Galilei, Galileo, Discourse concerning the natation of bodies, 1663

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb pagenum="414"/>
            <p type="head">
              <s>THEOREME
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                <arrow.to.target n="marg1416"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1416"/>
              The proportion
                <lb/>
              of water
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              ſite to make a
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              Solid ſwim.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              If a Solid, Cylinder, or Priſme, leſſe grave ſpecifically
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              than the Water, being put into a Veſſel, as above, of
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              whatſoever greatneſſe, and the Water, be afterwards
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              infuſed, the Solid ſhall reſt in the bottom, unraiſed, till
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              the Water arrive to that part of the Altitude, of the
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              ſaid Priſme, to which its whole Altitude hath the
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              ſame proportion, that the Specificall Gravity of the
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              Water, hath to the Specificall Gravity of the ſaid
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              Solid: but infuſing more Water, the Solid ſhall aſcend.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Let the Veſſell be M L G N of any bigneſs, and let there be
                <lb/>
              ced in it the Solid Priſme D F G E, leſs grave
                <emph type="italics"/>
              in ſpecie
                <emph.end type="italics"/>
              than the
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              water; and look what proportion the
                <emph type="italics"/>
              S
                <emph.end type="italics"/>
              pecificall Gravity of
                <lb/>
              the water, hath to that of the Priſme, ſuch let the Altitude D F, have
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              to the Altitude F B. </s>
              <s>I ſay, that infuſing water to the Altitude F B,
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              the Solid D G ſhall not float, but ſhall ſtand in
                <emph type="italics"/>
              Equilibrium,
                <emph.end type="italics"/>
              ſo, that
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              that every little quantity of water, that is infuſed, ſhall raiſe it. </s>
              <s>Let
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              the water, therefore, be infuſed to the Levell A B C, and, becauſe
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              the Specifick Gravity of the Solid D G, is to the Specifick Gravity of
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              the water, as the altitude B F is to the altitude F D; that is, as the Maſs
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              B G to the Maſs G D; as the proportion of the Maſs B G is to the
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              Maſs G D, as the proportion of the Maſs G D is to the Maſs A F, they
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              compoſe the Proportion of the Maſs B G to the Maſs A F. Therefore,
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              the Maſs B G is to the Maſs A F, in a proportion compounded of the
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              proportions of the Specifick Gravity of the Solid G D, to the
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              fick Gravity of the water, and of the Maſs G D
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              to the Maſs A F: But the ſame proportions
                <lb/>
                <figure id="fig266" number="5"/>
                <lb/>
              of the Specifick Gravity of G D, to the Specifick
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              Gravity of the water, and of the Maſs G D to
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              the Maſs A F, do alſo by the precedent
                <emph type="italics"/>
              Lemma,
                <emph.end type="italics"/>
                <lb/>
              compound the proportion of the abſolute
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              vity of the Solid D G, to the abſolute Gravity
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              of the Maſs of the water A F: Therefore,
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              as the Maſs B G is to the Maſs A F, ſo is the
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              Abſolute Gravity of the Solid D G, to the
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              ſolute Gravity of the Maſs of the water A F. </s>
              <s>But as the Maſs B G
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              is to the Maſs A F; ſo is the Baſe of the Priſme D E, to the Surface
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              of the water AB; and ſo is the deſcent of the water A B, to the
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              Elevation of the Priſme D G; Therefore, the deſcent of the </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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