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

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb pagenum="451"/>
              Therefore, the abſolute weight of B, to the abſolute weight of C, is
                <lb/>
              as the Maſs A C to the Maſs
                <emph type="italics"/>
              C
                <emph.end type="italics"/>
              : But as the Maſs AC, is to the Maſs C,
                <lb/>
              ſo is the abſolute weight of A C, to the abſolute weight of C:
                <lb/>
              fore the abſolute weight of B, hath the ſame proportion to the
                <lb/>
              lute weight of C, that the abſolute weight of A C, hath to the
                <lb/>
              ſolute weight of C: Therefore, the two Solids A C and B are equall
                <lb/>
              in abſolute Gravity: which was to be demonſtrated. </s>
              <s>Having
                <lb/>
              monſtrated this, I ſay,</s>
            </p>
            <p type="head">
              <s>THEOREME X.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              That it is poſſible of any aſſigned Matter, to form a Pi-
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1516"/>
                <lb/>
                <emph type="italics"/>
              ramide or Cone upon any Baſe, which being put upon
                <lb/>
              the Water ſhall not ſubmerge, nor wet any more than
                <lb/>
              its Baſe.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1516"/>
              There may be
                <lb/>
              Cones and
                <lb/>
              mides of any
                <lb/>
                <emph type="italics"/>
              M
                <emph.end type="italics"/>
              atter, which
                <lb/>
              demittedinto the
                <lb/>
              water, reſt only
                <lb/>
              their Baſes.</s>
            </p>
            <p type="main">
              <s>Let the greateſt poſſible Altitude of the Rampart be the Line D B,
                <lb/>
              and the Diameter of the Baſe of the Cone to be made of any
                <lb/>
              ter aſſigned B C, at right angles to D B: And as the Specificall Gravity
                <lb/>
              of the Matter of the Piramide or Cone to be made, is to the Specificall
                <lb/>
              Gravity of the water, ſo let the Altitude of the
                <lb/>
                <figure id="fig274" number="13"/>
                <lb/>
              Rampart D B, be to the third part of the Piramide
                <lb/>
              or Cone A B C, deſcribed upon the Baſe, whoſe
                <lb/>
              Diameter is B C: I ſay, that the ſaid Cone A B C,
                <lb/>
              and any other Cone, lower then the ſame, ſhall reſt
                <lb/>
              upon the Surface of the water B C without ſinking.
                <lb/>
              </s>
              <s>Draw D F parallel to B C, and ſuppoſe the Priſme
                <lb/>
              or Cylinder E C, which ſhall be tripple to the Cone
                <lb/>
              A B C. And, becauſe the Cylinder D C hath the ſame proportion
                <lb/>
              to the Cylinder C E, that the Altitude D B, hath to the Altitude B E:
                <lb/>
              But the Cylinder C E, is to the Cone A B C, as the Altitude E B is to
                <lb/>
              the third part of the Altitude of the Cone: Therefore, by Equality of
                <lb/>
              proportion, the Cylinder D C is to the Cone A B C, as D B is to the
                <lb/>
              third part of the Altitude B E: But as D B is to the third part of B E,
                <lb/>
              ſo is the Specificall Gravity of the Cone A B C, to the Specificall
                <lb/>
              vity of the water: Therefore, as the Maſs of the Solid D C, is to the
                <lb/>
              Maſs of the Cone A
                <emph type="italics"/>
              B
                <emph.end type="italics"/>
              C, ſo is the Specificall Gravity of the ſaid Cone,
                <lb/>
              to the Specificall Gravity of the water: Therefore, by the precedent
                <lb/>
              Lemma, the Cone A B C weighs in abſolute Gravity as much as a
                <lb/>
              Maſs of Water equall to the Maſs D C: But the water which by the
                <lb/>
              impoſition of the Cone A B C, is driven out of its place, is as much
                <lb/>
              as would preciſely lie in the place D C, and is equall in weight to the
                <lb/>
              Cone that diſplaceth it: Therefore, there ſhall be an
                <emph type="italics"/>
              Equilibrium,
                <emph.end type="italics"/>
                <lb/>
              and the Cone ſhall reſt without farther ſubmerging. </s>
              <s>And its
                <lb/>
              nifeſt,</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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