Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/067.jpg" pagenum="401"/>
              the Right Line R Y in Y. </s>
              <s>We
                <lb/>
                <figure id="id.073.01.067.1.jpg" xlink:href="073/01/067/1.jpg" number="72"/>
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              ſhall demonſtrate G H to be double
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              to H I, as it hathbeen demonſtra­
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              ted, that O G is double to G X.
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              </s>
              <s>Then draw G
                <foreign lang="grc">ω</foreign>
              touching the Section
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              A G Q L in G; and G C perpen di­
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              cular to B D; and drawing a Line
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              from A to I, prolong it to
                <expan abbr="q.">que</expan>
              Now
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              A I ſhall be equall to I
                <expan abbr="q;">que</expan>
              and
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              A Q parallel to G
                <foreign lang="grc">ω.</foreign>
              It is to be
                <lb/>
              demonſtrated, that the Portion being
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              demitted into the Liquid, and inclined, ſo, as that its Baſe touch
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              the Liquid, it ſhall ſtand ſo incli­
                <lb/>
                <figure id="id.073.01.067.2.jpg" xlink:href="073/01/067/2.jpg" number="73"/>
                <lb/>
              ned, as that its Axis ſhall make
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              an Angle with the Surface of the
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              Liquid leſſe than the Angle
                <foreign lang="grc">φ;</foreign>
                <lb/>
              and its Baſe ſhall not in the leaſt
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              touch the Liquids Surface. </s>
              <s>For
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              let it be demitted into the Liquid,
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              and let it ſtand, ſo, as that its Baſe
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              do touch the Surface of the Liquid
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              in one Point only: and the Portion
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              being cut thorow the Axis by a
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              Plane erect unto the Surface of the Liquid, let the Section of
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                <figure id="id.073.01.067.3.jpg" xlink:href="073/01/067/3.jpg" number="74"/>
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              the Portion be A N Z L, the Section
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              of a Rightangled Cone; that of
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              the Surface of the Liquid A Z; and
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              the Axis of the Portion and Dia­
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              meter of the Section B D; and let
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              B D be cut in the Points K and R
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              as hath been ſaid above; and draw
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              N F parallel to A Z, and touching
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              the Section of the Cone in the Point
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              N; and N T parallel to B D; and
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              N S perpendicular to the ſame. </s>
              <s>Be­
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              cauſe, now, that the Portion is in Gravity to the Liquid, as
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              the Square made of
                <foreign lang="grc">ψ</foreign>
              is to the Square B D; and ſince that as the
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              Portion is to the Liquid in Gravity, ſo is the Square N T to the
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              Square B D, by the things that have been ſaid; it is plain, that
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              N T is equall to the Line
                <foreign lang="grc">ψ</foreign>
              : And, therefore, alſo, the Portions
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              A N Z and A G Q are equall. </s>
              <s>And, ſeeing that in the Equall and
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              Like Portions A G Q L and A N Z L; there are drawn from the
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              Extremities of their Baſes, A Q and A Z which cut off equall Porti­
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              ons: It is obvious, that with the Diameters of the Portions they </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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