Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="073/01/045.jpg" pagenum="375"/>
            <p type="head">
              <s>PROP. IX. THE OR. IX.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              The Right Portion of a Rightangled Conoid, when it
                <lb/>
              ſhall have its Axis greater than Seſquialter of the
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              Semi-parameter, but leſſer than to be unto the ſaid
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              Semi-parameter in proportion as fifteen to four, and
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              hath greater proportion in Gravity to the Liquid, than
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              the exceſs by which the Square made of the Axis is
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              greater than the Square made of the Exceſs, by which
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              the Axis is greater than Seſquialter of the Semi­
                <lb/>
              parameter, hath to the Square made of the Axis,
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              being demitted into the Liquid, ſo as that its Baſe
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              be wholly within the Liquid, and being ſet inclining
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              it ſhall neither turn about, ſo as that its Axis ſtand
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              according to the Perpendicular, nor remain inclined,
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              ſave only when the Axis makes an Angle with
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              the Surface of the Liquid, equall to that aßigned
                <lb/>
              as before.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Let there be a Portion as was ſaid; and ſuppoſe D B equall to
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              the Axis of the
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion: and let B K be double to K D; and
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              K R equall to the Semi-parameter: and C B Seſquialter of
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              B R. </s>
              <s>And as the Portion is to the Liquid in Gravity, ſo let the Ex­
                <lb/>
              ceſſe by which the Square B D exceeds the Square F Q be to the
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              Square B D: and let F be double to Q: It is manifeſt, therefore,
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              that the Exceſſe by which the
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                <figure id="id.073.01.045.1.jpg" xlink:href="073/01/045/1.jpg" number="44"/>
                <lb/>
              Square B D is greater than the
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              Square B C hath leſser proportion
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              to the Square B D, than the Exceſs
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              by which the Square B D is greater
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              than the Square F Q hath to the
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              Square B D; for B C is the Exceſs
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              by which the Axis of the Portion is
                <lb/>
              greater than Seſquialter of the
                <lb/>
              Semi-parameter: And, therefore, </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1280"/>
                <lb/>
              the Square B D doth more exceed
                <lb/>
              the Square F Q, than doth the
                <lb/>
              Square B C: And, conſequently, the Line F Q is leſs than B C; </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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