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

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb pagenum="446"/>
              Priſme, or Cylinder, to wit, that hath its two flat Superficies,
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              our and inferiour, alike and equall, and at Right Angles with the
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              ther laterall Superficies, and let its thickneſs I O be equall to the
                <lb/>
              greateſt Altitude of the Banks of water: I ſay, that if it be put upon
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              the water, it will not ſubmerge: for the Altitude
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              A I being equall to the Altitude I O, the
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              of the Air A B C I ſhall be equall to the Maſs
                <lb/>
                <figure id="fig271" number="10"/>
                <lb/>
              the Solid C I O S: and the whole Maſs A O S
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              double to the Maſs I S; And ſince the
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              of the Air A C, neither encreaſeth nor
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              niſheth the Gravity of the Maſs I S, and the Solid I S was
                <lb/>
              double in Gravity to the water; Therefore as much water as
                <lb/>
              Maſs ſubmerged A O S B, compounded of the Air A I C B, and of
                <lb/>
              the Solid I O S C, weighs juſt as much as the ſame ſubmerged Maſs
                <lb/>
              A O S B: but when ſuch a Maſs of water, as is the ſubmerged part
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              the Solid, weighs as much as the ſaid Solid, it deſcends not farther,
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                <arrow.to.target n="marg1508"/>
                <lb/>
              but reſteth, as by
                <emph type="italics"/>
              (a) Archimedes,
                <emph.end type="italics"/>
              and above by us, hath been de­>
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              monſtrated: Therefore, I S ſhall deſcend no farther, but ſhall reſt.
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              And if the Solid I S ſhall be Seſquialter in Gravity to the water, it
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              ſhall float, as long as its thickneſs be not above twice as much as the
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              greateſt Altitude of the Ramparts of water, that is, of A I. </s>
              <s>For I S
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              being Seſquialter in Gravity to the water, and the Altitude O I
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              being double to I A, the Solid ſubmerged A O S B, ſhall be alſo
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              Seſquialter in Maſs to the Solid I S. </s>
              <s>And becauſe the Air A C,
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              neither increaſeth nor diminiſheth the ponderoſity of the Solid I S:
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              Therefore, as much water in quantity as the ſubmerged Maſs AOSB,
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              weighs as much as the ſaid Maſs ſubmerged: And, therefore, that
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              Maſs ſhall reſt. </s>
              <s>And briefly in generall.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1508"/>
              Of Natation
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              Lib. 1. Prop. </s>
              <s>3.</s>
            </p>
            <p type="head">
              <s>THEOREME. VI.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              When ever the exceſs of the Gravity of the Solid above
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1509"/>
                <lb/>
                <emph type="italics"/>
              the Gravity of the Water, ſhall have the ſame
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              portion to the Gravity of the Water, that the
                <lb/>
              tude of the Rampart, hath to the thickneſs of the
                <lb/>
              Solid, that Solid ſhall not ſink, but being never ſo
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              tle thicker it ſhall.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1509"/>
              The
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              on of the
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              eſt thickneſs of
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              Solids, beyond
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              which
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              ſed they ſink.</s>
            </p>
            <p type="main">
              <s>Let the Solid I S be ſuperior in Gravity to the water, and of ſuch
                <lb/>
              thickneſs, that the Altitude of the Rampart A I, be in
                <lb/>
              on to the thickneſs of the Solid I O, as the exceſs of the
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              ty of the ſaid Solid I S, above the Gravity of a Maſs of water equall
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              to the Maſs I S, is to the Gravity of the Maſs of water equall to the </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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