Archimedes, Natation of bodies, 1662

Table of figures

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[Figure 61]
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[Figure 63]
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[Figure 68]
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[Figure 69]
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[Figure 70]
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[Figure 72]
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[Figure 73]
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[Figure 75]
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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/065.jpg" pagenum="399"/>
              but one Point only. </s>
              <s>For let it be de­
                <lb/>
                <figure id="id.073.01.065.1.jpg" xlink:href="073/01/065/1.jpg" number="68"/>
                <lb/>
              mitted into the Liquid, as hath been
                <lb/>
              ſaid; and let it firſt be ſo inclined, as
                <lb/>
              that its Baſe do not in the leaſt
                <lb/>
              touch the Surface of the Liquid. </s>
              <s>And
                <lb/>
              then it being cut thorow the Axis,
                <lb/>
              by a Plane erect unto the Surface of
                <lb/>
              the Liquid, let the Section of the
                <lb/>
              Portion be A N Z G; that of the
                <lb/>
              Liquids Surface E Z; the Axis of
                <lb/>
              the Portion and Diameter of the
                <lb/>
              Section B D; and let B D be cut in
                <lb/>
              the Points K and R, as before; and
                <lb/>
              draw N L parallel to E Z, and touching the Section A N Z G
                <lb/>
              in N, and N S perpendicular to
                <lb/>
                <figure id="id.073.01.065.2.jpg" xlink:href="073/01/065/2.jpg" number="69"/>
                <lb/>
              B D. Now, ſeeing that the Por­
                <lb/>
              tion is in Gravity unto the Liquid,
                <lb/>
              as the Square made of the Line
                <lb/>
              is to the Square B D;
                <foreign lang="grc">ψ</foreign>
              ſhall
                <lb/>
              be equall to N T: Which is to
                <lb/>
              be demonſtrated as above: And,
                <lb/>
              therefore, N T is alſo equall to
                <lb/>
              V I: The Portions, therefore,
                <lb/>
              A V Q and E N Z are equall to
                <lb/>
              one another. </s>
              <s>And, ſince that in
                <lb/>
              the Equall and like Portions A V
                <lb/>
              Q L and A N Z G, there are drawn A Q and E Z, cutting off
                <lb/>
              equall Portions, that from the
                <lb/>
                <figure id="id.073.01.065.3.jpg" xlink:href="073/01/065/3.jpg" number="70"/>
                <lb/>
              Extremity of the Baſe, this not
                <lb/>
              from the Extreme, that which is
                <lb/>
              drawn from the Extremity of the
                <lb/>
              Baſe, ſhall make the Acute Angle
                <lb/>
              with the Diameter of the Portion
                <lb/>
              leſſer: and in the Triangles N L S
                <lb/>
              and V
                <foreign lang="grc">ω</foreign>
              C, the Angle at L is
                <lb/>
              greater than the Angle at
                <foreign lang="grc">ω</foreign>
              :
                <lb/>
              Therefore, B S ſhall be leſſer
                <lb/>
              than B C; and S R leſſer than
                <lb/>
              C R: and, conſequently, N X
                <lb/>
              greater than V H; and X T leſſer than H I. Seeing, therefore,
                <lb/>
              that V Y is double to Y I; It is manifeſt, that N X is greater than
                <lb/>
              double to X T. </s>
              <s>Let N M be double to M T: It is manifeſt, from what
                <lb/>
              hath been ſaid, that the Portion ſhall not reſt, but will incline, untill
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              that its Bafe do touch the Surface of the Liquid: and it toucheth it in
                <lb/>
              one Point only, as appeareth in the Figure: And other things </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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