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

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb pagenum="456"/>
              Cone weighing equally with the water, the part ſubmerged S B D T,
                <lb/>
              becomes indifferent to move downwards or upwards; and the Cone
                <lb/>
                <emph type="italics"/>
              A S T,
                <emph.end type="italics"/>
              being equall in Maſs to the water that would be contained in
                <lb/>
              the concave of the Rampart
                <emph type="italics"/>
              E S T O,
                <emph.end type="italics"/>
              ſhall be alſo equall unto it in
                <lb/>
              Gravity: and, therefore, there ſhall be a perfect
                <emph type="italics"/>
              Equilibrium,
                <emph.end type="italics"/>
              and,
                <lb/>
              conſequently, a Reſt. </s>
              <s>Now here ariſeth a doubt, whether the
                <lb/>
              Cone
                <emph type="italics"/>
              A B D
                <emph.end type="italics"/>
              may be made heavier, in ſuch ſort, that when it is put
                <lb/>
              wholly under water, it goes to the bottom, but yet not in ſuch ſort,
                <lb/>
              as to take from the Rampart the vertue of ſuſtaining it that it ſink not,
                <lb/>
              and, the reaſon of the doubt is this: that although at ſuch time as
                <lb/>
              the Cone
                <emph type="italics"/>
              A B D
                <emph.end type="italics"/>
              is ſpecifically as grave as the water, the Rampart
                <lb/>
                <emph type="italics"/>
              E S T O
                <emph.end type="italics"/>
              ſuſtaines it, not only when the point
                <emph type="italics"/>
              A S T
                <emph.end type="italics"/>
              is tripple in
                <lb/>
              height to the Altitude of the Rampart
                <emph type="italics"/>
              E S,
                <emph.end type="italics"/>
              but alſo when a leſſer
                <lb/>
              part is above water; [for although in the Deſcent of the Cone the
                <lb/>
              Point
                <emph type="italics"/>
              A S T
                <emph.end type="italics"/>
              by little and little diminiſheth, and ſo likewiſe the
                <lb/>
              Rampart
                <emph type="italics"/>
              E S T O,
                <emph.end type="italics"/>
              yet the Point diminiſheth in
                <lb/>
                <figure id="fig278" number="17"/>
                <lb/>
              greater proportion than the Rampart, in that
                <lb/>
              it diminiſheth according to all the three
                <lb/>
              menſions, but the Rampart according to two
                <lb/>
              only, the Altitude ſtill remaining the ſame;
                <lb/>
              or, if you will, becauſe the Cone
                <emph type="italics"/>
              S T
                <emph.end type="italics"/>
              goes
                <lb/>
              miniſhing, according to the proportion of the
                <lb/>
              cubes of the Lines that do ſucceſſively become
                <lb/>
              the Diameters of the Baſes of emergent Cones,
                <lb/>
              and the Ramparts diminiſh according to the proportion of the
                <lb/>
              Squares of the ſame Lines; whereupon the proportions of the Points
                <lb/>
              are alwayes Seſquialter of the proportions of the Cylinders,
                <lb/>
              tained within the Rampart; ſo that if, for Example, the height of
                <lb/>
              the emergent Point were double, or equall to the height of the
                <lb/>
              Rampart, in theſe caſes, the Cylinder contained within the
                <lb/>
              part, would be much greater than the ſaid Point, becauſe it would be
                <lb/>
              either ſeſquialter or tripple, by reaſon of which it would perhaps
                <lb/>
              ſerve over and above to fuſtain the whole Cone, ſince the part
                <lb/>
              merged would no longer weigh any thing;] yet, nevertheleſs, when
                <lb/>
              any Gravity is added to the whole Maſs of the Cone, ſo that alſo the
                <lb/>
              part ſubmerged is not without ſome exceſſe of Gravity above the
                <lb/>
              Gravity of the water, it is not manifeſt, whether the Cylinder
                <lb/>
              tained within the Rampart, in the deſcent that the Cone ſhall make,
                <lb/>
              can be reduced to ſuch a proportion unto the emergent Point, and to
                <lb/>
              ſuch an exceſſe of Maſs above the Maſs of it, as to compenſate the
                <lb/>
              exceſſe of the Cones Specificall Gravity above the Gravity of the
                <lb/>
              ter: and the Scruple ariſeth, becauſe that howbeit in the deſcent
                <lb/>
              made by the Cone, the emergent Point
                <emph type="italics"/>
              A S T
                <emph.end type="italics"/>
              diminiſheth, whereby
                <lb/>
              there is alſo a diminution of the exceſs of the Cones Gravity above </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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