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

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb pagenum="453"/>
              Point downwards, it ſhall not deſcend to the
                <lb/>
              tom: for the Aeriall Cylinder contained betwixt
                <lb/>
                <figure id="fig275" number="14"/>
                <lb/>
              the Ramparts
                <emph type="italics"/>
              D A C E,
                <emph.end type="italics"/>
              is equall in Maſs to the
                <lb/>
              Cone
                <emph type="italics"/>
              A B C
                <emph.end type="italics"/>
              ; ſo that the whole Maſs of the Solid
                <lb/>
              compounded of the Air
                <emph type="italics"/>
              D A C E,
                <emph.end type="italics"/>
              and of the Cone
                <lb/>
                <emph type="italics"/>
              A B C,
                <emph.end type="italics"/>
              ſhall be double to the Cone
                <emph type="italics"/>
              A C B:
                <emph.end type="italics"/>
              And,
                <lb/>
              becauſe the Cone
                <emph type="italics"/>
              A B C
                <emph.end type="italics"/>
              is ſuppoſed to be of Matter double in
                <lb/>
              vity to the water, therefore as much water as the whole Maſſe
                <lb/>
                <emph type="italics"/>
              D A B C E,
                <emph.end type="italics"/>
              placed beneath the Levell of the water, weighs as much
                <lb/>
              as the Cone
                <emph type="italics"/>
              A B C
                <emph.end type="italics"/>
              : and, therefore, there ſhall be an
                <emph type="italics"/>
              Equilibrium,
                <emph.end type="italics"/>
                <lb/>
              and the Cone
                <emph type="italics"/>
              A B C
                <emph.end type="italics"/>
              ſhall deſcend no lower. </s>
              <s>Now, I ſay farther,
                <lb/>
              that the ſame Cone placed with the Baſe downwards, ſhall ſink to
                <lb/>
              the bottom, without any poſſibility of returning again, by any means
                <lb/>
              to ſwimme.</s>
            </p>
            <p type="main">
              <s>Let, therefore, the Cone be
                <emph type="italics"/>
              A B D,
                <emph.end type="italics"/>
              double in Gravity to the
                <lb/>
              water, and let its height be tripple the height
                <lb/>
                <figure id="fig276" number="15"/>
                <lb/>
              of the Rampart of water L B: It is already
                <lb/>
              manifeſt, that it ſhall not ſtay wholly out of
                <lb/>
              the water, becauſe the Cylinder being
                <lb/>
              prehended betwixt the Ramparts
                <emph type="italics"/>
              L B D P,
                <emph.end type="italics"/>
                <lb/>
              equall to the Cone
                <emph type="italics"/>
              A B D,
                <emph.end type="italics"/>
              and the Matter of
                <lb/>
              the Cone, beig double in Gravity to the
                <lb/>
              water, it is evident that the weight of the ſaid
                <lb/>
              Cone ſhall be double to the weight of the Maſs of water equall to the
                <lb/>
              Cylinder
                <emph type="italics"/>
              L B D P
                <emph.end type="italics"/>
              : Therefore it ſhall not reſt in this ſtate, but
                <lb/>
              ſhall deſcend.</s>
            </p>
            <p type="head">
              <s>COROLARY I.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              I ſay farther; that much leſſe ſhall the ſaid Cone stay afloat, if one
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1520"/>
                <lb/>
                <emph type="italics"/>
              immerge a part thereof.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1520"/>
              Much leſs ſhall
                <lb/>
              the ſaid Cone
                <lb/>
              ſwim, if one
                <lb/>
              merge a part
                <lb/>
              thereof.</s>
            </p>
            <p type="main">
              <s>Which you may ſee, comparing with the water as well the part
                <lb/>
              that ſhall immerge as the other above water. </s>
              <s>Let us therefore
                <lb/>
              of the Cone A B D, ſubmergeth part N T O S, and advance the
                <lb/>
              Point N S F above water. </s>
              <s>The Altitude of the Cone F N S, ſhall
                <lb/>
              either be more than half the whole Altitude of the Cone F T O, or
                <lb/>
              it ſhall not be more: if it ſhall be more than half, the Cone F N S
                <lb/>
              ſhall be more than half of the Cylinder E N S C: for the Altitude
                <lb/>
              of the Cone F N S, ſhall be more than Seſquialter of the Altitude
                <lb/>
              of the Cylinder E N S C: And, becauſe the Matter of the Cone is
                <lb/>
              ſuppoſed to be double in Specificall Gravity to the water, the water
                <lb/>
              which would be contained within the Rampart E N S C, would be
                <lb/>
              leſs grave abſolutely than the Cone F N S; ſo that the whole Cone
                <lb/>
              F N S cannot be ſuſtained by the Rampart: But the part immerged
                <lb/>
              N T O S, by being double in Specificall Gravity to the water, ſhall </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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