Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/040.jpg" pagenum="370"/>
                <figure id="id.073.01.040.1.jpg" xlink:href="073/01/040/1.jpg" number="38"/>
                <lb/>
              wards according to that ſame Perpendicular
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              which paſſeth thorow B; and the Portion
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              which is within the Liquid ſhall move up­
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              wards acording to that paſſing thorow G:
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              From whence it is manifeſt that the Solid
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              ſhall turn about in ſuch manner, as that
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              its Baſe ſhall in no wiſe touch the Surface
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              of the Liquid; for that now when it touch­
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              eth but in one Point only, it moveth down­
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              wards on the part towards L. </s>
              <s>And though
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              N O ſhould not cut
                <foreign lang="grc">ω</foreign>
              K, yet ſhall the ſame hold true.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1237"/>
              (a)
                <emph type="italics"/>
              By 10 of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>PROP. VIII. THE OR. VIII.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              The Right Portion of a Rightangled Conoid, when it
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              ſhall have its Axis greater than ſeſquialter of the Se­
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              mi-parameter, but leſſe than to be unto the ſaid Semi­
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              parameter, in proportion as fifteen to fower, if it
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              have a leſſer proportion in Gravity to the Liquid, than
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              the Square made of the Exceſſe by which the Axis is
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              greater than Seſquialter of the Semi-parameter hath
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              to the Square made of the Axis, being demitted into
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              the Liquid, ſo as that its Baſe touch not the Liquid,
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              it ſhall neither return to Perpendicularity, nor conti­
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              nue inclined, ſave only when the Axis makes an
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              Angle with the Surface of the Liquid, equall to that
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              which we ſhall preſently ſpeak of.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Let there be a Portion as hath been ſaid; and let B D be equall
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              to the Axis: and let B K be double to K D; and R K equall
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                <arrow.to.target n="marg1238"/>
                <lb/>
              to the Semi-parameter: and let C B be Seſquialter of B R:
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              C D ſhall be alſo Sefquialter of K R. </s>
              <s>And as the Portion is to the
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              Liquid in Gravity, ſo let the Square F Q be to the Square D B;
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              and let F be double to Q: It is manifeſt, therefore, that F Q hath
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              to D B, leſs proportion than C B hath to B D; For C B is the
                <lb/>
              Exceſs by which the Axis is greater than Seſquialter of the Semi­
                <lb/>
                <arrow.to.target n="marg1239"/>
                <lb/>
              parameter: And, therefore, F Q is leſs than B C; and, for the
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                <arrow.to.target n="marg1240"/>
                <lb/>
              ſame reaſon, F is leſs than B R. </s>
              <s>Let R
                <foreign lang="grc">ψ</foreign>
              be equall to F; and draw
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                <foreign lang="grc">ψ</foreign>
              E perpendicular to B D; which let be in power or contence the
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              half of that which the Lines K R and
                <foreign lang="grc">ψ</foreign>
              B containeth; and
                <lb/>
              draw a Line from B to E: It is to be demonſtrated, that the </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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