Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
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            <p type="main">
              <s>
                <pb xlink:href="073/01/056.jpg" pagenum="386"/>
                <emph type="italics"/>
              Again, by dividing, I D ſhall be to D Z, as one to two: But Z D was to D A, that is, to D L,
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              as two to five: Therefore,
                <emph.end type="italics"/>
              ex equali,
                <emph type="italics"/>
              and Converting, L D is to D I, as five to one: and, by
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              Converſion of Proportion, D L is to D I, as five to four: But D Z was to D L, as two to
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              five: Therefore, again,
                <emph.end type="italics"/>
              ex equali,
                <emph type="italics"/>
              D Z is to L I, as two to four: Therefort L I is double
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              of D Z: Which was to be demonſtrated.
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1347"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1346"/>
              P</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1347"/>
              Q</s>
            </p>
            <p type="main">
              <s>And, A D is to D I, as five to one.]
                <emph type="italics"/>
              This we have but juſt now demon­
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              ſtrated.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1348"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1348"/>
              R</s>
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            <p type="main">
              <s>For it hath been demonſtrated, above, that the Portion whoſe
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              Axis is greater than Seſquialter of the Semi-parameter, if it have
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              not leſſer proportion in Gravity to the Liquid, &c.]
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              He hath demonstra­
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              ted this in the fourth Propoſition of this Book.
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              </s>
            </p>
            <p type="head">
              <s>CONCLVSION II.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              If the Portion have leſſer proportion in Gravity to the
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1349"/>
                <lb/>
                <emph type="italics"/>
              Liquid, than the Square S B hath to the Square
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              B D, but greater than the Square X O hath to the
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              Square B D, being demitted into the Liquid, ſo in­
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              clined, as that its Baſe touch not the Liquid, it ſhall
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              continue inclined, ſo, as that its Baſe ſhall not in the
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              leaſt touch the Surface of the Liquid, and its Axis
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              ſhall make an Angle with the Liquids Surface, greater
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              than the Angle X.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1349"/>
              A</s>
            </p>
            <p type="main">
              <s>Therfore repeating the firſt figure, let the Portion have unto
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              the Liquid in Gravitie a proportion greater than the Square
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              X O hath to the ſquare B D, but leſſer than the Square made of
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              the Exceſſe by which the Axis is greater than Seſquialter of the Semi­
                <lb/>
                <figure id="id.073.01.056.1.jpg" xlink:href="073/01/056/1.jpg" number="54"/>
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              Parameter, that is, of S B, hath to
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              the Square B D: and as the Portion
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              is to the Liquid in Gravity, ſo let
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              the Square made of the Line
                <foreign lang="grc">ψ</foreign>
              be
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              to the Square B D:
                <foreign lang="grc">ψ</foreign>
              ſhall be great­
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                <arrow.to.target n="marg1350"/>
                <lb/>
              er than X O, but leſſer than the
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              Exceſſe by which the Axis is grea­
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              ter than Seſquialter of the Semi­
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              parameter, that is, than S B. </s>
              <s>Let
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              a Right Line M N be applyed to
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              fall between the Conick-Sections
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              A M Q L and A
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              X
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              D, [
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              parallel to
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              B D falling betwixt O X and B D,
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              ] and equall to the Line
                <foreign lang="grc">ψ</foreign>
              : and let
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              it cut the remaining Conick Section A H I in the point H, and the
                <lb/>
                <arrow.to.target n="marg1351"/>
                <lb/>
              Right Line R G in V. </s>
              <s>It ſhall be demonſtrated that M H is double to
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              H N, like as it was demonſtrated that O G is double to G X. </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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