Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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DE CENTRO GRAVIT. SOLID.
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              <pb o="36" file="0183" n="183" rhead="DE CENTRO GRAVIT. SOLID."/>
            grauitatis magnitudinis, quæ ex utriſque pyramidibus cõ
              <lb/>
            ſtat; </s>
            <s xml:space="preserve">hoc eſt ipſius fruſti. </s>
            <s xml:space="preserve">Sed fruſti centrum eſt etiam in a-
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            xe g h. </s>
            <s xml:space="preserve">ergo in puncto φ, in quo lineæ z u, g h conueniunt.
              <lb/>
            </s>
            <s xml:space="preserve">Itaque u φ ad φ z eam proportionem habet, quam pyramis
              <lb/>
              <anchor type="note" xlink:label="note-0183-01a" xlink:href="note-0183-01"/>
            b c f e d ad pyramidem a b c d. </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">componendo u z ad z φ
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            eam habet, quam fruſtum ad pyramidem a b c d. </s>
            <s xml:space="preserve">Vtuero
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            u z ad z φ, ita o p ad p φ ob ſimilitudinem triangulorum,
              <lb/>
            u o φ, z p φ. </s>
            <s xml:space="preserve">quare o p ad p φ eſt ut fruſtum ad pyramidem
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            a b c d. </s>
            <s xml:space="preserve">ſed ita erat o p ad p q. </s>
            <s xml:space="preserve">æquales igitur ſunt p φ, p q: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">
              <lb/>
              <anchor type="note" xlink:label="note-0183-02a" xlink:href="note-0183-02"/>
            q φ unum atque idem punctum. </s>
            <s xml:space="preserve">ex quibus ſequitur lineam
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            z u ſecare o p in q: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">propterea pũctum q ipſius fruſti gra-
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            uitatis centrum eſſe.</s>
            <s xml:space="preserve"/>
          </p>
          <div type="float" level="2" n="1">
            <figure xlink:label="fig-0181-01" xlink:href="fig-0181-01a">
              <image file="0181-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0181-01"/>
            </figure>
            <note position="right" xlink:label="note-0181-01" xlink:href="note-0181-01a" xml:space="preserve">3. diffi. hu
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            ius.</note>
            <note position="right" xlink:label="note-0181-02" xlink:href="note-0181-02a" xml:space="preserve">Vltima e-
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            auſdẽ libri
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            Archime-
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            dis.</note>
            <figure xlink:label="fig-0182-01" xlink:href="fig-0182-01a">
              <image file="0182-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0182-01"/>
            </figure>
            <note position="left" xlink:label="note-0182-01" xlink:href="note-0182-01a" xml:space="preserve">2. ſexti.</note>
            <note position="right" xlink:label="note-0183-01" xlink:href="note-0183-01a" xml:space="preserve">8. prim I
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            libri Ar-
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            chimedis
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            de cẽtro
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            grauita-
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            tis plano
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            runi</note>
            <note position="right" xlink:label="note-0183-02" xlink:href="note-0183-02a" xml:space="preserve">7. quinti.</note>
          </div>
          <p>
            <s xml:space="preserve">Sit fruſtum a g à pyramide, quæ quadrangularem baſim
              <lb/>
            habeat abſciſſum, cuius maior baſis a b c d, minor e f g h,
              <lb/>
            & </s>
            <s xml:space="preserve">axis k l. </s>
            <s xml:space="preserve">diuidatur autem primũ _k_ l, ita ut quam propor-
              <lb/>
            tionem habet duplum lateris a b unà cum latere e f ad du
              <lb/>
            plum lateris e f unà cum a b; </s>
            <s xml:space="preserve">habeat k m ad m l. </s>
            <s xml:space="preserve">deinde à
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            púcto m ad k ſumatur quarta pars ipſius m k, quæ ſit m n.
              <lb/>
            </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">rurſus ab l ſumatur quarta pars totius axis l k, quæ ſit
              <lb/>
            l o. </s>
            <s xml:space="preserve">poſtremo fiat o n ad n p, ut fruſtum a g ad pyramidẽ,
              <lb/>
            cuius baſis ſit eadem, quæ fruſti, & </s>
            <s xml:space="preserve">altitudo æqualis. </s>
            <s xml:space="preserve">Dico
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            punctum p fruſti a g grauitatis centrum eſſe. </s>
            <s xml:space="preserve">ducantur
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            enim a c, e g: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">intelligantur duo fruſta triangulares ba-
              <lb/>
            ſes habentia, quorum alterum l f ex baſibus a b c, e f g cõ-
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            ſtet; </s>
            <s xml:space="preserve">alterum l h ex baſibus a c d, e g h. </s>
            <s xml:space="preserve">Sitq; </s>
            <s xml:space="preserve">fruſti l f axis
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            q r; </s>
            <s xml:space="preserve">in quo grauitatis centrum s: </s>
            <s xml:space="preserve">fruſti uero l h axis t u, & </s>
            <s xml:space="preserve">
              <lb/>
            x grauitatis centrum: </s>
            <s xml:space="preserve">deinde iungantur u r, t q, x s. </s>
            <s xml:space="preserve">tranſi-
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            bit u r per l: </s>
            <s xml:space="preserve">quoniam l eſt centrum grauitatis quadran-
              <lb/>
            guli a b c d: </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">puncta r u grauitatis centra triangulorum
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            a b c, a c d; </s>
            <s xml:space="preserve">in quæ quadrangulum ipſum diuiditur. </s>
            <s xml:space="preserve">eadem
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            quoque ratione t q per punctum _k_ tranſibit. </s>
            <s xml:space="preserve">At uero pro
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            portiones, ex quibus fruſtorum grauitatis centra inquiri-
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            mus, eædem ſunt in toto ſruſto a g, & </s>
            <s xml:space="preserve">in fruſtis l f, l h. </s>
            <s xml:space="preserve">Sunt
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            enim per octauam huius quadrilatera a b c d, e f g h ſimilia:</s>
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