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

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb pagenum="412"/>
            <p type="head">
              <s>THEOREME III.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              A Priſme or regular Cylinder, of a ſubſtance ſpecifically
                <lb/>
              leſs grave than Water, if it ſhould be totally ſubmerged
                <lb/>
              in Water, ſtayes not underneath, but riſeth, though the
                <lb/>
              Water circumfuſed be very little, and in abſolute
                <lb/>
              Gravity, never ſo much inferiour to the Gravity of the
                <lb/>
              ſaid Priſme.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Let then the Priſme A E F B, be put into the Veſſell C D F B, the
                <lb/>
              ſame being leſs grave
                <emph type="italics"/>
              in ſpecie
                <emph.end type="italics"/>
              than the Water: and let the
                <lb/>
              Water infuſed riſe to the height of the Priſme: I ſay, that the
                <lb/>
              Priſme left at liberty, it ſhall riſe, being born up
                <lb/>
              by the Water circumfuſed C D E A. </s>
              <s>For the
                <lb/>
                <figure id="fig264" number="3"/>
                <lb/>
              Water C E being ſpecifically more grave than
                <lb/>
              the Solid A F, the abſolute weight of the water
                <lb/>
              C E, ſhall have greater proportion to the
                <lb/>
              lute weight of the Priſme A F, than the Maſs
                <lb/>
              C E hath to the Maſs A F (in regard the Maſs
                <lb/>
              hath the ſame proportion to the Maſs, that the
                <lb/>
              weight abſolute hath to the weight abſolute,
                <lb/>
              in caſe the Maſſes are of the ſame Gravity
                <emph type="italics"/>
              in ſpecie.
                <emph.end type="italics"/>
              ) But
                <lb/>
              the Maſs C E is to the Maſs A F, as the Surface of the water A C, is
                <lb/>
              to the Superficies, or Baſe of the Priſme A B; which is the ſame
                <lb/>
              portion as the aſcent of the Priſme when it riſeth, hath to the deſcent
                <lb/>
              of the water circumfuſed C E.</s>
            </p>
            <p type="main">
              <s>Therefore, the abſolute Gravity of the water C E, hath greater
                <lb/>
              proportion to the abſolute Gravity of the Priſme A F; than the
                <lb/>
              Aſcent of the Priſme A F, hath to the deſcent of the ſaid
                <lb/>
              water C E. </s>
              <s>The Moment, therefore, compounded of the abſolute
                <lb/>
              Gravity of the water C E, and of the Velocity of its deſcent, whilſt
                <lb/>
              it forceably repulſeth and raiſeth the Solid A F, is greater than the
                <lb/>
              Moment compounded of the abſolute Gravity of the Priſme A F, and
                <lb/>
              of the Tardity of its aſcent, with which Moment it contraſts and
                <lb/>
              fiſts the repulſe and violence done it by the Moment of the water:
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              Therefore, the Priſme ſhall be
                <lb/>
                <arrow.to.target n="marg1414"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1414"/>
              The Proportion
                <lb/>
              according to
                <lb/>
              which the
                <lb/>
              merſion & Na
                <lb/>
              tation of Solids
                <lb/>
              is made.</s>
            </p>
            <p type="main">
              <s>It followes, now, that we proceed forward to demonſtrate more
                <lb/>
              particularly, how much ſuch Solids ſhall be inferiour in Gravity to
                <lb/>
              the water elevated; namely, what part of them ſhall reſt ſubmerged,
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              and what ſhall be viſible above the Surface of the water: but firſt
                <lb/>
              it is neceſſary to demonſtrate the ſubſequent Lemma.</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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