Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/014.jpg" pagenum="343"/>
              the Superficies that proceed according to the Circumferences X O
                <lb/>
              and P O are equally preſſed; whereby the Gravity preſſed is equal.
                <lb/>
                <figure id="id.073.01.014.1.jpg" xlink:href="073/01/014/1.jpg" number="7"/>
                <lb/>
              But the Gravity of the
                <lb/>
              Liquid which is in the
                <lb/>
                <arrow.to.target n="marg1135"/>
                <lb/>
              firſt Pyramid ^{*} without
                <lb/>
              the Solid B H T G, is
                <lb/>
              equal to the Gravity of
                <lb/>
              the Liquid which is in
                <lb/>
              the other Pyramid with­
                <lb/>
              out the Liquid R S C Y:
                <lb/>
              It is manifeſt, therefore,
                <lb/>
              that the Gravity of the Solid E Z H T, is equal to the Gravity of
                <lb/>
              the Liquid R S C Y: Therefore it is manifeſt that a Maſs of Liquor
                <lb/>
              equal in Maſs to the part of the Solid ſubmerged is equal in Gra­
                <lb/>
              vity to the whole Solid.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1135"/>
              *
                <emph type="italics"/>
              Without, i.e.
                <emph.end type="italics"/>
              that
                <lb/>
              being deducted.</s>
            </p>
            <p type="main">
              <s>RIC. </s>
              <s>This was a pretty Demonſtration, and becauſe I very well underſtand
                <lb/>
              it, let us loſe no time, but proceed to the ſixth
                <emph type="italics"/>
              Propoſition,
                <emph.end type="italics"/>
              ſpeaking thus.</s>
            </p>
            <p type="head">
              <s>PROP. VI. THEOR. VI.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Solid Magnitudes lighter than the Liquid being thruſt
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              into the Liquid, are repulſed upwards with a Force
                <lb/>
              as great as is the exceſs of the Gravity of a Maſs
                <lb/>
              of Liquor equal to the Magnitude above the Gra­
                <lb/>
              vity of the ſaid Magnitude.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>NIC. </s>
              <s>This ſixth
                <emph type="italics"/>
              Propoſition
                <emph.end type="italics"/>
              ſaith, that the Solids lighter than the Liquid
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              demitted, thruſt, or trodden by Force underneath the Liquids Sur­
                <lb/>
              face, are returned or driven upwards with ſo much Force, by
                <lb/>
              how much a quantity of the Liquid equal to the. </s>
              <s>Solid ſhall
                <lb/>
              exceed the ſaid Solid in Gravity.</s>
            </p>
            <p type="main">
              <s>And to delucidate this
                <emph type="italics"/>
              Propoſition,
                <emph.end type="italics"/>
              let the Solid A be lighter
                <lb/>
              than the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid, and let us ſuppoſe that the Gravity of the ſaid
                <lb/>
              Solid A is B: and let the Gravity of a
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid, equal in Maſs to A,
                <lb/>
              be B G. </s>
              <s>I ſay, that the Solid A depreſſed or demitted with Force
                <lb/>
              into the ſaid
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid, ſhall be returned and repulſed upwards with
                <lb/>
              a Force equal to the Gravity G. </s>
              <s>And to demonſtrate this
                <emph type="italics"/>
              Propo­
                <lb/>
              ſition,
                <emph.end type="italics"/>
              take the Solid D, equal in Gravity to the ſaid G. </s>
              <s>Now
                <lb/>
              the Solid compounded of the two Solids A and D will be lighter
                <lb/>
              than the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid: for the Gravity of the Solid compounded of
                <lb/>
              them both is BG, and the Gravity of as much Liquor as equal­
                <lb/>
              leth in greatneſs the Solid A, is greater than the ſaid Gravity BG, </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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