"/>
<arrow.to.target n="fig80"></arrow.to.target><lb/>uertendo CA ad AB, vt FD ad DE. & per conuer <lb/>&longs;ionem rationis AC ad CB, vt DF ad FE. & rur&longs;us
<arrow.to.target n="marg335"></arrow.to.target><lb/>conuertendo CB ad CA, vt FE ad FD. quod <expan abbr="demõ-&longs;trare">demon­<lb/>&longs;trare</expan> oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg335"></margin.target><emph type="italics"/>co.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="quĩti">quinti</expan>.<emph.end type="italics"/></s></p>
<figure id="fig80"></figure>
<p type="head">
<s>ALITER.</s></p>
<p type="main">
<s>Quoniam enim AB e&longs;t ad AC, vt DE ad DF, erit conuer­<lb/>tendo AC ad AB, vt DF ad DE. diuidendoquè CB ad BA, vt <lb/>FE ad ED. e&longs;t autem AB ad AC, vt DE ad DF, erit igitur
<arrow.to.target n="marg336"></arrow.to.target><lb/>ex æquali BC ad CA, vt EF ad FD. quod demon&longs;trare opor
<arrow.to.target n="marg337"></arrow.to.target><lb/>tebat. </s></p>
<p type="margin">
<s><margin.target id="marg336"></margin.target>17.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg337"></margin.target>22,<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="head">
<s>LEMMA IIII.</s></p>
<figure></figure>
<p type="main">
<s>Si fuerint quotcunque magnitudines ABC, & nli&ecedil; ip&longs;is nu <lb/>mero æquales DEF, & in <expan abbr="ead&etilde;">eadem</expan> proportione. Dico vtramque <lb/>&longs;imul AD ad vtramque &longs;imul BE, & vtramque &longs;imul BE ad v­<lb/>tramque &longs;imul CF eandem habere proportionem, quam ha­<lb/>bet A ad B, & B ad C. </s></p>
<pb pagenum="178"/>
<p type="main">
<s>
<arrow.to.target n="marg338"></arrow.to.target> Quoniam enim e&longs;t A ad B, ut D ad E; erit AD &longs;imul ad <lb/>BE &longs;imul, vt A ad B. &longs;imiliter quoniam B ad C e&longs;t, vt E ad <lb/>F, erit BE &longs;imul ad CF &longs;imul, vt B ad C. in eadem igitur &longs;unt <lb/>proportione AD &longs;imul, & BE &longs;imul, & CF &longs;imul, vt ABC. <lb/>quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg338"></margin.target>12.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="head">
<s>LEMMA. V.</s></p>
<p type="main">
<s>Si magnitudo magnitudinis fuerit &longs;e&longs;quialtera ad tres quin <lb/>tas eiu&longs;dem erit duplex &longs;e&longs;quialtera. </s></p>
<figure></figure>
<p type="main">
<s>Sit AB ip&longs;ius CD &longs;e&longs;quialtera. &longs;it uerò CE tres quintæ <lb/>ip&longs;ius CD. Dico AB ad CE ita e&longs;&longs;e, vt quinque ad duo. Fiat EF <lb/>&ecedil;qualis EC, erit CF &longs;ex quin tæ ip&longs;ius CD. & quoniam AB i­<lb/>p&longs;ius CD e&longs;t &longs;e&longs;quialtera, &longs;uperabit AB ip&longs;am CD dimidia <lb/>ip&longs;ius CD. erit igitur AB &longs;eptem quintæ cum dimidia i­<lb/>p&longs;ius CD. quare CF minor e&longs;t AB. fiat igitur AG æqua­<lb/>lis CF. erit vtique AG &longs;ex quin t&ecedil; ip&longs;ius CD. & ob id GB <lb/>ip&longs;ius CD quinta e&longs;t pars cum dimidia. & quoniam CE e&longs;t <lb/>eiu&longs;dem CD tres quintæ, erit BG dimidia ip&longs;ius CE. qua­<lb/>re GB ip&longs;am CE bis metietur. Et quoniam EF e&longs;t æqua­<lb/>lis ip&longs;i EC, ip&longs;a BG bis quoque metietur ip&longs;am EF. quare
<pb pagenum="179"/>totam CF, hoc e&longs;t ip&longs;am AG quater metietur. at verò GB &longs;ei­<lb/>p&longs;am &longs;emel metitur ip&longs;a igitur GB totam AB quinquies metie­<lb/>tur. Ex quibus liquet GB ip&longs;arum ABCE communem e&longs;&longs;e <lb/>men&longs;uram. Et e&longs;t quidem AB quintupla ip&longs;ius BG; ip&longs;a verò <lb/>CE eiu&longs;dem BG dupla. erit AB ad CE, vt quintupla ad <expan abbr="duplã">duplam</expan>. <lb/>hoc e&longs;t duplex &longs;e&longs;quialtera. quod demon&longs;trare oportebat. </s></p>
<p type="head">
<s>PROPOSITIO. VIIII.</s></p>
<p type="main">
<s>Si quatuor lineæ in continua fuerint proportio­<lb/>ne, & quam proportionem habet minima ad exce&longs; <lb/>&longs;um, quo maxima minimam &longs;uperat; eandem ha­<lb/>beat quædam a&longs;&longs;umpta linea ad tres quintas exce&longs;­<lb/>&longs;us, quo maxima proportionalium tertiam exce­<lb/>dit: quam verò proportionem habet linea æ qualis <lb/>duplæ maximæ proportionalium, & quadruplæ &longs;e <lb/>cundæ, & &longs;extuplæ tertiæ, & triplæ quartæ ad <expan abbr="lineã">lineam</expan> <lb/>æqualem quintuplæ maximæ, & decuplæ &longs;ecundæ, <lb/>& decuplæ tertiæ, & quintuplæ quartæ, ean-­<lb/>dem habeat quædam a&longs;&longs;umpta linea ad ex ce&longs;&longs;um, <lb/>quo maxima proportionalium tertiam &longs;uperat; <lb/>vtræque &longs;imul a&longs;&longs;umptæ lineæ erunt duæ quin­<lb/>tæ maximæ. <lb/></s></p>
<pb pagenum="180"/>
<p type="main">
<s><emph type="italics"/>Sint quatuor lineæ proportionales AB BC BD BE,<emph.end type="italics"/> ita vt AB <lb/>ad BC &longs;it, vt BC ad BD. & vt BC ad BD, ita &longs;it BD ad BE. <emph type="italics"/>& <lb/>quam proportionem habet BE ad E A, eandem habeat FG adtres quin <lb/>tas ip&longs;ius AD. quam autem proportionem habet linea æqualis duplæ i­<lb/>p&longs;ius AB, & quidruplæ ip&longs;ius BC, & &longs;extuplæ ip&longs;i^{9} BD, & triplæ ip&longs;i^{9} <lb/>BE, ad <expan abbr="lineã">lineam</expan> <expan abbr="æqual&etilde;">æqualem</expan> <expan abbr="quĩtuplæ">quintuplæ</expan> ip&longs;i^{9} AB, ot decuplæ ip&longs;i^{9} CB, & decuplæ <lb/>ip&longs;i^{9} B D, & quintuplæ ip&longs;ius BE, eandem habeat GH ad AD. O&longs;teden <lb/>dum est FH duasquintas e&longs;&longs;e ip&longs;ius AB. Quoniam enim proportiona­<lb/>les &longs;unt AB BC BD BE, &<emph.end type="italics"/> ip&longs;arum exce&longs;&longs;us <emph type="italics"/>AC CD DE in<emph.end type="italics"/><lb/>
<arrow.to.target n="fig81"></arrow.to.target><lb/>
<arrow.to.target n="marg339"></arrow.to.target> <emph type="italics"/>eadem erunt proportione. &<emph.end type="italics"/> quoniam magnitudines AB BC BD <lb/>in continua &longs;unt proportione, & earum exce&longs;&longs;us AC CD DE <lb/>in eadem erunt proportione. quia verò tres &longs;unt magnitudi­<lb/>nes proportionales AB BC BD; & ali&ecedil; ip&longs;is numero çquales, & <lb/>
<arrow.to.target n="marg340"></arrow.to.target> in eadem proportione AC CD DE, erit in primis magnitu­<lb/>dinibus prima, & &longs;ecunda ad tertiam, vt in &longs;ecundis magni­<lb/>tudinibus prima, & &longs;ecunda ad tertiam; hoc e&longs;t <emph type="italics"/>vtraque &longs;imul <lb/>AB BC ad BD eandem habebit proportionem, quam<emph.end type="italics"/> vtraque &longs;imul <lb/>
<arrow.to.target n="marg341"></arrow.to.target> AC CD, hoc e&longs;t <emph type="italics"/>AD ad DE; &<emph.end type="italics"/> ob eandem rationem cum <lb/>
<arrow.to.target n="marg342"></arrow.to.target> tres &longs;int proportionales magnitudines AC CD DE, aliçquè <lb/>eodem modo proportionales BC BD BE; crit vtraque &longs;imul
<pb pagenum="181"/>AC CD, hoc e&longs;t AD ad DE, vt <emph type="italics"/>vtraque &longs;imul BC BD ad EB. <lb/>& omnes adomnes,<emph.end type="italics"/> quoniam &longs;cilicet e&longs;t vtraque &longs;imul AB BC <lb/>ad BD, vt horum dupla; erit vtraque &longs;imul AB BC ad BD, vt <lb/>dupla vtriu&longs;que &longs;imul AB BC ad duplam ip&longs;ius BD. e&longs;t <expan abbr="aut&etilde;">autem</expan> <lb/>vtraque &longs;imul AB BC ad BD, vt AD ad DE. eritigitur AD ad <lb/>DE, vt dupla vtriu&longs;que &longs;imul AB BC ad duplam ip&longs;ius BD. <lb/>quia veròita etiam e&longs;t AD ad DE, vtvtraque &longs;imul CB BD ad <lb/>BE; erit dupla vtriu&longs;que &longs;imul AB BC ad duplam ip&longs;ius BD, vt
<arrow.to.target n="marg343"></arrow.to.target><lb/>vtraque &longs;imul CB BD ad BE. & vtraque an tecedentia ad
<arrow.to.target n="marg344"></arrow.to.target> vtra­<lb/>que con&longs;equentia in eadem erunt proportione: eruntquè in <lb/>antecedenti du&ecedil; AB, tres BC, & &longs;ola BD. in con&longs;equenti verò <lb/>erunt duæ BD cum &longs;ola BE. erit igitur dupla ip&longs;ius AB, & tri <lb/>pla ip&longs;ius CB cum &longs;ola BD ad duplam ip&longs;ius BD cum &longs;ola BE, <lb/>vt vtraque &longs;imul CB BD ad BE. vtraque verò &longs;imul CB BD <lb/>ad BE e&longs;t, vt AD ad DE. <emph type="italics"/>eandem ergo proportionem habet AD ad<emph.end type="italics"/>
<arrow.to.target n="marg345"></arrow.to.target><lb/><emph type="italics"/>DE, quam linea æ qualis duplæ ip&longs;ius AB, & triplæip&longs;ius CB, &<emph.end type="italics"/> &longs;oli <lb/><emph type="italics"/>DB adlineam æqualem duplæ ip&longs;ius BD &<emph.end type="italics"/> &longs;oli <emph type="italics"/>BE.<emph.end type="italics"/> Quoniam au­<lb/>tem linea compo&longs;ita ex dupla ip&longs;ius AB, & quadrupla ip&longs;ius <lb/>CB, & quadrupla ip&longs;ius BD, & dupla ip&longs;ius BE, maior e&longs;t ea, <lb/>quæ compo&longs;ita e&longs;t ex dupla ip&longs;ius AB, & tripla ip&longs;ius CB, & <lb/>&longs;ola BD; maiorem habebit proportionem compo&longs;ita ex
<arrow.to.target n="marg346"></arrow.to.target> du­<lb/>pla ip&longs;ius AB, & quadrupla ip&longs;ius CB, & quadrupla ip&longs;ius BD, <lb/>& dupla ip&longs;ius BE ad compo&longs;itam ex dupla ip&longs;ius BD cum <lb/>&longs;ola BE, quam compo&longs;ita ex dupla ip&longs;ius AB, & tripla ip&longs;ius <lb/>CB cum &longs;ola BD ad eandem compo&longs;itam ex dupla ip&longs;ius BD <lb/>cum &longs;ola EB. compo&longs;ita verò ex dupla ip&longs;ius AB, & tripla <lb/>ip&longs;ius BC cum &longs;ola BD ad duplam ip&longs;ius BD cum &longs;ola BE ita <lb/>o&longs;ten&longs;a e&longs;t &longs;e habere AD ad DE. compo&longs;ita igitur ex dupla i­<lb/>p&longs;ius AB, & quadrupla ip&longs;ius BC, & quadrupla ip&longs;ius BD, & <lb/>dupla ip&longs;ius BE ad compo&longs;itam ex dupla ip&longs;ius BD cum &longs;ola <lb/>BE maiorem habebit proportionem, quam AD ad DE. <emph type="italics"/>Quam <lb/>itaque proportionem habet linea æqualis duplæ ip&longs;ius AB, & quadruplæ <lb/>ip&longs;ius BC, & quadruplæ ip&longs;ius BD, & duplæ ip&longs;ius BE ad <expan abbr="lineã">lineam</expan> <expan abbr="æqual&etilde;">æqualem</expan> <lb/>duplæ ip&longs;ius DB, & ad EB, eandem habebit AD adminorem ip&longs;a DE.<emph.end type="italics"/>
<arrow.to.target n="marg347"></arrow.to.target><lb/><emph type="italics"/>habeat igitur ad DO.<emph.end type="italics"/> & <expan abbr="quoniãita">quonianita</expan> &longs;e habet AD ad DO, vt <expan abbr="cõpo">compo</expan> <lb/>&longs;ita ex dupla ip&longs;ius AB, & quadrupla ip&longs;ius BC, & quadrupla <lb/>ip&longs;ius BD, & dupla ip&longs;ius BE, hoc e&longs;t <expan abbr="cõpo&longs;ita">compo&longs;ita</expan> ex dupla vtriu&longs;-
<pb pagenum="182"/>que &longs;imul AB BE, & quadrupla vtriu&longs;que &longs;imul BC BD. (bis <lb/>enim a&longs;&longs;umitur AB, & bis BE, quater verò BC, & quater BD) <lb/>
<arrow.to.target n="marg348"></arrow.to.target> ad compo&longs;itam ex dupla ip&longs;ius BD cum &longs;ola BE; erit conuer­<lb/>rendo, ut OD ad DA, ita compo&longs;ita ex dupla ip&longs;ius BD <expan abbr="cũ">cum</expan> &longs;o­<lb/>la BE ad <expan abbr="cõpo&longs;itam">compo&longs;itam</expan> ex dupla utriu&longs;que &longs;imul AB BE, & qua­<lb/>
<arrow.to.target n="marg349"></arrow.to.target> drupla vtriu&longs;que &longs;imul BCBD. <emph type="italics"/>et vtræque ad primas eandem habe <lb/>bunt proportionem.<emph.end type="italics"/> hoc e&longs;t componendo erit OA ad AD, vt <expan abbr="cõ-po&longs;ita">con­<lb/>po&longs;ita</expan> ex dupla ip&longs;ius BD cum &longs;ola BE, & dupla vtriu&longs;que &longs;i­<lb/>mul AB BE, & quadrupla vtriu&longs;que &longs;imul BC BD ad compo­<lb/>
<arrow.to.target n="fig82"></arrow.to.target><lb/>&longs;itam ex dupla vtriu&longs;que &longs;imul AB BE, & quadrupla <expan abbr="vtrius&qacute;">vtriusque</expan>; <lb/>&longs;imul BC BD. In hocautem an tecedente bis&longs;umitur AB, qua <lb/>ter BC, &longs;exies verò BD, & ter BE. <emph type="italics"/>habebit igitur OA ad AD ean­<lb/>demproportionem, quam linea æqualis duplæip&longs;ius AB, et quadruplæi­<lb/>p&longs;ius CB, et &longs;extuplæ ip&longs;ius BD, ettriplæ ip&longs;ius BE ad lineam compo&longs;i­<lb/>tam ex dupla vtriu&longs;que &longs;imul AB EB, et quadrupla vtriu&longs;que &longs;imul <lb/>CB BD. babet autem<emph.end type="italics"/> (vt &longs;uppo&longs;itum e&longs;t) GH ad AD eandem <lb/>proportionem, quam linea æ qualis duplæ ip&longs;ius AB, & qua­<lb/>druplæ ip&longs;ius BC, & &longs;extuplæ ip&longs;ius BD, & triplæ ip&longs;ius BE <lb/>ad lineam æ qualem quintuplæ ip&longs;ius AB, & decuplæ ip&longs;ius <lb/>CB, & decuplæ ip&longs;ius BD, & quintuplæ ip&longs;ius BE, hoc e&longs;t ad
<pb pagenum="183"/><expan abbr="quintuplã">quintuplam</expan> vtriu&longs;que&longs;imul AB BE <expan abbr="cũ">cum</expan> decupla vtriu&longs;què &longs;imul <lb/>CB BD. In <expan abbr="cõ&longs;equ&etilde;ti">con&longs;equenti</expan>.n.quinquies <expan abbr="a&longs;sũpta">a&longs;sumpta</expan> e&longs;t AB, & quinquies <lb/>BE, decies CB, & decies BD. & conuettendo habebit <emph type="italics"/>AD ad<emph.end type="italics"/>
<arrow.to.target n="marg350"></arrow.to.target><lb/><emph type="italics"/>GH eandem proportionem, quam quintupla vtriu&longs;que &longs;imul AB BE <lb/><expan abbr="cũ">cum</expan> decupla vtriu&longs;que &longs;imul CB BD ad lineam compo&longs;itam ex dupla i­<lb/>p&longs;ius AB, & quadrupla ip&longs;ius CB, & &longs;extuplaip&longs;ius BD, & triplai­<lb/>p&longs;ius EB. Di&longs;similiter autem quàm in proportionibus ordinatis, hocest <lb/>in perturbata proportione<emph.end type="italics"/> quoniam in primis magnitudinibus ita <lb/>&longs;e habet an tecedens OA ad con&longs;equens AD, vt in &longs;ecundis ma <lb/>gnitudinibus antecedens compo&longs;ita nempè ex dupla ip&longs;ius <lb/>AB, & quadrupla ip&longs;ius BC, & &longs;extupla ip&longs;ius BD, & tripla <lb/>ip&longs;ius BE, ad con&longs;equens lineam &longs;cilic et compo&longs;itam ex du­<lb/>pla vtriu&longs;que &longs;imul AB BE, & quadrupla vtriu&longs;que &longs;imul CB <lb/>BD: ut autem in primis magnitudinibus con&longs;equens AD ad <lb/>aliud quippiam GH, ita in &longs;ecundis magnitudinibus aliud <lb/>quippiam, nempèlinea compo&longs;ita ex quintupla vtriu&longs;que &longs;i­<lb/>mul AB BE cum decupla vtriu&longs;que &longs;imul CB BD ad antece­<lb/>dens, hoc e&longs;t ad compo&longs;itam ex dupla ip&longs;ius AB, & quadru­<lb/>pla ip&longs;ius CB, & &longs;extupla ip&longs;ius BD, & tripla ip&longs;ius BE. quare <lb/><emph type="italics"/>ex æquali eandemhabet proportionem OA ad GH, quam quintupla v-<emph.end type="italics"/>
<arrow.to.target n="marg351"></arrow.to.target><lb/><emph type="italics"/>triu&longs;que &longs;imul AB BE cum decupla<emph.end type="italics"/> vtriu&longs;que &longs;imul <emph type="italics"/>CB BD ad <lb/><expan abbr="cõpo&longs;itã">compo&longs;itam</expan> ex dupla <expan abbr="vtrius&qacute;">vtriusque</expan>; &longs;imul AB BE, et quadrupla <expan abbr="vtrius&qacute;">vtriusque</expan>; &longs;imul <lb/>CB BD. At verò<emph.end type="italics"/> quoniam quintupla ip&longs;ius AB ad duplam <lb/>eiu&longs;dem AB e&longs;t, vt quinque ad duo; &longs;imiliter quintupla ip&longs;i^{9} <lb/>BE ad duplam eiu&longs;dem BE e&longs;t, vt quinque ad duo, erit quin­<lb/>tupla vtriu&longs;que &longs;imul AB BE ad duplam vtriu&longs;que &longs;imul AB <lb/>BE, vt quinque ad duo. pariquè ratione decupla vtriu&longs;que &longs;i­<lb/>mul CB BD ad quadruplam vtriu&longs;que &longs;imul CB BD e&longs;t, vt <lb/>decem ad quatuor, hoc e&longs;t vt quinque ad duo. & <expan abbr="anteced&etilde;tia">antecedentia</expan>
<arrow.to.target n="marg352"></arrow.to.target><lb/>ad con&longs;equentia in eadem erunt proportione, hoce&longs;t <emph type="italics"/>compo&longs;i­<lb/>ta ex quintupla vtriu&longs;que &longs;imul AB BE cum decupla vtriu&longs;que &longs;imul <lb/>CB BD ad compo&longs;itam ex dupla vtriu&longs;que &longs;imul AB BE, & quadru <lb/>pla vtriu&longs;que &longs;imul CB BD proportionem habet, quam quinque ad duo <lb/>Quare OA ad GH proportionem habet, quam quinque ad duo. Rur&longs;us<emph.end type="italics"/><lb/>factum fuit AD ad DO, vt compo&longs;ita ex dupla vtriu&longs;que &longs;i­<lb/>mul AB BE cum quadrupla vtriu&longs;que &longs;imul CB BD ad <expan abbr="lineã">lineam</expan> <lb/>BE vnà cum dupla ip&longs;ius BD. conuertendo etiam <emph type="italics"/>quoniam<emph.end type="italics"/>
<arrow.to.target n="marg353"></arrow.to.target>
<pb pagenum="184"/>in primis magnitudinibus an tecedens <emph type="italics"/>OD ad<emph.end type="italics"/> con&longs;equens <emph type="italics"/>DA <lb/>eandem habet proportionem, quam<emph.end type="italics"/> in &longs;ecundis magnitudinibus an <lb/>tecedens <emph type="italics"/>EB cum dupla ip&longs;ius BD ad<emph.end type="italics"/> con&longs;equens, <emph type="italics"/>lineam<emph.end type="italics"/> &longs;cilicet <emph type="italics"/>æ­<lb/>qualem lineæ compo&longs;itæ ex dupla vtriu&longs;que &longs;imul AB BE cum quadru­<lb/>pla vtriu&longs;que &longs;imul CB BD; est autem<emph.end type="italics"/> (vt antea o&longs;ten&longs;um e&longs;t) & <lb/>in primis magnitudinibus con&longs;equens <emph type="italics"/>AD ad<emph.end type="italics"/> aliud <expan abbr="quippiã">quippiam</expan> <lb/><emph type="italics"/>DE, vt<emph.end type="italics"/> in &longs;ecundis magnitudinibus aliud quippiam, linea <lb/>&longs;cilicet <emph type="italics"/>compo&longs;ita ex dupla ip&longs;ius AB, & tripla ip&longs;ius CB, &<emph.end type="italics"/> &longs;ola <emph type="italics"/>BD <lb/>ad<emph.end type="italics"/> antecedens, nempè <emph type="italics"/>lineam <expan abbr="æqual&etilde;">æqualem</expan> ip&longs;i EB, & duplæ ip&longs;ius BD.<emph.end type="italics"/><lb/>
<arrow.to.target n="fig83"></arrow.to.target><lb/><emph type="italics"/>Non igitur perinde, vt in proportione ordinata; hoc est, perturbata <expan abbr="exi&longs;t&etilde;">exi&longs;tem</expan><emph.end type="italics"/><lb/>
<arrow.to.target n="marg354"></arrow.to.target> <emph type="italics"/>te proportione, ex æqualiest OD ad DE, vt duplaip&longs;ius AB cum tripla <lb/>ip&longs;ius BC &<emph.end type="italics"/> &longs;ola <emph type="italics"/>BD ad <expan abbr="cõpo&longs;itam">compo&longs;itam</expan> ex dupla vtriu&longs;que &longs;imul AB BE, <lb/>& quadrupla vtriu&longs;que &longs;imul CB BD.<emph.end type="italics"/> &longs;uperat verò DE ip&longs;am <lb/>DO exce&longs;&longs;u OE; linea verò <expan abbr="cõpo&longs;ita">compo&longs;ita</expan> ex dupla vtriu&longs;que &longs;imul <lb/>AB BE, & quadrupla vtriu&longs;que &longs;imul CB BD lineam excedit <lb/>compo&longs;itam ex dupla ip&longs;ius AB cum tripla ip&longs;ius BC, ac &longs;ola <lb/>BD, exce&longs;&longs;u lineæ, quæ &longs;it æqualis &longs;oli CB cum tripla ip&longs;ius <lb/>
<arrow.to.target n="marg355"></arrow.to.target> BD, & dupla ip&longs;ius BE. <emph type="italics"/>Quare est EO ad ED, vt CB cum tripla <lb/>ip&longs;ius BD, & dupla ip&longs;ius EB ad duplam vtriu&longs;que &longs;imul AB BE, <lb/>& quadruplam vtriu&longs;que &longs;imul CB BD. est autem<emph.end type="italics"/> in lineis pro-
<pb pagenum="185"/>portionalibus initio expo&longs;itis; cùm in continua &longs;int propor­<lb/>tione, tertia in ordine BD ad quartam BE, vt prima AB ad <lb/>&longs;ecundam BC, quare diuidendo vt DE ad EB, ita AC ad
<arrow.to.target n="marg356"></arrow.to.target><lb/>CB. Rur&longs;us quoniam in lineis proportionalibus ob eandem <lb/>cau&longs;am CB ad BD ita e&longs;t, vt DB ad BE; erit diuidendo, vt <lb/>CD ad DB, ita DE ad EB. ego <emph type="italics"/>vt DE ad EB, ita AC ad<emph.end type="italics"/>
<arrow.to.target n="marg357"></arrow.to.target><lb/><emph type="italics"/>CB, & CD ad DB. ac propterea &longs;ecundum<emph.end type="italics"/> multiplicem <emph type="italics"/>compo&longs;itio <lb/>nemtripla ip&longs;ius CD, adtriplam ip&longs;ius DB<emph.end type="italics"/> e&longs;t, vt &longs;ola CD ad &longs;o­<lb/>lam DB. <emph type="italics"/>& dupla ip&longs;ius DE ad duplam ip&longs;ius EB<emph.end type="italics"/> e&longs;t, <lb/>vt DE ad EB. e&longs;t verò CD ad DB, vt DE ad <lb/>EB, & AC ad CB; erit igitur AC ad CB, vt tripla ip&longs;ius <lb/>CD ad triplam ip&longs;ius DB; & vt dupla ip&longs;ius DE ad <lb/>duplam ip&longs;ius EB. <emph type="italics"/>Quare &<emph.end type="italics"/> tria antecedentia &longs;imul ad
<arrow.to.target n="marg358"></arrow.to.target><lb/>tria &longs;imul con&longs;equentia, hoc e&longs;t, <emph type="italics"/>compo&longs;ita ex AC, & <lb/>tripla ip&longs;ius CD, & dupla ip&longs;ius DE ad compo&longs;itam ex CB, <lb/>& tripla ip&longs;ius DB, & dupla ip&longs;ius EB<emph.end type="italics"/> ita erit, vt AC <lb/>ad CB, hoc e&longs;t, DE ad EB. <emph type="italics"/>Rur&longs;us itaque di&longs;simili modo, <lb/>quàm in proportionibus ordinatis, hoc est in perturbata proportione,<emph.end type="italics"/><lb/>quoniam e&longs;t in primis magnitudinibus antecedens OE ad <lb/>con&longs;equens ED, ita in &longs;ecundis magnitudinibus an <expan abbr="teced&etilde;s">tecedens</expan> <lb/>compo&longs;ita &longs;cilicet ex CB, cum tripla ip&longs;ius BD, & dupla ip­<lb/>&longs;ius EB, ad con&longs;equens nem pè compo&longs;itam ex dupla vtriu&longs;­<lb/>que &longs;imul AB BE, cum quadrupla vtriu&longs;que &longs;imul CB BD: <lb/>in primis verò magnitudinibus con&longs;equens DE ad aliud quip <lb/>piam EB e&longs;t, vt in &longs;ecundis magnitudinibus aliud quippia, <lb/>hoc e&longs;t compo&longs;ita ex AC cum tripla ip&longs;ius CD, & dupla ip­<lb/>&longs;ius DE ad an tecedens, lineam &longs;cilicet compo&longs;itam ex CB cum <lb/>tripla ip&longs;ius BD, & dupla ip&longs;ius EB. <emph type="italics"/>ex æquali eandem<emph.end type="italics"/>
<arrow.to.target n="marg359"></arrow.to.target><lb/><emph type="italics"/>habebit proportionem EO ad EB, quam AC cum tri <lb/>pla ip&longs;ius CD, & dupla ip&longs;ius DE ad duplam vtriu&longs; <lb/>que &longs;imul AB BE cum qnadrupla vtriu&longs;que &longs;imul CB <lb/>BD.<emph.end type="italics"/> & componendo erit OB ad BE, vtlinea AC
<arrow.to.target n="marg360"></arrow.to.target><lb/>cum tripla ip&longs;ius CD, & dupla ip&longs;ius DE, & dupla <lb/>vtriu&longs;que &longs;imul AB BE, & quadrupla vtriu&longs;que &longs;i­<lb/>mul CB BD, ad duplam vtriu&longs;que &longs;imul AB BE <lb/>cum quadrupla vtriu&longs;que &longs;rmul CB BD. In hoc autem
<pb pagenum="186"/>antecedente a&longs;&longs;umitur &longs;ola AC, ter CD, bis DE, bis AB, <lb/>bis BE, quater CB, & quater BD. Duæ verò AB vnà <lb/>cum &longs;ola AC, & &longs;ola. CB, ex quatuor vicibus, quibus ip­<lb/>&longs;a CB &longs;umitur, &longs;unt æquales tribus AB. tres autem CB, <lb/>quæ relictæ &longs;unt, vnà cum tribus CD, & tribus BD <lb/>ex quatuor vicibus, quibus ip&longs;a BD &longs;umitur, &longs;unt æ­<lb/>quales &longs;ex CB. &longs;ola verò BD, quæ relicta fuit, vnà <lb/>cum duabus DE, & duabus BE, e&longs;t æqualis tribus <lb/>BD. linea nimirum AC cum tripla ip&longs;ius CD, & <lb/>dupla ip&longs;ius DE, & dupla vtriu&longs;que &longs;imul AB BE, <lb/>& quadrupla vtriu&longs;que &longs;imul CB BD, æqualis erit tri­<lb/>plæ ip&longs;ius AB, cum &longs;extupla ip&longs;ius CB, & tripla ip­<lb/>&longs;ius BD. <emph type="italics"/>Tota igitur OB ad EB eandem habet proportio­<lb/>nem, quam linea æqualis triplæ ip&longs;ius AB cum &longs;extupla ip­<lb/>&longs;ius CB & tripla ip&longs;ius BD ad duplam vtriu&longs;que &longs;imul <lb/>AB BE cum quadrupla vtriu&longs;que &longs;imul CB BD. & <lb/>quoniam<emph.end type="italics"/> initio o&longs;ten&longs;um fuit lineas AC CD DE in eadem <lb/>e&longs;&longs;e proportione, vt &longs;unt quatuor lineæ continuè pro­<lb/>portionales AB BC BD BE; erunt tres AC CD <lb/>DE, & tres AB BC BD, & tres BC BD BE <lb/>
<arrow.to.target n="marg361"></arrow.to.target> in eadem proportione. conuertendo igitur in eadem quo­<lb/>que erunt proportione. quare tres <emph type="italics"/>ED DC CA,<emph.end type="italics"/> & <lb/>tres BE BD BC, & tres BD BC BA <emph type="italics"/>in eadem &longs;unt proportione.<emph.end type="italics"/><lb/>
<arrow.to.target n="marg362"></arrow.to.target> Quoniam autem BE BD BC ita &longs;e habent, vt BD BC BA; <lb/>vtraque &longs;imul BE BD advtramque &longs;imul BD BC, & <lb/>vtraque &longs;imul BD BC ad vtramque &longs;imul BC BA <lb/>ita &longs;e habebunt, vt BE BD BC. hæ verò <emph type="italics"/>B<emph.end type="italics"/>E <emph type="italics"/>B<emph.end type="italics"/>D <lb/>BC &longs;unt, vt ED DC CA. ergo <emph type="italics"/>& vtraque &longs;imul <lb/>vnaqueque ip&longs;arum EB BD, DB BC, CB BA<emph.end type="italics"/>, ita &longs;e <lb/>
<arrow.to.target n="marg363"></arrow.to.target> habebunt, vt ED DC CA. quare <emph type="italics"/>erit &<emph.end type="italics"/> antecedens <lb/><emph type="italics"/>ED<emph.end type="italics"/> ad &longs;uas con&longs;equentes DC CA &longs;imul &longs;umptas, <lb/>hoc e&longs;t <emph type="italics"/>ad DA, vt<emph.end type="italics"/> antecedens <emph type="italics"/>vtraque &longs;imul EB BD<emph.end type="italics"/><lb/>ad &longs;uas con&longs;equentes, nempè <emph type="italics"/>ad <expan abbr="vtrāque">vtranque</expan> &longs;imul DB BC<emph.end type="italics"/><lb/>
<arrow.to.target n="marg364"></arrow.to.target> <emph type="italics"/>cum vtraque &longs;imul CB BA. & componendo EA ad AD, <lb/>vt vtraque &longs;imul EB BD cum vtraque &longs;imul AB BC, <lb/>& vtraque &longs;imul CB BD<emph.end type="italics"/> ad vtramque &longs;imul BD BC
<pb pagenum="187"/>cum vtra<gap/>ue &longs;imul CB BA. In hoc autem antecedenti &longs;emel <lb/>&longs;umitur EB, & &longs;emel AB, bis BD, & bis BC. in con&longs;equentive <lb/>rò &longs;umitur <gap/>ola BD, &longs;olaquè BA, & bis BC. Proportio igitur <lb/>ip&longs;arum EA AD e&longs;t eadem, <emph type="italics"/>quæ est vtraque &longs;imul EB BA cum du <lb/>pla vtriu&longs;que &longs;imul DB BC ad vtramque &longs;imul BD BA cum dupla <lb/>ip&longs;ius BC. Quare & dupla ad duplam eandem habebit <expan abbr="proportion&etilde;">proportionem</expan> hoc <lb/>est, vt EA ad AD, ita dupla vtriu&longs;que &longs;imul EB BA cum quadru­<lb/>pla vtriu&longs;que &longs;imul CB BD ad duplam vtriu&longs;que &longs;imul AB BD cum<emph.end type="italics"/><lb/>
<arrow.to.target n="fig84"></arrow.to.target><lb/><emph type="italics"/>quadrupla ip&longs;ius CB. Quapropter EA adtres quintas ip&longs;ius AD e&longs;t, vt <lb/>compo&longs;ita ex dupla vtriu&longs;que &longs;imul AB BE, & qua-<emph.end type="italics"/>
<arrow.to.target n="marg365"></arrow.to.target><lb/><emph type="italics"/>drupla utrivsque &longs;imul CB BD ad tres quintas lineæ com <lb/>po&longs;itæ ex dupla vtriu&longs;que &longs;imul AB BD, & quadruplaip&longs;ius CB. Ve­<lb/>rùm<emph.end type="italics"/> quia initio a&longs;&longs;umptum fuitita e&longs;&longs;e BE ad EA, vt FG ad <lb/>tres quintas ip&longs;ius AD, erit conuertendo EA ad EB, vt
<arrow.to.target n="marg366"></arrow.to.target><lb/>tres quintæ ip&longs;ius AD ad FG; permutandoquè <emph type="italics"/>vt EA ad <lb/>tres quintasip&longs;ius AD, &longs;ic e&longs;t EB ad FG, vtigitur EB ad FG, <lb/>&longs;ic dupla vtriu&longs;que &longs;imul AB BE cum quadrupla vtriu&longs;que<emph.end type="italics"/>
<pb pagenum="188"/><emph type="italics"/>&longs;imul DB BC ad tres quintas lineæ compo&longs;itæ ex dupla vtriu&longs;que &longs;i­<lb/>mul AB BD cum quadrupla ip&longs;ius CB. osten &longs;um e&longs;t aut<gap/> OB ad EB <lb/>ita e&longs;&longs;e, vt<emph.end type="italics"/> tripla ip&longs;ius AB cum &longs;extupla ip&longs;ius CB, & tripla i­<lb/>p&longs;ius BD ad duplam vtriu&longs;que &longs;imul AB BE cum quadrupla <lb/>vtriu&longs;que &longs;imul CB BD. Atin hoc antecedente ter a&longs;&longs;umpta <lb/>e&longs;t AB, terquè BD, & &longs;exies CB. erititaque in primis magni­<lb/>tudinibus antecedens OB ad con&longs;equens EB, vtin &longs;ecundis <lb/>magnitudinibus an recedens <emph type="italics"/>tripla<emph.end type="italics"/> &longs;cilicet <emph type="italics"/>vtriu&longs;que &longs;imul AB <lb/>BD cum &longs;extupla ip&longs;ius CB ad<emph.end type="italics"/> con&longs;equens nempè <emph type="italics"/>duplam v­<lb/>triu&longs;que &longs;imul AB BE, & quadruplam vtriu&longs;que &longs;imul CB BD.<emph.end type="italics"/><lb/>
<arrow.to.target n="fig85"></arrow.to.target><lb/>in primis verò magnitudinibus e&longs;t con&longs;equens EB ad aliud <lb/>quippiam FG, ut in &longs;ecundis magnitudinibus con&longs;equens, <lb/>hoc e&longs;t dupla vtriu&longs;que &longs;imul AB BE cum quadrupla vtriu&longs;­<lb/>que &longs;imul DB BC ad aliud quippiam, nempè ad tres quintas <lb/>lineæ <expan abbr="cõpo&longs;it&ecedil;">compo&longs;it&ecedil;</expan> ex dupla vtri^{9} <expan abbr="&qacute;">que</expan>; &longs;imul AB BD <expan abbr="cũ">cum</expan> quadrupla i­<lb/>
<arrow.to.target n="marg367"></arrow.to.target> p&longs;i^{9} CB. <emph type="italics"/>Ex æquali igitur e&longs;t, ut OB ad FG, ita linea compo&longs;ita ex tripla<emph.end type="italics"/><lb/>
<arrow.to.target n="marg368"></arrow.to.target> <emph type="italics"/><expan abbr="utrius&qacute;">utriusque</expan> &longs;imul AB BD, et &longs;extuplaip&longs;i^{9} CB ad tres <expan abbr="quĩtas">quintas</expan> lineæ <expan abbr="cõpo&longs;i">compo&longs;i</expan> <lb/>tæ ex dupla utri^{9} <expan abbr="&qacute;">que</expan>; &longs;imul AB BD, & quadrupla ip&longs;ius CB. at uerò<emph.end type="italics"/> tri <lb/>pla ip&longs;ius AB ad <expan abbr="duplã">duplam</expan> <expan abbr="eiu&longs;d&etilde;">eiu&longs;dem</expan> AB e&longs;t, vt tria ad duo. &longs;imiliter <lb/>tripla ip&longs;ius BD ad duplam eiu&longs;dem BD e&longs;t, vt tria ad duo.
<pb pagenum="189"/>pariquè ratione &longs;extupla ip&longs;ius CB ad quadruplam ciu&longs;dem, <lb/>CB ita &longs;e habet, vt &longs;ex ad quatuor, hoce&longs;t tria ad duo, & om­<lb/>nesad omnes, hoc e&longs;t <emph type="italics"/>compo&longs;ita ex tripla vtriu&longs;que &longs;imul AB BD, <lb/>et &longs;extupla ip&longs;ius CB ad compo&longs;itam ex dupla vtriu&longs;que &longs;imul AB BD, <lb/>& quadrupla ip&longs;ius CB proportionem habet, quam tria ad duo.<emph.end type="italics"/> vt exem
<arrow.to.target n="marg369"></arrow.to.target><lb/>pli gratia quindecim ad decem, <emph type="italics"/>&longs;ed<emph.end type="italics"/> eadem compo&longs;ita ex tri­<lb/>pla vtriu&longs;que &longs;imul AB BD, & &longs;extupla ip&longs;ius CB <emph type="italics"/>ad tres quin­<lb/>tas eiu&longs;dem<emph.end type="italics"/> compofitæ ex dupla vtriu&longs;que &longs;imul AB BD, & qua
<arrow.to.target n="marg370"></arrow.to.target><lb/>drupla ip&longs;ius, CB, quæ po&longs;ita e&longs;t decem, <emph type="italics"/>proportionem habet, quam <lb/>quinque ad duo.<emph.end type="italics"/> hoc e&longs;t ut quindecim ad &longs;ex, tres enim quintæ <lb/>ip&longs;ius decem &longs;unt &longs;ex. at verò proportio, quam habet linea <expan abbr="cõ">com</expan> <lb/>po&longs;ita ex tripla vtriu&longs;que &longs;imul AB BD, & &longs;extupla ip&longs;ius CB <lb/>ad tres quintas lineæ compo&longs;it&ecedil; ex dupla vtriu&longs;que &longs;imul AB <lb/>BD cum quadrupla ip&longs;ius CB, e&longs;t æqualis ei, quam habet OB <lb/>ad FG. ergo erit OB ad FG, vtquinque ad duo. <emph type="italics"/><expan abbr="Demonstratū">Demonstratum</expan> <lb/>autem e&longs;t, & AO ad GH proportionem habere, quam quinque ad duo; <lb/>totaigitur BA ad totam FH proportionem habet, quam quinque ad duo.<emph.end type="italics"/>
<arrow.to.target n="marg371"></arrow.to.target><lb/><emph type="italics"/>&longs;iautem hoc, e&longs;t quidem FH duæ quintæ ip&longs;ius AB. Quod oportebat <lb/>demon&longs;trare.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg339"></margin.target>1.<emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> hu­<lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg340"></margin.target>2. <emph type="italics"/>lemma <lb/>buius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg341"></margin.target>1.<emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> hu­<lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg342"></margin.target>2. <emph type="italics"/>lemma <lb/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg343"></margin.target>11. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg344"></margin.target>12. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg345"></margin.target>11, <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg346"></margin.target>8.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg347"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 8. <emph type="italics"/><expan abbr="quĩti">quinti</expan><emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg348"></margin.target><emph type="italics"/>co.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="quĩti">quinti</expan><emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg349"></margin.target>18, <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg350"></margin.target><emph type="italics"/>co.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="quĩti">quinti</expan>.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg351"></margin.target>23. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg352"></margin.target>12, <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg353"></margin.target><emph type="italics"/>co.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="quĩti">quinti</expan><emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg354"></margin.target>23.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg355"></margin.target>3.<emph type="italics"/><expan abbr="l&etilde;ma">lemma</expan> hu <lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg356"></margin.target>17. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg357"></margin.target>A</s></p>
<p type="margin">
<s><margin.target id="marg358"></margin.target>12.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg359"></margin.target>23.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg360"></margin.target>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg361"></margin.target><emph type="italics"/>cor.4.quĩ <lb/>ti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg362"></margin.target>4.<emph type="italics"/>lema hu <lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg363"></margin.target><emph type="italics"/>cor.2.lem. <lb/>in<emph.end type="italics"/> 13. <emph type="italics"/>pri­<lb/>mi huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg364"></margin.target>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg365"></margin.target>B</s></p>
<p type="margin">
<s><margin.target id="marg366"></margin.target><emph type="italics"/>co.<emph.end type="italics"/>4.<emph type="italics"/><expan abbr="quĩti">quinti</expan>.<emph.end type="italics"/><lb/>16,<emph type="italics"/>quinti.<emph.end type="italics"/><lb/>11. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg367"></margin.target>22.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg368"></margin.target>C</s></p>
<p type="margin">
<s><margin.target id="marg369"></margin.target>D</s></p>
<p type="margin">
<s><margin.target id="marg370"></margin.target>5.<emph type="italics"/>lemma <lb/>huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg371"></margin.target>12.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<figure id="fig81"></figure>
<figure id="fig82"></figure>
<figure id="fig83"></figure>
<figure id="fig84"></figure>
<figure id="fig85"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>Græcus codex po&longs;t ea verba, <emph type="italics"/>vt DE ad EB, ita AC ad CB,<emph.end type="italics"/>
<arrow.to.target n="marg372"></arrow.to.target><lb/>non habet, <emph type="italics"/>& CD ad DB,<emph.end type="italics"/> quæ ob ea, quæ &longs;equuntur, omninò <lb/>nece&longs;&longs;aria videntur. ideo po&longs;t gr&ecedil;ca verba, <foreign lang="greek">e)/side\ka<gap/> w(s de w_<10>o\s eb, <lb/>ou)/tws a)/te ag w_<10>o\s, gb</foreign> de&longs;iderarividentur. <foreign lang="greek">ka<gap/> a(= gd w_<10>o/s db. </foreign></s></p>
<p type="margin">
<s><margin.target id="marg372"></margin.target><emph type="italics"/>A<emph.end type="italics"/></s></p>
<p type="main">
<s>Vbiautem &longs;untverba, <emph type="italics"/>vt <expan abbr="cõpo&longs;ita">compo&longs;ita</expan> ex dupla vtriu&longs;que &longs;imul,<emph.end type="italics"/> Græ
<arrow.to.target n="marg373"></arrow.to.target><lb/>cus codex tantùm habet, <foreign lang="greek">ou)\tws a) sugkeime/na e)\xte ta=s suuamfote<10>ou</foreign>. <lb/>In quibus de&longs;ideratur illa particula, <emph type="italics"/>dupla,<emph.end type="italics"/> ideo corrigendus e&longs;t <lb/>hoc modo, <foreign lang="greek">ou)/tws a\ sugkeime/na e)\kte ta=s b suuamfote/<10>ou</foreign>, &c. </s></p>
<pb pagenum="190"/>
<p type="margin">
<s><margin.target id="marg373"></margin.target>B</s></p>
<p type="main">
<s>Præterea cùm inquit, <emph type="italics"/>ex æqualiigitur e&longs;t vt OB ad FG,<emph.end type="italics"/> Græ­<lb/>
<arrow.to.target n="marg374"></arrow.to.target> cus non habet, <emph type="italics"/>ad FG,<emph.end type="italics"/> idcirco po&longs;t ea verba <foreign lang="greek">kai\ di\<gap/>sou a)/<10>a e)sin co(s <lb/>a<gap/> ob</foreign> addenda &longs;unt <foreign lang="greek">w_<10>o\s zk. </foreign></s></p>
<p type="margin">
<s><margin.target id="marg374"></margin.target>C</s></p>
<p type="main">
<s>Similiter quando in quit <emph type="italics"/>ad compo&longs;itam ex dupla vtriu&longs;que &longs;imul<emph.end type="italics"/><lb/>
<arrow.to.target n="marg375"></arrow.to.target> <emph type="italics"/>AB BD, & quadrupla ip&longs;ius CB,<emph.end type="italics"/> græca verba &longs;unt <foreign lang="greek">w_<10>o=s me\n ta\n sug­<lb/>keim<gap/>nan e)/kte ta=s b sunamfote\<10>ou ta=s ab bd ta=s *gb</foreign>, in quib^{9} &longs;imiliter deli­<lb/>deratur, <emph type="italics"/>& quadrupla.<emph.end type="italics"/> quare ita corrigendus videtur. <foreign lang="greek">w_<10>o\s me\n ta/n <lb/>sugkeime\nan e)/ k te tas b sunamfote/<10>ou ta=s ab bd, kai\ d ta)/s *gb</foreign>, </s></p>
<p type="margin">
<s><margin.target id="marg375"></margin.target>D</s></p>
<p type="main">
<s>Po&longs;tremum theorema, & &longs;i non habeat <expan abbr="tãtam">tantam</expan> <expan abbr="ob&longs;curitat&etilde;">ob&longs;curitatem</expan>, <lb/>veluti pr&ecedil;cedens, non e&longs;t tamen &longs;ine aliqua ob&longs;curitate, ob cu <lb/>ius intelligentiam hanc priùs propo &longs;itionem o&longs;tendemus. </s></p>
<p type="head">
<s>PROPOSITIO.</s></p>
<p type="main">
<s>Si duæ fuerint rectæ line&ecedil; in para bolc ad diametrum ordi <lb/>natim applicatæ, erit maior parabole ad <expan abbr="minor&etilde;">minorem</expan>, vt cubus ex <lb/>dimidia line&ecedil; maioris ad cubum ex dimidia minoris. </s></p>
<figure></figure>
<p type="main">
<s>In parabole ABC, cuius diameter BF, duæ &longs;int rectæ lineæ <lb/>ad diametrum applicatæ AC DE. Dico parabolen ABC ad <lb/>parabolen DBE eandem habere proportionem, quam cub^{9} <lb/>ex AF ad cubum ex DG. lungantur AB BC DB BE; &longs;ecet-
<pb pagenum="191"/>què AB ip&longs;am DG in H. Quoniam enim parabole ABC
<arrow.to.target n="marg376"></arrow.to.target><lb/>&longs;e&longs;quitertia e&longs;t trianguli ABC, itidemquè parabole DBE <lb/>trianguli DBE &longs;e&longs;quitertia exi&longs;tit, erit parabole ABC ad trian <lb/>gulum ABC, vt parabole DBE ad triangulum DBE. &
<arrow.to.target n="marg377"></arrow.to.target> per­<lb/>mutando parabole ABC ad parabolen DBE, vt triangulum <lb/>ABC ad triangulum DBE. Quoniam autem AC ordina­<lb/>tim e&longs;t applicata, vnde AF ip&longs;i FC e&longs;t æqualis, ac per con&longs;e­<lb/>quens AF e&longs;t ip&longs;ius AC dimidia. erit triangulum ABF dimi­<lb/>dium trianguli ABC. cùm vtraquè &longs;ub eadem &longs;int altitudine.
<arrow.to.target n="marg378"></arrow.to.target><lb/>eademquè ratione triangulum DBG trianguli DBE dimi­<lb/>dium exi&longs;tit. quare vt triangulum ABF ad triangulum <lb/>DBG, ita e&longs;t triangulum ABC ad DBE triangulum, ac pro­<lb/>pterea triangulum ABF ad DBG triangulum e&longs;t, vt parabo­<lb/>le ABC ad parabolen DBE. Cùm autem &longs;it HG æquidi&longs;tans <lb/>ip&longs;i AF, &longs;iquidem &longs;unt ordinatim applicatæ, ob <expan abbr="triangulorũ">triangulorum</expan>
<arrow.to.target n="marg379"></arrow.to.target><lb/>&longs;imilitudinem ABF HBG, ita erit FB ad BG, vt AF ad HG <lb/>vtautem FB ad BG, ita quadratum ex AF ad quadratum ex
<arrow.to.target n="marg380"></arrow.to.target><lb/>DG, eritigitur quadratum ex AF ad quadratum ex DG, vt AF <lb/>ad HG. quare line&ecedil; AF DG HG &longs;unt proportionales. Pro­<lb/>ducatur FB, ducaturquè à puncto D ip&longs;i AB æquidi&longs;tans <lb/>DK, erit vtiquè triangulorum ABF DKG anguli ABF <lb/>DHG æquales, & angulus AFB angulo DGK e&longs;t æqualis, erit <lb/>igitur, & reliquus BAF reliquo KDG æqualis, ac propterea <lb/>triangulum ABF e&longs;t triangulo DKG &longs;imile. quare triangu­<lb/>lum ABF ad triangulum DKG eam habet proportionem, <lb/>quàm AF ad DG duplicatam, hoc e&longs;t quàm AF ad HG, qu&ecedil; <lb/>e&longs;t ea, quàm habet FB ad BG. atqui triangulum ABF ad <lb/>DKG eam quoque habet proportionem, quam FB ad GK <lb/>duplicatam. tres igitur line&ecedil; FB GK GB &longs;unt proportiona­<lb/>les. ex quibus &longs;equiturita e&longs;&longs;e FB ad GK, vt AF ad DG; & <lb/>GK ad GB, vt DG ad GH. &longs;ed quoniam triangulum GDK <lb/>ad GDB (cùm &longs;int &longs;ub eadem altitudine) ita e&longs;t, vt KG ad
<arrow.to.target n="marg381"></arrow.to.target><lb/>BG, &longs;i igitur fiat HG ad L, vt KG ad BG, erit triangulum <lb/>GDK ad triangulum GDB, vt HG ad L. Cùm autem &longs;it <expan abbr="triã">triam</expan> <lb/>gulum ABF ad DKG, vt AF ad HG, e&longs;tquè <expan abbr="triangulũ">triangulum</expan> DKG <lb/>ad DBG, vt HG ad L, erit ex &ecedil;quali triangulum ABF ad <lb/>triangulum DBG, vt AF ad L. ac propterea parabole ABC
<pb pagenum="192"/>ad parabolen DBE eam habet proportionem, quam linea <lb/>AF ad lineam L. Quoniam autem ita e&longs;t KG ad GB, vt <lb/>HG ad L, & vt eadem KG ad GB, ita e&longs;t DG ad GH. vt <lb/>verò DG ad GH, ita e&longs;t AF ad DG; crunt quatuor lineæ AF <lb/>DG HG L in continua proportione. & quoniam cubi in tri­<lb/>pla &longs;unt proportione laterum, erit cubus ex AF ad cubum ex <lb/>DG, vt AF ad L. cubus ergo ex AF ad cubum ex DG eam <lb/>habet proportionem, quam parabole ABC ad parabolen <lb/>DBE. quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg376"></margin.target>17.34. A<emph type="italics"/>r <lb/>ch.de qua. <lb/>par.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg377"></margin.target>16. <emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg378"></margin.target><emph type="italics"/>ex prima <lb/>&longs;extt.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg379"></margin.target><emph type="italics"/>ex<emph.end type="italics"/> 4.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg380"></margin.target>20. <emph type="italics"/>primi <lb/>conicorum <lb/>Apoll. & <lb/>ex<emph.end type="italics"/>3. A<emph type="italics"/>rch. <lb/>de quad. <lb/>parab. <lb/>ex cor.<emph.end type="italics"/> 20. <lb/><emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg381"></margin.target>1.<emph type="italics"/>&longs;exti.<emph.end type="italics"/><lb/>11.<emph type="italics"/>quintl.<emph.end type="italics"/></s></p>
<p type="main">
<s>Oportet autem banc quoquè <expan abbr="propo&longs;ition&etilde;">propo&longs;itionem</expan> nobis e&longs;&longs;e cogni <lb/>tam, nem pè quòd &longs;olida parallelepipeda in eadem ba&longs;i con&longs;ti <lb/>tuta eam inter&longs;e proportionem habent, quam ip&longs;arum alti­<lb/>tudines. </s></p>
<p type="main">
<s>Hoc quidem à Federico Commandino in eius libro decen <lb/>tro grauitatis &longs;olidorum propo&longs;itione decimanona demon­<lb/>&longs;tratum fuit. </s></p>
<p type="head">
<s>PROPOSITIO. X.</s></p>
<p type="main">
<s>Omnis fru&longs;ti à rectanguli coni portione ab&longs;ci&longs;&longs;i <lb/>centrum grauitatis e&longs;t in recta linea, quæ fru&longs;ti dia­<lb/>meter exi&longs;tit, ita po&longs;itum, vt diui&longs;a linea in quin­<lb/>que partes æquales, &longs;it in quinta parte media; ita <lb/>vt ip&longs;ius portio propinquior minoriba&longs;i fru&longs;ti ad <lb/>reliquam portionem eandem habeat proportio­<lb/>nem, quam habet &longs;olidum ba&longs;im habens quadra­<lb/>tumex dimidia maioris ba&longs;is fru&longs;ti, altitudinem au <lb/>tem lineam æqualem vtrique &longs;imul duplæ mino­<lb/>ris ba&longs;is, & maiori ad &longs;olidum ba&longs;im habens qua­<lb/>dratum ex dimidia minoris ba&longs;is fru&longs;ti, <expan abbr="altitudin&etilde;">altitudinem</expan> <lb/>autem lineam æqualem vtrique duplæ maioris, & <lb/>minori. </s></p>
<pb pagenum="193"/>
<figure></figure>
<p type="main">
<s><emph type="italics"/>Sit in rectanguli coni portione<emph.end type="italics"/> ABC <emph type="italics"/>duæ rectæ lineæ AC DE<emph.end type="italics"/><lb/>æquidi&longs;tantes. <emph type="italics"/>diameter verò portionis ABC &longs;it BF.<emph.end type="italics"/> Intelli­<lb/>gaturquè fru&longs;tum ADEC à portione ABC ab&longs;ci&longs;&longs;um. om­<lb/>nes vtique lineæ ip&longs;is AC DE æquidi&longs;tantes in fru&longs;to AD <lb/>EC ductæ, erunt à linea GF bifartam diui&longs;æ, ex quo <emph type="italics"/>pa­<lb/>tet quidem & ip&longs;ius ADEC diametrum e&longs;&longs;e GF, lineasquè AC <lb/>DE lineæ portionem in B contingenti æquidistantes e&longs;&longs;e. Recta<emph.end type="italics"/>
<arrow.to.target n="marg382"></arrow.to.target><lb/><emph type="italics"/>verò linea FG in quinque partes æquales diui&longs;a, quinta pars me­<lb/>dia &longs;it HK. atque<emph.end type="italics"/> diuidatur HK in I, ita vt <emph type="italics"/>HI ad <lb/>IK eandem habeat proportionem, quam habet &longs;olidum ba&longs;im habens <lb/>quadratum ex AF, altitudinem verò lineam æqualem vtri&longs;que <lb/>&longs;imul duplæ ip&longs;ius DG, & ip&longs;i AF, ad &longs;olidum, quod <lb/>ba&longs;im habeat quadratum ex DG, altitudinem autem lineam æqua-<emph.end type="italics"/>
<pb pagenum="194"/>
<arrow.to.target n="fig86"></arrow.to.target><lb/><emph type="italics"/>lem vtri&longs;que duplæ ip&longs;ius AF, & ip&longs;i DG. ostenden­<lb/>dum est frusti ADEC centrum grauitatis e&longs;&longs;e punctum 1.<emph.end type="italics"/><lb/>
<arrow.to.target n="marg383"></arrow.to.target> <emph type="italics"/>&longs;it quidem ip&longs;i FB æqualis MN, ip&longs;i verò GB æqualis NO. <lb/>&longs;umaturquè ip&longs;arum MN NO media proportionalis NX. <lb/>quarta verò proportionalis TN.<emph.end type="italics"/> lineæ nimirum MN NX <lb/>NO NT in continua erunt proportione. <emph type="italics"/>& vt TM <lb/>ad TN, ita<emph.end type="italics"/> fiat <emph type="italics"/>FH ad quandam lineam à puncto I, vt <gap/>R, vbi­<lb/>cunque perueniat alterum punctum<emph.end type="italics"/> R. <emph type="italics"/>nihil enim refert, &longs;iue inter <lb/>FG, &longs;iue inter GB cadat. & quoniam in portione rectanguli coni<emph.end type="italics"/><lb/>ABC <emph type="italics"/>diameter portionis est FB; at verò BF, vel prin­<lb/>cipalis est diameter portionis, vel ducta diametro æquidistans. <lb/>lineæ verò AF DG ad ip&longs;am ordinatim &longs;unt ap­<lb/>plicatæ, cùm &longs;int æquidistantes contingenti portionem<emph.end type="italics"/>
<pb pagenum="195"/><emph type="italics"/>in puncto B. &longs;i autem hoc, est vt AF ad DG potentia,<emph.end type="italics"/>
<arrow.to.target n="marg384"></arrow.to.target><lb/><emph type="italics"/>&longs;ic FB ad BG longitudine, hoc est MN ad NO. <lb/>vt autem MN ad NO longitudine, itaest MN ad Nx potentia.<emph.end type="italics"/><lb/>quandoquidem treslineæ MN NX NO &longs;unt proportio­<lb/>nales. <emph type="italics"/>vt igitur AF ad DG potentia, ita est MN ad N X<emph.end type="italics"/>
<arrow.to.target n="marg385"></arrow.to.target><lb/><emph type="italics"/>potentia. quare, & longitudine in eadem &longs;unt proportione<emph.end type="italics"/>; vt &longs;cili <lb/>cet AF ad DG, ita MN ad NX. <emph type="italics"/>&longs;ieist itaque cubus ex AF<emph.end type="italics"/>
<arrow.to.target n="marg386"></arrow.to.target><lb/><emph type="italics"/>ad cubum ex DG, ita cubus ex MN ad cubum ex NX. Verùm<emph.end type="italics"/>
<arrow.to.target n="marg387"></arrow.to.target><lb/><emph type="italics"/>vt cubus ex AF adcubum ex DG, &longs;ic portio ABC ad portio­<lb/>nem DBE.<emph.end type="italics"/> vtigitur cubus ex MN ad cubum ex NX, ita <lb/>portio ABC ad portionem DBE. <emph type="italics"/>&longs;icut autem cubus ex MN <lb/>ad culum ex Nx, ita MN ad NT.<emph.end type="italics"/> cùm &longs;int quatuor lineæ <lb/>MN NX NO NT in continua proportione. ac propterea <lb/>eritportio ABC ad portionem DBE, vt MN ad NT. <lb/><emph type="italics"/>Quare & diuidendo frustum ADEC ad portionem DBE e&longs;t, vt<emph.end type="italics"/>
<arrow.to.target n="marg388"></arrow.to.target><lb/><emph type="italics"/>MT ad NT.<emph.end type="italics"/> Quia vero, vt factum fuit, ità e&longs;t MT ad TN, <lb/>vt FH ad IR, e&longs;t verò FH ip&longs;ius FG tresquintæ, erit fru­<lb/>&longs;tum ADEC ad portionem DBE, vt FH ad IR <emph type="italics"/>hoc est <lb/>tres quintæ ip&longs;ius GF ad IR. & quoniam &longs;olidum ba&longs;im habens qua­<lb/>dratum ex AF, altitudinem verò lineam compo&longs;itam ex dupla ip&longs;ius <lb/>DG, & ip&longs;a AF, ad cubum ex AF proportionem habet,<emph.end type="italics"/> quam &longs;o <lb/>lidi altitudo ad altitudinem cubi, &longs;iquidem &longs;untin eadem ba <lb/>&longs;i. tàm emm &longs;olidum, quàm cubus ba&longs;im habet quadratum <lb/>ex AF. idcirco &longs;olidum ba&longs;im habens quadratum ex AF, <lb/>altitudinem verò lineam compo&longs;itam ex dupla ip&longs;ius DG, & <lb/>ip&longs;a AF ad cubum ex AF eam proportio nem habebit, <emph type="italics"/>quam<emph.end type="italics"/><lb/>&longs;olidi altitudo, <emph type="italics"/>dupla,<emph.end type="italics"/> &longs;cilicet <emph type="italics"/>ip&longs;ius DG cumlinea AF<emph.end type="italics"/> ad alci­<lb/>tudinem cubi, hoc e&longs;t <emph type="italics"/>ad FA.<emph.end type="italics"/> Atverò quoniam o&longs;ten&longs;um e&longs;t <lb/>ita e&longs;&longs;e AF ad DG, vt MN ad NX, eritconuertendo DG <lb/>ad AF, vt NX ad MN, & antecedentium dupla, hoc e&longs;t du <lb/>pla ip&longs;ius DG ad AF, vt dupla ip&longs;ius NX ad MN. & com­<lb/>ponendo dupla ip&longs;ius DG cum AF ad AF, vt dupla ip&longs;ius
<arrow.to.target n="marg389"></arrow.to.target><lb/>NX cum MN ad MN. <emph type="italics"/>Quare & vt<emph.end type="italics"/> &longs;olidum ba&longs;im habens <lb/>quadratum ex AF, altitudinem verò lineam compo&longs;itam ex <lb/>dupla ip&longs;ius DG cum AF ad cubum ex AF, ita <emph type="italics"/>dupla ip&longs;ius NX <lb/>cum linea NM ad NM. est autem<emph.end type="italics"/> cubus ex AF adcubum <lb/>ex DG, vt cubus ex MN ad cubum ex NX, vt o&longs;ten&longs;um e&longs;t,
<pb pagenum="196"/>
<arrow.to.target n="fig87"></arrow.to.target><lb/><emph type="italics"/>cubusverò ex MN ad cubum ex NX e&longs;t, vt MN ad N<emph.end type="italics"/>T; <lb/>erit <emph type="italics"/>& vt cubus ex AF ad cubum ex DG, ita MN ad NT. <lb/>&longs;icut autem cubus ex DG ad &longs;olidum ba&longs;im habens quadratum ex DG, <lb/>altitudinem verò lineam compo&longs;itam ex dupla ip&longs;ius AF, cum linea <lb/>DG,<emph.end type="italics"/> ita altitudo cubi ad altitudinem &longs;olidi, cum &longs;int in ea­<lb/>dem ba&longs;i, quadrato nempè ex DG. erit igitur vt cubus ex <lb/>DG ad &longs;olidum ba&longs;im habens quadratum ex DG, altitudi­<lb/>nem verò lineam compo&longs;itam ex dupla ip&longs;ius AF cum linea <lb/>DG, <emph type="italics"/>ita<emph.end type="italics"/> cubi altitudo <emph type="italics"/>DG ad<emph.end type="italics"/> altitudinem &longs;olidi, ad <lb/>lineam &longs;cilicet <emph type="italics"/>compo&longs;itam ex dupla ip&longs;ius AF, & linea <lb/>DG.<emph.end type="italics"/> Quoniam autem ita e&longs;t AF ad DG, vt <lb/>MN ad NX, vt verò MN ad NX, ita NO <lb/>ad NT. cùm &longs;int MN NX NO NT in continua proportio <lb/>
<arrow.to.target n="marg390"></arrow.to.target> ne, crit AF ad DG, vt NO ad NT. & antecedentium dupla,
<pb pagenum="197"/>hoc e&longs;t, dupla ip&longs;ius AF ad DG, vt dupla ip&longs;ius NO ad <lb/>NT, & componendo, dupla ip&longs;ius AF cum DG ad
<arrow.to.target n="marg391"></arrow.to.target><lb/>DG, vt dupla ip&longs;ius NO cum NT ad NT. & conuer­<lb/>tendo DG ad duplam ip&longs;ius AF cum DG, vt NT ad
<arrow.to.target n="marg392"></arrow.to.target> du­<lb/>plam ip&longs;ius NO cum NT. <emph type="italics"/>Quare & vt<emph.end type="italics"/> &longs;e habet cubus ex <lb/>DG ad &longs;olidum ba&longs;im habens quadratum ex DG, altitu­<lb/>dinem verò compo&longs;itam ex dupla ip&longs;ius AF cum DG, ita <lb/>e&longs;t <emph type="italics"/>TN ad compo&longs;itam ex dupla ip&longs;ius ON, & linea TN.<emph.end type="italics"/> Ita­<lb/>que ex ijs, quæ dicta &longs;unt, ita &longs;e habet &longs;olidum ba&longs;im ha­<lb/>bens quadratum ex AF, altitudinem verò lineam com­<lb/>po&longs;itam ex dupla ip&longs;ius DG, & linea AF ad cubum <lb/>ex AF, vt dupla ip&longs;ius NX cum NM ad MN, <lb/>cubus verò ex AF ad cubum ex DG e&longs;t, vt MN ad <lb/>NT; ita deinde &longs;e habetcubus ex DG ad &longs;olidum ba­<lb/>&longs;im habens quadratum ex DG, altitudinem verò lineam <lb/>compo&longs;itam ex dupla ip&longs;ius AF, & ip&longs;a DG, vt <lb/>NT ad compo&longs;itam ex dupla ip&longs;ius NO, & ip&longs;a NT. <lb/><emph type="italics"/>Sunt igitur quatuor magnitudines &longs;olidum ba&longs;im habens quadratum <lb/>ex AF, altitudinem verò lineam compo&longs;itam ex dupla ip&longs;ius <lb/>DG, & linea AF, & cubus ex AF, & cubus ex <lb/>DG, & &longs;olidum ba&longs;im habens quadratum ex DG, altitu <lb/>dinem verò lineam compo&longs;itam: ex dupla ip&longs;ius AF, & ip&longs;a <lb/>DG, quatuor magnitudinibus proportionales, duabus &longs;imul &longs;umptis <lb/>tineæ compo&longs;itæ ex dupla ip&longs;ius NX<gap/> & ip&longs;a NM; & alte­<lb/>ri magnitudini MN; aliiquè deinceps NT, ac tandem lineæ <lb/>compo&longs;itæ ex duplaip&longs;ius NO, & ip&longs;a NT. ex æquali igitur <lb/>erit, vt &longs;olidum ba&longs;im habens quadratum ex AF, altitudinem<emph.end type="italics"/>
<arrow.to.target n="marg393"></arrow.to.target><lb/><emph type="italics"/>autem lineam compo&longs;itam ex dupla ip&longs;ius DG, & ip&longs;a AE, ad <lb/>&longs;olidum ba&longs;im habens quadratum ex DG, altitudinem verò lt­<lb/>neam compo&longs;itam ex dupla ip&longs;ius AF, & ip&longs;a DG, ita <lb/>compo&longs;ita ex dupla ip&longs;ius NX, & ip&longs;a MN ad compo&longs;itam <lb/>ex dupla ip&longs;ius NO, & ip&longs;a NT &longs;ed vt præfatum &longs;oii­<lb/>dum<emph.end type="italics"/> ba&longs;im habens quadratum ex AF, altitudinem verò <lb/>lineam compo&longs;itam ex dupla ip&longs;ius DG, & ip&longs;a AF <emph type="italics"/>ad <lb/>dictum &longs;olidum<emph.end type="italics"/> ba&longs;im habens quadratum ex DG, altitudi­<lb/>nem verò compo&longs;itam ex dupla ip&longs;ius AF & ip&longs;a DG,
<arrow.to.target n="marg394"></arrow.to.target><lb/><emph type="italics"/>ita<emph.end type="italics"/> factum fuit <emph type="italics"/>HI ad IK. vt igitur HI ad IK, &longs;u<emph.end type="italics"/>
<pb pagenum="198"/>
<arrow.to.target n="fig88"></arrow.to.target><lb/><emph type="italics"/>compo&longs;ita<emph.end type="italics"/> ex dupla ip&longs;ius NX cum MN <emph type="italics"/>ad compo&longs;itam<emph.end type="italics"/> ex dupla <lb/>
<arrow.to.target n="marg395"></arrow.to.target> ip&longs;ius NO cum NT. <emph type="italics"/>quare & componendo<emph.end type="italics"/> HK ad KI, vt <lb/>dupla ip&longs;ius NX cum MN, & dupla ip&longs;ius NO cum NT ad <lb/>compo&longs;itam ex dupla ip&longs;ius NO cum NT, quia verò in hoc <lb/>antecedenti &longs;emel &longs;umitur MN, & &longs;emel NT, bis verò NX, <lb/>& bis NO, erit HK ad KI, vt vtraque &longs;imul MN NT, & du­<lb/>pla vtriu&longs;que &longs;imul NX NO ad duplam ip&longs;ius NO, & ip&longs;am <lb/>NT. <emph type="italics"/>& antecedentium quintupla.<emph.end type="italics"/> quintupla verò antecedentis <lb/>HK e&longs;t FG, quintupla verò alterius antecedentis MN NT, <lb/>& duplæ vtriu&longs;que &longs;imul NX NO e&longs;t quintupla vtriu&longs;que &longs;i­<lb/>mul MN NT, & decupla vtriu&longs;que &longs;imul NX NO. decu­<lb/>pla enim e&longs;t quintupla duplæ. <emph type="italics"/>e&longs;t igitur FG ad IK, vt quintupla <lb/>vtriu&longs;que &longs;imul MN NT, & decupla vtriu&longs;que &longs;imul NX NO ad du <lb/>plam ip&longs;ius ON, & ip&longs;am NT. & vt FG ad FK, quæe&longs;t duæ quin <lb/>tæip&longs;ius<emph.end type="italics"/> FG, <emph type="italics"/>ita quintupla vtriu&longs;que &longs;imul MN NT, & decupla <lb/>vtriu&longs;que &longs;imul NX NO ad duplam vtriu&longs;que &longs;imul MN NT,<emph.end type="italics"/>
<pb pagenum="199"/><emph type="italics"/>& quadruplam vtriu&longs;que &longs;imul NX NO.<emph.end type="italics"/> cùm hoc quidem con <lb/>&longs;equens &longs;itduæ quintæ ip&longs;ius antecedentis. etenim dupla v­<lb/>triu&longs;que &longs;imul MN NT quintuplæ earumdem &longs;imul MN <lb/>NT duæ quintæ exi&longs;tit. & quadrupla vtriu&longs;que &longs;imul NX <lb/>NO e&longs;t duæ quintæ decuplæ earumdem NX NO. quadru­<lb/>pla enim decuplæ e&longs;t duæ quintæ. Quoniam itaque ita e&longs;t FG <lb/>ad FK, vt quintupla vtriu&longs;que &longs;imul MN NT, & decupla <lb/>vtriu&longs;que &longs;imul NX NO ad duplam vtriu&longs;que &longs;imul MN <lb/>NT, & quadruplam vtriu&longs;que &longs;imul NX NO, & vt FG ad <lb/>KI, ita quintupla vtriu&longs;que &longs;imul MN NT, & decupla vtriu&longs; <lb/>que &longs;imul NX NO ad duplam ip&longs;ius ON, & ip&longs;am NT: <lb/>erit FG ad &longs;uas con&longs;equentes &longs;imul &longs;umptas FK KI, hoc
<arrow.to.target n="marg396"></arrow.to.target><lb/>e&longs;t FI, vt quintupla vtriu&longs;que &longs;imul MN NT, & decupla <lb/>vtriu&longs;que &longs;imul NX NO ad duplam vtriu&longs;que &longs;imul MN <lb/>NT, & quadruplam vtriu&longs;que &longs;imul NX NO, & duplam <lb/>ip&longs;ius ON, & ip&longs;am NT. &longs;ed in hoc con&longs;equenti bis &longs;umi­<lb/>tur MN, quater NX, &longs;exies NO, & ter NT. <emph type="italics"/>erit igitur vt <lb/>FG æd FI, ita quintupla vtriu&longs;que &longs;imul MN NT, & decupla v­<lb/>triu&longs;que &longs;imul NX NO ad compo&longs;itam ex dupla ip&longs;ius MN, & qua­<lb/>drupla ip&longs;ius NX, & &longs;extupla ip&longs;ius NO, & tripla ip&longs;ius NT.<emph.end type="italics"/> & <lb/>conuertendo FI ad FG, vt compo&longs;ita ex dupla ip&longs;ius MN,
<arrow.to.target n="marg397"></arrow.to.target><lb/>& quadrupla ip&longs;ius NX, & &longs;extupla ip&longs;rus NO, & tripla ip­<lb/>&longs;iús NT ad quintuplam vtriu&longs;que &longs;imul MN NT, & decu­<lb/>plam vtriu&longs;que &longs;imul NX NO. <emph type="italics"/>Quoniam itaque quatuor rectæ li <lb/>neæ MN NX NO NT &longs;unt continuè proportionales.<emph.end type="italics"/> factaquè <lb/>fuit MN æqualis ip&longs;i FB, & NO ip&longs;i GB; crit reliqua OM <lb/>ip&longs;i FG æqualis. & vt TM ad TN ita factum fuit FH, <lb/>hoc e&longs;t tres quintæ ip&longs;ius FG, tres &longs;cilicet quintæ ip&longs;ius MO <lb/>ad IR. quare & conuertendo <emph type="italics"/>vt NT ad TM, ita quædam a&longs;&longs;um­<lb/>pta linea NI ad tres quintas ip&longs;ius FG, hoc e&longs;t ip&longs;ius MO. vt autem <lb/>compo&longs;ita ex dupla ip&longs;ius NM, & quadrupla ip&longs;ius NX, & &longs;extupla ip­<lb/>&longs;ius NO & tripla ip&longs;ius NT ad lineam compo&longs;itam ex quintupla vtrius­<lb/>que &longs;imul MN NT, & decupla vtriu&longs;que &longs;imul XN NO, &longs;ic altera quæ <lb/>dam a&longs;&longs;umpta linea IF ad FG, hoc est ad MO, erit ex &longs;uperioribus RF<emph.end type="italics"/>
<arrow.to.target n="marg398"></arrow.to.target><lb/><emph type="italics"/>duæ quintæ ip&longs;ius MN, hoc est ip&longs;ius FB.<emph.end type="italics"/> ac propterea reliqua RB <lb/>erit tres quintæ ip&longs;ius FB. & obid BR ad. RF e&longs;t, vt tria ad
<arrow.to.target n="marg399"></arrow.to.target><lb/>duo. <emph type="italics"/>Quare punctum R centrum est grauitatis portionis ABC. &longs;it<emph.end type="italics"/>
<pb pagenum="200"/>
<arrow.to.target n="fig89"></arrow.to.target><lb/>
<arrow.to.target n="marg400"></arrow.to.target> <emph type="italics"/>quidem portionis DBE centrum grauitatis punctum Q frusti AD <lb/>EC centrum grauitatis erit in linea QR<emph.end type="italics"/> producta, <emph type="italics"/>quæ<emph.end type="italics"/> quiden QR <lb/><emph type="italics"/>adip&longs;ain<emph.end type="italics"/> productam <emph type="italics"/>eandem habeat proportionem quam habet fru&longs;ium<emph.end type="italics"/><lb/>ADEC <emph type="italics"/>ad reliquam portionem<emph.end type="italics"/> DBE. <emph type="italics"/>est autem punctum I. nam.<emph.end type="italics"/><lb/>cùm &longs;it tota FB ad totam BR, vt ablata BG ad ablatam <lb/>
<arrow.to.target n="marg401"></arrow.to.target> BQ, &longs;unt enim vt quinque ad tria, erit & reliqua FG ad reli­<lb/>quam QR, vt FB ad BR. itaque <emph type="italics"/>quoniam tres quintæ ip&longs;ius FB <lb/>linea e&longs;i BR; ip&longs;ius verò GB tres quintæ linea est <expan abbr="Bq.">Bque</expan> & reliquæ <lb/>igitur GF est tres quintæ QR. quoniamigitur est, vt fru&longs;tum AD <lb/>EC adportionem DBE, ita MT ad NT,<emph.end type="italics"/> vt o&longs;ten&longs;um fuit; <emph type="italics"/>&longs;ed vt <lb/>MN ad NT, &longs;ic<emph.end type="italics"/> factum fuit FH ad IR, hoc e&longs;t <emph type="italics"/>tres quintæ ip&longs;ius <lb/>GF; quæ est QR ad RI. erit igitur vt fru&longs;tum ADEC adportionem <lb/>DBE, ita QR ad RI. & est quidem totius portionis<emph.end type="italics"/> ABC <emph type="italics"/>centrum<emph.end type="italics"/><lb/>
<arrow.to.target n="marg402"></arrow.to.target> <emph type="italics"/>grauitatis punctum R; ip&longs;ius verò DBE centrum grauitatis punctum <lb/>Q: manife&longs;tum est igitur fru&longs;ti ADEC centrum grauitatis e&longs;&longs;e <expan abbr="pun-ctũ">pun­<lb/>ctum</expan> l.<emph.end type="italics"/> quod <expan abbr="quid&etilde;">quidem</expan> e&longs;t in quinta parte media HK ip&longs;ius FG ab
<pb pagenum="201"/>eo ita diui&longs;a, vt HI ad IK &longs;it, vt&longs;olidum ba&longs;im habens qua­<lb/>dratum ex AF, altitudinem autem duplam ip&longs;ius DG cum <lb/>AF ad &longs;olidum ba&longs;im habens quadratum ex DG, altitudinem <lb/>verò duplam ip&longs;ius AF <expan abbr="cũ">cum</expan> DG. quod demon&longs;trare oportebat. </s></p>
<p type="margin">
<s><margin.target id="marg382"></margin.target>1 <emph type="italics"/>Arch de <lb/>quad. pa­<lb/>rab. & <lb/><expan abbr="&longs;ecũdi">&longs;ecundi</expan> coni <lb/>corum A­<lb/>poll.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg383"></margin.target>13.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg384"></margin.target>3.<emph type="italics"/>Arch.de <lb/>quad. pa­<lb/>rab. &<emph.end type="italics"/> 20. <lb/><emph type="italics"/>pilmi coni <lb/>corum A­<lb/>poil.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg385"></margin.target>2.<emph type="italics"/>cor.<emph.end type="italics"/> 20. <lb/><emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg386"></margin.target>22.<emph type="italics"/>&longs;exti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg387"></margin.target>37. <emph type="italics"/>vndeci <lb/>mi.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg388"></margin.target>17.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg389"></margin.target>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg390"></margin.target>11.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg391"></margin.target>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg392"></margin.target><emph type="italics"/>cor<emph.end type="italics"/> 4.<emph type="italics"/>quin <lb/>ti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg393"></margin.target>22.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg394"></margin.target>11.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg395"></margin.target>18.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg396"></margin.target><emph type="italics"/>cor.<emph.end type="italics"/>2.<emph type="italics"/>lem­<lb/>in<emph.end type="italics"/> 13. <emph type="italics"/>pri­<lb/>mi huius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg397"></margin.target><emph type="italics"/>cor.<emph.end type="italics"/>4.<emph type="italics"/>quin <lb/>ti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg398"></margin.target><emph type="italics"/>ex præce­<lb/>denti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg399"></margin.target>8.<emph type="italics"/>buius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg400"></margin.target>8.<emph type="italics"/>prim hu <lb/>ius.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg401"></margin.target>19.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="margin">
<s><margin.target id="marg402"></margin.target>8 <emph type="italics"/>prim.hu <lb/>ius.<emph.end type="italics"/></s></p>
<figure id="fig86"></figure>
<figure id="fig87"></figure>
<figure id="fig88"></figure>
<figure id="fig89"></figure>
<p type="head">
<s>SCHOLIVM.</s></p>
<p type="main">
<s>In hoc Theoremate primùm ob&longs;eruanda occurrunt verba <lb/>propo&longs;itionis, quibus Archimedes pr&ecedil;cipit pottionem HK <lb/>in I ita diui&longs;am e&longs;&longs;e oportere, vt HI ad IK eam habeat pro­<lb/>portionem, quam habet &longs;olidum ba&longs;im habens quadratum <lb/>ex dimidia maioris ba&longs;is fru&longs;ti, altitudinem autem lineam æ­<lb/>qualem vtrique &longs;imul duplæ minoris ba&longs;is, & maiori ad &longs;oli­<lb/>dum ba&longs;im habens quadratum ex dimidia minoris ba&longs;is fru­<lb/>&longs;ti, altitudinem autem lineam æqualem vtri&longs;que, duplæ &longs;cili­<lb/>cet ba&longs;is maioris, & minori. hoc e&longs;t &longs;it HI ad IK, vt &longs;olidum <lb/>ba&longs;im habens quadratum ex AF, altitudinem verò lineam æ­<lb/>qualem duplæ ip&longs;ius DE cum AC ad &longs;olidum ba&longs;im habens <lb/>quadratum ex DG, altitudinem verò lineam æqualem <expan abbr="vtriq;">vtrique</expan> <lb/>&longs;imul duplæ ip&longs;ius AC, & ip&longs;i DE. In con&longs;tructione autem <lb/>hunc propo&longs;itionis locum explicans, & in pergre&longs;&longs;u totius <expan abbr="de-mõ&longs;trationis">de­<lb/>mon&longs;trationis</expan>, inquit HI ad IK <expan abbr="eã">eam</expan> debere proportionem habe­<lb/>re, quam habet &longs;olidum ba&longs;im habens quadratum ex AF, alti <lb/>tudinem verò lineam æqualem <expan abbr="vtriq;">vtrique</expan> &longs;imul duplæ ip&longs;ius DG, <lb/>& ip&longs;i AF ad &longs;olidum ba&longs;im habens quadratum ex DG, al­<lb/>titudinem verò lineam æqualem vtrique &longs;imul duplæ ip&longs;ius <lb/>AF, & DG. Quoniam autem &longs;olida parallelepipeda (vt præ­<lb/>fata &longs;olida &longs;unt) in eadem ba&longs;i exi&longs;tentia ita &longs;e habentin ter&longs;e, <lb/>vt corum altitudine; &longs;olidum, quod ba&longs;im habet quadratum <lb/>ex AF, altitudinem autem duplam ip&longs;ius DE cum AC, du <lb/>plum erit &longs;olidi ba&longs;im habentis quadratum ex AF, altitudi­<lb/>nem verò duplam ip&longs;ius DG cum AF. Nam hæc &longs;olida can <lb/>dem habent ba&longs;im, quadratum nempè ex AF; ip&longs;orumquè <lb/>alterum habet altitudinem duplam. quia cùm &longs;it DE dupla <lb/>ip&longs;ius DG, erit dupla ip&longs;ius DE dupla ip&longs;ius duplæ DG;
<pb pagenum="202"/>& AC dupla e&longs;t ip&longs;ius AF. altitudines igitur horum <expan abbr="&longs;olidorũ">&longs;olidorum</expan> <lb/>in dupla &longs;unt proportione. hoc e&longs;t altitudo, linea &longs;cilicet du­<lb/>pla ip&longs;ius DE cum AC altitudinis nempè lineæ duplæ ip&longs;ius <lb/>DG cum AF dupla exi&longs;tit. Quare &longs;olidum ba&longs;im habens qua­<lb/>dratum ex AF, altitudinem verò duplam ip&longs;ius DE cum AC <lb/>duplum e&longs;t &longs;olidi, quod ba&longs;im habeatidem quadratum ex AF, <lb/>altitudinem verò duplam ip&longs;ius DG cum AF. cademquè ratio <lb/>neo&longs;tendetur <expan abbr="&longs;olidũ">&longs;olidum</expan> ba&longs;im habens quadratum ex DG, altitu <lb/>dinem verò duplam ip&longs;ius AC cum DE duplum e&longs;&longs;e &longs;olidi ba <lb/>&longs;im habentis quadratum ex eadem DG, altitudinem autem du <lb/>plam ip&longs;ius AF cum DG. &longs;olidum igitur ba&longs;im habens qua­<lb/>dratum ex AF, altitudinem autem duplam ip&longs;ius DE cum AC <lb/>ad &longs;olidum quadtatum habens ba&longs;im ex AF, altitudinent verò <lb/>duplam ip&longs;ius DG cum AF eam habet proportionem, quam <lb/>habet &longs;olidum ba&longs;im habens quadratum ex DG, altitudinem <lb/>verò duplam ip&longs;ius AC cum AE ad &longs;olidum ba&longs;im <expan abbr="hab&etilde;s">habens</expan> qua <lb/>dratum ex DG, altitudinem verò duplam ip&longs;ius AF cum DG. <lb/>
<arrow.to.target n="marg403"></arrow.to.target> quare permutando <expan abbr="primũ">primum</expan> &longs;olidum ba&longs;im habens quadratum <lb/>ex AF, altitudinem verò duplam ip&longs;ius DE cum AC ad &longs;ecun­<lb/>dum &longs;olidum ba&longs;im habens quadratum ex DG, altitudinem <lb/>autem duplam ip&longs;ius AC cum DE eandem habet proportio­<lb/>nem, quam habet tertium &longs;olidum ba&longs;im habens quadratum <lb/>ex AF, altitudinem autem duplam ip&longs;ius DG cum AF ad quar <lb/>tum &longs;olidum ba&longs;im habens quadratum ex DG, altitudinem ve <lb/>rò duplam ip&longs;ius AF cum DG. Quapropter Archimedes loco <lb/>primi, & &longs;ecundi &longs;olidi in propo&longs;itione propo&longs;iti rectè potuit <lb/>in demon&longs;tratione accipere tertium, & quartum &longs;olidum. co <lb/>dem enim modo, & in eadem proportione linea HK in pun­<lb/>cto I diui&longs;a prouenit: quod quidem punctum fru&longs;ti ACED <lb/>centrum grauitatis exi&longs;tit. </s></p>
<p type="margin">
<s><margin.target id="marg403"></margin.target>16.<emph type="italics"/>quinti.<emph.end type="italics"/></s></p>
<p type="head">
<s>Secundi libri Finis.</s></p>
<pb/>
<p type="head">
<s>Erratorum quorundam re&longs;titutio.</s></p>
<p type="main">
<s>Pagina 8, ver&longs;u 18, Archimedes. <33> 10, 7, &longs;ione. <33> 18, 20, conducenti. <33> 21, 14, per <lb/>di&longs;cere ip&longs;um. <33> 39, 25, hoc e&longs;t AB. <33> 43, 19, lineam. <33> 47, 20, cúm inquit, <33> 63, <lb/>20, GD DK in. <33> 65, 21, DC. Ibidem, 27, ex DC. <33> 67, 29, in maiori. <33> 69, in <lb/>po&longs;til: ex proxima propo&longs;itione. <33> 70, 5, vt NL <33> 73, 1, de his, vel. <33> 84, 8, AEEB <lb/>CF FD. <33> 90, 17, totus. <33> 98, 1, quam VH. Ibidem, 7, aufertur. <33> 11<gap/>, 21, repo­<lb/>&longs;uit. <33> 124, 19, <expan abbr="&longs;ection&etilde;">&longs;ectionem</expan>, <33> 140, 1, <expan abbr="æquidi&longs;tãtes">æquidi&longs;tantes</expan> <33> 143, 11, e&longs;t CH <33> 147, 3, <expan abbr="cũ">cum</expan> EK ad EK, vt. <lb/>Ibide, 25, &longs;ta S 9, ad Y<foreign lang="greek">a</foreign> <33> 149, 19, ad <foreign lang="greek">xn</foreign>. Ibidem, 25, e&longs;t, vt OR. Ibidem, 27, L<foreign lang="greek">*g</foreign>, vt <lb/>OR ad. Ibidem, 31, vt OR ad <foreign lang="greek">zd</foreign> Ibidem, 32, vt <foreign lang="greek">d<10></foreign> ad <foreign lang="greek"><10>z</foreign> Ibidem, 34, BD ad B<foreign lang="greek">s</foreign>, <lb/>ita. Ibidem, 35, &longs;it BD ad D<foreign lang="greek">n</foreign> Ibidem, 36, BD ad D<foreign lang="greek">n</foreign> B<foreign lang="greek">s</foreign>. <33> 150, 5, vt OR ad O<foreign lang="greek">c</foreign> <33> 153, <lb/>13, ræ, vt. <33> 157, in po&longs;till ante 15, primi Ibidem, 17, maiorem. <33> 161, 24, erit KH. <lb/><33> 167, 34, efficax. <33> 170, 1, ip&longs;ius AC erit. <33> 181, 36, ex dupla ip&longs;ius AB, <33> 191, <lb/>21, erunt. Ibidem, 22, DKG æquales. </s></p>
<p type="head">
<s>REGISTRVM.</s></p>
<p type="main">
<s><12> ABCDEFGHIKLMNOPQRSTVXYZ, <lb/>AA BB. </s></p>
<p type="head">
<s>Omnes duerniones, præter, BB, ternionem.</s></p>
<p type="head">
<s>PISAVRI. <lb/>Apud Hieronymum Concordiam, <lb/>M. D. LXXXVII.</s></p>
</chap>
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</archimedes>