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

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb pagenum="459"/>
              the Gravity of the water, yet the caſe ſtands ſo, that the Rampart
                <lb/>
              doth alſo contract it ſelf, and the Cylinder contained in it doth
                <lb/>
              miniſh. </s>
              <s>Nevertheleſs it ſhall be demonſtrated, how that the Cone
                <lb/>
                <emph type="italics"/>
              A B D
                <emph.end type="italics"/>
              being of any ſuppoſed bigneſſe, and made at the firſt of a
                <lb/>
              Matter exactly equall in Gravity to the Water, if there may
                <lb/>
              be affixed to it ſome Weight, by means of which it may deſcend to
                <lb/>
              the bottom, when ſubmerged under water, it may alſo by vertue of
                <lb/>
              the Rampart ſtay above without ſinking.</s>
            </p>
            <p type="main">
              <s>Let, therefore, the Cone
                <emph type="italics"/>
              A B D
                <emph.end type="italics"/>
              be of any ſuppoſed greatneſſe,
                <lb/>
              and alike in ſpecificall Gravity to the water. </s>
              <s>It is manifeſt, that
                <lb/>
              being put lightly into the water, it ſhall reſt without deſcending;
                <lb/>
              and it ſhall advance above water, the Point
                <lb/>
                <figure id="fig279" number="18"/>
                <lb/>
                <emph type="italics"/>
              AS T,
                <emph.end type="italics"/>
              tripple in height to the height of the
                <lb/>
              Rampart
                <emph type="italics"/>
              E S
                <emph.end type="italics"/>
              : Now, ſuppoſe the Cone
                <emph type="italics"/>
              A B D
                <emph.end type="italics"/>
                <lb/>
              more depreſſed, ſo that it advance above
                <lb/>
              ter, only the Point
                <emph type="italics"/>
              A I R,
                <emph.end type="italics"/>
              higher by half than
                <lb/>
              the Point
                <emph type="italics"/>
              A S T,
                <emph.end type="italics"/>
              with the Rampart about it
                <lb/>
                <emph type="italics"/>
              C I R N.
                <emph.end type="italics"/>
              And, becauſe, the Cone
                <emph type="italics"/>
              A B D
                <emph.end type="italics"/>
              is
                <lb/>
              to the Cone
                <emph type="italics"/>
              A I R,
                <emph.end type="italics"/>
              as the cube of the Line
                <emph type="italics"/>
              S T
                <emph.end type="italics"/>
                <lb/>
              is to the cube of the Line
                <emph type="italics"/>
              I R,
                <emph.end type="italics"/>
              but the
                <lb/>
              der
                <emph type="italics"/>
              E S T O,
                <emph.end type="italics"/>
              is to the Cylinder
                <emph type="italics"/>
              C I R N,
                <emph.end type="italics"/>
              as the Square of
                <emph type="italics"/>
              S T
                <emph.end type="italics"/>
              to
                <lb/>
              the Square of
                <emph type="italics"/>
              I R,
                <emph.end type="italics"/>
              the Cone
                <emph type="italics"/>
              A S T
                <emph.end type="italics"/>
              ſhall be Octuple to the Cone
                <lb/>
                <emph type="italics"/>
              A I R,
                <emph.end type="italics"/>
              and the Cylinder
                <emph type="italics"/>
              E S T O,
                <emph.end type="italics"/>
              quadruple to the Cylinder
                <emph type="italics"/>
              C I R N
                <emph.end type="italics"/>
              :
                <lb/>
              But the Cone
                <emph type="italics"/>
              A S T,
                <emph.end type="italics"/>
              is equall to the Cylinder E
                <emph type="italics"/>
              S T O
                <emph.end type="italics"/>
              : Therefore,
                <lb/>
              the Cylinder
                <emph type="italics"/>
              C I R N,
                <emph.end type="italics"/>
              ſhall be double to the Cone
                <emph type="italics"/>
              A I R:
                <emph.end type="italics"/>
              and the
                <lb/>
              water which might be contained in the Rampart
                <emph type="italics"/>
              C I R N,
                <emph.end type="italics"/>
              would be
                <lb/>
              double in Maſs and in Weight to the Cone
                <emph type="italics"/>
              A I R,
                <emph.end type="italics"/>
              and, therefore,
                <lb/>
              would be able to ſuſtain the double of the Weight of the Cone
                <emph type="italics"/>
              AIR
                <emph.end type="italics"/>
              :
                <lb/>
              Therefore, if to the whole Cone
                <emph type="italics"/>
              A B D,
                <emph.end type="italics"/>
              there be added as much
                <lb/>
              Weight as the Gravity of the Cone
                <emph type="italics"/>
              A I R,
                <emph.end type="italics"/>
              that is to ſay, the eighth
                <lb/>
              part of the weight of the Cone
                <emph type="italics"/>
              A S T,
                <emph.end type="italics"/>
              it alſo ſhall be ſuſtained by
                <lb/>
              the Rampart
                <emph type="italics"/>
              C I R N,
                <emph.end type="italics"/>
              but without that it ſhall go to the bottome:
                <lb/>
              the Cone
                <emph type="italics"/>
              A B D,
                <emph.end type="italics"/>
              being, by the addition of the eighth part of the
                <lb/>
              weight of the Cone
                <emph type="italics"/>
              A S T,
                <emph.end type="italics"/>
              made ſpecifically more grave than the
                <lb/>
              water. </s>
              <s>But if the Altitude of the Cone
                <emph type="italics"/>
              A I R,
                <emph.end type="italics"/>
              were two thirds
                <lb/>
              of the Altitude of the Cone
                <emph type="italics"/>
              A S T,
                <emph.end type="italics"/>
              the Cone
                <emph type="italics"/>
              A S T
                <emph.end type="italics"/>
              would be to the
                <lb/>
              Cone
                <emph type="italics"/>
              A I R,
                <emph.end type="italics"/>
              as twenty ſeven to eight; and the Cylinder
                <emph type="italics"/>
              E S T O,
                <emph.end type="italics"/>
              to
                <lb/>
              the Cylinder
                <emph type="italics"/>
              C I R N,
                <emph.end type="italics"/>
              as nine to four, that is, as twenty ſeven to
                <lb/>
              twelve; and, therefore, the Cylinder
                <emph type="italics"/>
              C I R N,
                <emph.end type="italics"/>
              to the Cone
                <emph type="italics"/>
              A I R,
                <emph.end type="italics"/>
                <lb/>
              as twelve to eight; and the exceſs of the Cylinder
                <emph type="italics"/>
              C I R N,
                <emph.end type="italics"/>
              above
                <lb/>
              the Cone
                <emph type="italics"/>
              A I R,
                <emph.end type="italics"/>
              to the Cone
                <emph type="italics"/>
              A S T,
                <emph.end type="italics"/>
              as four to twenty ſeven:
                <lb/>
              fore if to the Cone
                <emph type="italics"/>
              A B D
                <emph.end type="italics"/>
              be added ſo much weight as is the four
                <lb/>
              twenty ſevenths of the weight of the Cone
                <emph type="italics"/>
              A S T,
                <emph.end type="italics"/>
              which is a little
                <lb/>
              more then its ſeventh part, it alſo ſhall continue to ſwimme, and </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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