Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/033.jpg" pagenum="363"/>
                <emph type="italics"/>
              the proportion of the whole Portion unto that part thereof which is above the Liquid ſhall not be
                <lb/>
              greater than that of the Square N O unto the Square M O: Which was to be demonſtrated.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>And hence it followeth that P F is not leſſe than O M, nor P B </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1213"/>
                <lb/>
              than O H.]
                <emph type="italics"/>
              This followeth by the 10 and 14 of the fifth, and by the 22 of the ſixth of
                <emph.end type="italics"/>
                <lb/>
              Euclid,
                <emph type="italics"/>
              as hath been ſaid above.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1213"/>
              B</s>
            </p>
            <p type="main">
              <s>A
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine, therefore, drawn from Hat Right Angles unto N O ſhall
                <lb/>
                <arrow.to.target n="marg1214"/>
                <lb/>
              meet with P B betwixt P and B.]
                <emph type="italics"/>
              Why this ſo falleth out, we will ſhew in the
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              next.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1214"/>
              C</s>
            </p>
            <p type="head">
              <s>PROP. VI. THEOR. VI.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              The Right Portion of a Rightangled Conoid lighter
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              than the Liquid, when it ſhall have its Axis greater
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              than ſeſquialter of the Semi-parameter, but leſſe than
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              to be unto the Semi-parameter in proportion as fifteen
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              to fower, being demitted into the Liquid ſo as that
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              its Baſe do touch the Liquid, it ſhall never stand ſo
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              enclined as that its Baſe toucheth the Liquid in one
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              Point only.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Let there be a Portion, as was ſaid; and demit it into the Li­
                <lb/>
              quid in ſuch faſhion as that its Baſe do touch the Liquid in
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              one only Point: It is to be demonſtrated that the ſaid Portion
                <lb/>
                <arrow.to.target n="marg1215"/>
                <lb/>
              ſhall not continue ſo, but ſhall turn about in ſuch manner as that
                <lb/>
              its Baſe do in no wiſe touch the Surface of the Liquid. </s>
              <s>For let it be
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              cut thorow its Axis by a Plane erect
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                <figure id="id.073.01.033.1.jpg" xlink:href="073/01/033/1.jpg" number="29"/>
                <lb/>
              upon the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquids Surface: and let
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              the Section of the Superficies of the
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              Portion be A P O L, the Section of
                <lb/>
              a Rightangled Cone; and the Sect­
                <lb/>
              ion of the Surface of the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid be
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              A S; and the Axis of the Portion
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              and Diameter of the Section N O:
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              and let it be cut in F, ſo as that O
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              F be double to F N; and in
                <foreign lang="grc">ω</foreign>
              ſo, as that N O may be to F
                <foreign lang="grc">ω</foreign>
              in the
                <lb/>
              ſame proportion as fifteen to four; and at Right Angles to N O
                <lb/>
              draw
                <foreign lang="grc">ω</foreign>
                <emph type="italics"/>
              N
                <emph.end type="italics"/>
              ow becauſe N O hath greater proportion unto F
                <foreign lang="grc">ω</foreign>
              than
                <lb/>
              unto the Semi-parameter, let the Semi-parameter be equall to F B:
                <lb/>
                <arrow.to.target n="marg1216"/>
                <lb/>
              and draw P C parallel unto A S, and touching the Section A P O L
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              in P; and P I parallel unto
                <emph type="italics"/>
              N O
                <emph.end type="italics"/>
              ; and firſt let P I cut K
                <foreign lang="grc">ω</foreign>
              in H. For­
                <lb/>
                <arrow.to.target n="marg1217"/>
                <lb/>
              aſmuch, therefore, as in the Portion A P O L, contained betwixt
                <lb/>
              the Right
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine and the Section of the Rightangled Cone, K
                <foreign lang="grc">ω</foreign>
              is
                <lb/>
              parallel to A L, and P I parallel unto the Diameter, and cut by the </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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