Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/057.jpg" pagenum="387"/>
                <figure id="id.073.01.057.1.jpg" xlink:href="073/01/057/1.jpg" number="55"/>
                <lb/>
              And from the Point M draw M Y
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              touching the Section A M Q L in M;
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              and M C perpendicular to B D: and
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              laſtly, having drawn A N & prolong­
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              ed it to Q, the Lines A N & N Q ſhall
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              be equall to each other. </s>
              <s>For in
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              regard that in the Like Portions
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                <arrow.to.target n="marg1352"/>
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              A M Q L and A
                <emph type="italics"/>
              X
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              D the Lines A Q
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              and A N are drawn from the Baſes
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              unto the Portions, which Lines
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              contain equall Angles with the ſaid
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              Baſes, Q A ſhall have the ſame proportion to A M that L A hath
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              to A D: Therefore A N is equall to N Q, and A Q parallel to M Y.
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                <arrow.to.target n="marg1353"/>
                <lb/>
              It is to be demonſtrated that the Portion being demitted into the
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              Liquid, and ſo inclined as that its Baſe touch not the Liquid, it
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              ſhall continue inclined ſo as that its Baſe ſhall not in the leaſt touch
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              the Surface of the Liquid, and its Axis ſhall make an Angle with
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              the Liquids Surface greater than the Angle X. </s>
              <s>Let it be demitted
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              into the Liquid, and let it ſtand, ſo, as that its Baſe do touch the
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              Surface of the Liquid in one Point only; and let the Portion be cut
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              thorow the Axis by a Plane erect unto the Surface of the Liquid,
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                <figure id="id.073.01.057.2.jpg" xlink:href="073/01/057/2.jpg" number="56"/>
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              and Let the Section of the Super­
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              ficies of the Portion be A P O L,
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              the Section of a Rightangled Cone,
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              and let the Section of the Liquids
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              Surface be A O; And let the Axis
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              of the Portion and Diameter of the
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              Section be
                <emph type="italics"/>
              B
                <emph.end type="italics"/>
              D: and let B D be
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                <arrow.to.target n="marg1354"/>
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              cut in the Points K and R as hath
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              been ſaid; alſo draw P G Parallel to
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              A O and touching the Section
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              A P O L in P; and from that Point
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              draw P T Parallel to B D, and P S perpendicular to the ſame B D.
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              Now, foraſmuch as the Portion is unto the Liquid in Gravity, as
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              the Square made of the Line
                <foreign lang="grc">ψ</foreign>
              is to the Square B D; and ſince that
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              as the portion is unto the Liquid in Gravitie, ſo is the part thereof
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              ſubmerged unto the whole Portion; and that as the part ſubmerged
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              is to the whole, ſo is the Square T P to the Square B D; It follow­
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              eth that the Line
                <foreign lang="grc">ψ</foreign>
              ſhall be equall to T P: And therefore the Lines
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              M N and P T, as alſo the Portions A M Q and A P O ſhall like­
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              wiſe be equall to each other. </s>
              <s>And ſeeing that in the Equall and
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              Like Portions A P O L and A M Q L the Lines A O and A Q
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                <arrow.to.target n="marg1355"/>
                <lb/>
              are drawn from the extremites of their Baſes, ſo, as that the Portions
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              cut off do make Equall Angles with their Diameters; as alſo the </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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