Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/013.jpg" pagenum="342"/>
              that of the firſt Pyramid is preſſed by the Solid R, and by the Liquid
                <lb/>
              which that containeth, that is, that which is in the place of the Py­
                <lb/>
              ramid according to A B O X: but that part which, in the other Py­
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              ramid, is preſſed by the Solid H, ſuppoſed to be of the ſame Li­
                <lb/>
              quid, and by the Liquid which that containeth, that is, that which
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              is in the place of the ſaid Pyramid according to P O B G: and the
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              Gravity of the Solid R is leſs than the Gravity of the Liquid
                <lb/>
              H, for that theſe two Magnitudes were ſuppoſed to be equal in
                <lb/>
              Maſs, and the Solid R was ſuppoſed to be lighter than the Liquid:
                <lb/>
              and the Maſſes of the two Pyramids of Liquor that containeth theſe
                <lb/>
                <arrow.to.target n="marg1134"/>
                <lb/>
              two Solids R and H are equal ^{*} by what was preſuppoſed: There­
                <lb/>
              fore the part of the Liquid that is under the Superficies that pro­
                <lb/>
              ceeds according to the Circumference O P is more preſſed; and,
                <lb/>
              therefore, by the
                <emph type="italics"/>
              Suppoſition,
                <emph.end type="italics"/>
              it ſhall repulſe that part which is leſs
                <lb/>
              preſſed, whereby the ſaid Liquid will not be ſetled: But it was be­
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              fore ſuppoſed that it was ſetled: Therefore that Solid R ſhall not
                <lb/>
              totally ſubmerge, but ſome part thereof will remain without the
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              Liquid, that is, above its Surface, Which was the
                <emph type="italics"/>
              Propoſition.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1133"/>
              * That is a Maſs of
                <lb/>
              the Liquid.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1134"/>
              * For that the Py­
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              ramids were ſuppo­
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              ſed equal.</s>
            </p>
            <p type="main">
              <s>RIC. </s>
              <s>I have very well underſtood you, therefore let us come to the fifth
                <emph type="italics"/>
              Pro­
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              poſition,
                <emph.end type="italics"/>
              which, as you know, doth thus ſpeak.</s>
            </p>
            <p type="head">
              <s>PROP. V. THEOR. V.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Solid Magnitudes that are lighter than the Liquid,
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              being demitted in the (ſetled) Liquid, will ſo far
                <lb/>
              ſubmerge, till that a Maſs of Liquor, equal to the
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              Part ſubmerged, doth in Gravity equalize the
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              whole Magnitude.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>NIC. </s>
              <s>It having, in the precedent, been demonſtrared that Solids lighter than
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              the Liquid, being demitted in the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid, alwaies a part of them remains
                <lb/>
              without the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid, that is above its Surface; In this fifth
                <emph type="italics"/>
              Propoſition
                <emph.end type="italics"/>
              it is
                <lb/>
              aſſerted, that ſo much of ſuch a Solid ſhall ſubmerge, as that a Maſs of the
                <lb/>
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid equal to the part ſubmerged, ſhall have equal Gravity with the whole
                <lb/>
              Solid.</s>
            </p>
            <p type="main">
              <s>And to demonſtrate this, let us aſſume all the ſame Schemes
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              as before, in
                <emph type="italics"/>
              Propoſition
                <emph.end type="italics"/>
              3. and likewiſe let the Liquid be ſet­
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              led, and let the Solid E Z H T be lighter than the Liquid.
                <lb/>
              </s>
              <s>Now if the ſaid Liquid be ſetled, the parts of it that are equija­
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              cent are equally preſſed: Therefore the Liquid that is beneath </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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