Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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DE CENTRO GRAVIT. SOLID.
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              <pb o="37" file="0185" n="185" rhead="DE CENTRO GRAVIT. SOLID."/>
            ducta fuerìnt, ira ut in unum punctum y coeant, erunt triã
              <lb/>
            gala u y l, x y p, t y _k_ inter ſe ſimilia: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">ſimilia etiam triangu
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            la l y r, p y s, _k_ y q. </s>
            <s xml:space="preserve">quare ut in 19 huius, demonſtrabitur
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            x p, ad p s: </s>
            <s xml:space="preserve">itemq; </s>
            <s xml:space="preserve">t k ad _k_ q èandem habere proportionẽ,
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            quam u l ad l r. </s>
            <s xml:space="preserve">Sed ut u l ad l r, ita eſt triangulum a b c ad
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            triangulum a c d: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">ut t k ad K q, ita triangulum e f g ad
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            triangulum e g h. </s>
            <s xml:space="preserve">Vt autem triangulum a b c ad triangu-
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            lum a c d, ita pyramis a b c y ad pyramidem a c d y. </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">ut
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            triangulum e f g ad triangulum e g h, ita pyramis e f g y
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            ad pyramidem e g h y; </s>
            <s xml:space="preserve">ergo ut pyramis a b c y ad pyramidẽ
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            a c d y, ita pyramis e f g y ad pyramidem e g h y. </s>
            <s xml:space="preserve">reliquum
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              <anchor type="note" xlink:label="note-0185-01a" xlink:href="note-0185-01"/>
            igitur fruſtũ l f ad reliquum fruſtũ l h eſt ut pyramis a b c y
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            ad pyramidem a c d y, hoc eſt ut u l ad l r, & </s>
            <s xml:space="preserve">ut x p ad p s.
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            </s>
            <s xml:space="preserve">Quòd cum fruſti l f centrum grauitatis ſit s: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">fruſti l h ſit
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            centrum x: </s>
            <s xml:space="preserve">conſtat punctum p totius fruſti a g grauitatis
              <lb/>
              <anchor type="note" xlink:label="note-0185-02a" xlink:href="note-0185-02"/>
            eſſe centrum. </s>
            <s xml:space="preserve">Eodem modo fiet demonſtratio etiam in
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            aliis pyramidibus.</s>
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          </p>
          <div type="float" level="2" n="2">
            <figure xlink:label="fig-0184-01" xlink:href="fig-0184-01a">
              <image file="0184-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0184-01"/>
            </figure>
            <note position="left" xlink:label="note-0184-01" xlink:href="note-0184-01a" xml:space="preserve">2. ſexti.</note>
            <note position="right" xlink:label="note-0185-01" xlink:href="note-0185-01a" xml:space="preserve">19. quinti</note>
            <note position="right" xlink:label="note-0185-02" xlink:href="note-0185-02a" xml:space="preserve">8. Archi-
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            medis.</note>
          </div>
          <p>
            <s xml:space="preserve">Sit fruſtum a d à cono, uel coni portione abſciſſum, cu-
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            ius maior baſis circulus, uel ellipſis circa diametrum a b;
              <lb/>
            </s>
            <s xml:space="preserve">minor circa diametrum c d: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">axis e f. </s>
            <s xml:space="preserve">diuidatur autẽ e f
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            in g, ita ut e g ad g f eandem proportionem habeat, quam
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            duplum diametri a b unà cum diametro c d ad duplum c d
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            unà cum a b. </s>
            <s xml:space="preserve">Sitq; </s>
            <s xml:space="preserve">g h quarta pars lineæ g e: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">ſit ſ K item
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            quarta pars totius f e axis. </s>
            <s xml:space="preserve">Rurfus quam proportionem
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            habet fruſtum a d ad conum, uel coni portionem, in eadẽ
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            baſi, & </s>
            <s xml:space="preserve">æquali altitudine, habeat linea _k_ h ad h l. </s>
            <s xml:space="preserve">Dico pun-
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            ctum l fruſti a d grauitatis centrum eſſe. </s>
            <s xml:space="preserve">Si enim fieri po-
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            teſt, ſit m centrum: </s>
            <s xml:space="preserve">producaturq; </s>
            <s xml:space="preserve">l m extra fruſtum in n: </s>
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              <lb/>
            & </s>
            <s xml:space="preserve">ut n l ad l m, ita fiat circulus, uel ellipſis circa diametrũ
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            a b ad aliud ſpacium, in quo ſit o. </s>
            <s xml:space="preserve">Itaque in circulo, uel
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            ellipſi circa diametrum a b rectilinea figura plane deſcri-
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            batur, ita ut quæ relinquuntur portiones ſint o ſpacio mi-
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            nores: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">inteiligatur pyramis a p b, baſim habens rectili-
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            neam figuram in circulo, uel ellipſi a b deſcriptam: </s>
            <s xml:space="preserve">à qua</s>
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