Archimedes, Archimedis De iis qvae vehvntvr in aqva libri dvo

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FED. COMMANDINI
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              <pb file="0172" n="172" rhead="FED. COMMANDINI"/>
            Dico eas proportion ales eſſe in proportione, quæ eſt la-
              <lb/>
            teris a b adlatus d e, itaut earum maior ſit a b c e, me-
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            dia a d c e, & </s>
            <s xml:space="preserve">minor d e f c. </s>
            <s xml:space="preserve">Quoniam enim lineæ d e,
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            a b æquidiſtant; </s>
            <s xml:space="preserve">& </s>
            <s xml:space="preserve">interipſas ſunt triangula a b e, a d e;
              <lb/>
            </s>
            <s xml:space="preserve">erit triangulum a b e
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              <anchor type="figure" xlink:label="fig-0172-01a" xlink:href="fig-0172-01"/>
              <anchor type="note" xlink:label="note-0172-01a" xlink:href="note-0172-01"/>
            ad triangulum a d e,
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            ut linea a b ad lineam
              <lb/>
            d e. </s>
            <s xml:space="preserve">ut autem triangu
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            lum a b e ad triangu-
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            lum a d e, ita pyramis
              <lb/>
              <anchor type="note" xlink:label="note-0172-02a" xlink:href="note-0172-02"/>
            a b e c ad pyramidem
              <lb/>
            a d e c: </s>
            <s xml:space="preserve">habent enim
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            altitudinem eandem,
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            quæ eſt à puncto c ad
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            planum, in quo qua-
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            drilaterum a b e d. </s>
            <s xml:space="preserve">er-
              <lb/>
              <anchor type="note" xlink:label="note-0172-03a" xlink:href="note-0172-03"/>
            go ut a b ad d e, ita pyramis a b e c ad pyramidem a d e c.
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            </s>
            <s xml:space="preserve">Rurſus quoniam æquidiſtantes ſunt a c, d f; </s>
            <s xml:space="preserve">erit eadem
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            ratione pyramis a d c e ad pyramidem c d f e, ut a c ad
              <lb/>
              <anchor type="note" xlink:label="note-0172-04a" xlink:href="note-0172-04"/>
            d f. </s>
            <s xml:space="preserve">Sed ut a c a l d f, ita a b ad d e, quoniam triangula
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            a b c, d e f ſimilia ſunt, ex nona huius. </s>
            <s xml:space="preserve">quare ut pyramis
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            a b c e ad pyramidem a d c e, ita pyramis a d c e ad ipſam
              <lb/>
            d e f c. </s>
            <s xml:space="preserve">fruſtum igitur a b c d e f diuiditur in tres pyramides
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            proportionales in ea proportione, quæ eſt lateris a b ad d e
              <lb/>
            latus, & </s>
            <s xml:space="preserve">earum maior eſt c a b e, media a d c e, & </s>
            <s xml:space="preserve">minor
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            d e f c. </s>
            <s xml:space="preserve">quod demonſtrare oportebat.</s>
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            <figure xlink:label="fig-0172-01" xlink:href="fig-0172-01a">
              <image file="0172-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0172-01"/>
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            <note position="left" xlink:label="note-0172-01" xlink:href="note-0172-01a" xml:space="preserve">1. ſextí.</note>
            <note position="left" xlink:label="note-0172-02" xlink:href="note-0172-02a" xml:space="preserve">5. duodeci
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            mi.</note>
            <note position="left" xlink:label="note-0172-03" xlink:href="note-0172-03a" xml:space="preserve">11. quinti.</note>
            <note position="left" xlink:label="note-0172-04" xlink:href="note-0172-04a" xml:space="preserve">4 ſexti.</note>
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        </div>
        <div type="section" level="1" n="89">
          <head xml:space="preserve">PROBLEMA V. PROPOSITIO XXIIII.</head>
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              <emph style="sc">Qvodlibet</emph>
            fruſtum pyramidis, uel coni,
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            uel coni portionis, plano baſi æquidiſtanti ita ſe-
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            care, ut ſectio ſit proportionalis inter maiorem,
              <lb/>
            & </s>
            <s xml:space="preserve">minorem baſim.</s>
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