Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="073/01/031.jpg" pagenum="361"/>
            <p type="head">
              <s>PROP. V. THEOR. V.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              The Right Portion of a Right-Angled Conoid lighter
                <lb/>
              than the Liquid, when it ſhall have its Axis great­
                <lb/>
              er than
                <emph.end type="italics"/>
              Seſquialter
                <emph type="italics"/>
              of the Semi-parameter, if it have
                <lb/>
              not greater proportion in Gravity to the Liquid [of
                <lb/>
              equal Maſs] than the Exceſſe by which the Square
                <lb/>
              made of the Axis is greater than the Square made
                <lb/>
              of the Exceſſe by which the Axis is greater than
                <emph.end type="italics"/>
                <lb/>
              ſeſquialter
                <emph type="italics"/>
              of the Semi-Parameter hath to the
                <lb/>
              Square made of the Axis being demitted into the Li­
                <lb/>
              quid, ſo as that its Baſe be wholly within the Liquid,
                <lb/>
              and being ſet inclining, it ſhall not remain ſo inclined,
                <lb/>
              but ſhall turn about till that its Axis ſhall be accor­
                <lb/>
              ding to the Perpendicular.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>For let any Portion be demitted into the Liquid, as hath been
                <lb/>
              ſaid; and let its Baſe be wholly within the Liquid, And being
                <lb/>
              cut thorow its Axis by a Plain erect upon the Surface of the
                <lb/>
              Liquid; its Section ſhall be the Section
                <lb/>
                <figure id="id.073.01.031.1.jpg" xlink:href="073/01/031/1.jpg" number="28"/>
                <lb/>
              of a Rightangled Cone: Let it be
                <lb/>
              A P O L, and let the Axis of the Por­
                <lb/>
              tion and Diameter of the Section be
                <lb/>
              N O; and the Section of the Surface of
                <lb/>
              the Liquid I S. </s>
              <s>And becauſe the Axis
                <lb/>
              is not according to the Perpendicu­
                <lb/>
              lar, N O will not be at equall angles
                <lb/>
              with I S. </s>
              <s>Draw K
                <foreign lang="grc">ω</foreign>
              touching the Se­
                <lb/>
              ction A P O L in P, and parallel unto
                <lb/>
              I S: and thorow P, draw P F parallel unto N O: and take the
                <lb/>
              Centres of Gravity; and of the Solid A P O L let the Centre be
                <lb/>
              R; and of that which lyeth above the Liquid let the Centre be B;
                <lb/>
              and draw a Line from B to R, prolonging it to G; which let be the
                <lb/>
              Centre of Gravity of the Solid demerged within the Liquid: and
                <lb/>
              moreover, take R H equall to the Semi-parameter, and let O H be
                <lb/>
              double to H M; and do in the reſt as hath been ſaid
                <emph type="italics"/>
              (a)
                <emph.end type="italics"/>
              above.
                <lb/>
                <arrow.to.target n="marg1206"/>
                <lb/>
              Now foraſmuch as it was ſuppoſed that the Portion hath not greater
                <lb/>
              proportion in Gravity to the Liquid, than the Exceſſe by which
                <lb/>
              the Square N O is greater than the Square M O, hath to the ſaid
                <lb/>
              Square N O: And in regard that whatever proportion in Gravity </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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