Archimedes, Natation of bodies, 1662

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/042.jpg" pagenum="372"/>
              the ſame manner we might demon­
                <lb/>
                <figure id="id.073.01.042.1.jpg" xlink:href="073/01/042/1.jpg" number="40"/>
                <lb/>
              ſtrate the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine T H to be perpendi­
                <lb/>
              cular unto the Surface of the Liquid:
                <lb/>
              and that the Portion demerged with­
                <lb/>
              in the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid moveth or aſcend­
                <lb/>
              eth out of the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid according to
                <lb/>
              the Perpendicular that ſhall be
                <lb/>
              drawn thorow Z unto the Surface
                <lb/>
              of the Liquid; and that the part
                <lb/>
              that is above the Liquid deſcendeth
                <lb/>
              into the Liquid according to that
                <lb/>
              drawn thorow G: therefore the Portion will not continue ſo inclined
                <lb/>
              as was ſuppoſed: But neither ſhall it return to Rectitude or Per­
                <lb/>
              pendicularity; For that of the Perpendiculars drawn thorow Z and
                <lb/>
              G, that paſſing thorow Z doth fall on thoſe parts which are to­
                <lb/>
              wards L; and that that paſſeth thorow G on thoſe towards A:
                <lb/>
              Wherefore it followeth that the Centre Z do move upwards,
                <lb/>
              and G downwards: Therefore the parts of the whole Solid which
                <lb/>
              are towards A ſhall move downwards, and thoſe towards L up­
                <lb/>
              wards. </s>
              <s>Again let the Propoſition run in other termes; and let
                <lb/>
              the Axis of the Portion make an Angle with the Surface of the
                <lb/>
                <arrow.to.target n="marg1251"/>
                <lb/>
              Liquid leſſe than that which is at B. </s>
              <s>Therefore the Square P I
                <lb/>
              hath leſſer Proportion unto the Square
                <lb/>
                <figure id="id.073.01.042.2.jpg" xlink:href="073/01/042/2.jpg" number="41"/>
                <lb/>
              I Y, than the Square E
                <foreign lang="grc">Ψ</foreign>
              hath to the
                <lb/>
              Square
                <foreign lang="grc">Ψ</foreign>
              B: Wherefore K R hath
                <lb/>
              leſſer proportion to I Y, than the half
                <lb/>
              of K R hath to
                <foreign lang="grc">Ψ</foreign>
              B: And, for the
                <lb/>
              ſame reaſon, I Y is greater than dou­
                <lb/>
              ble of
                <foreign lang="grc">Ψ</foreign>
              B: but it is double of O I:
                <lb/>
              Therefore O I ſhall be greater than
                <lb/>
                <foreign lang="grc">Ψ</foreign>
              B: But the Totall O
                <foreign lang="grc">ω</foreign>
              is equall
                <lb/>
              to R B, and the Remainder
                <foreign lang="grc">ω</foreign>
              I leſſe
                <lb/>
              than
                <foreign lang="grc">ψ</foreign>
              R: Wherefore P H ſhall alſo
                <lb/>
              be leſſe than F. And, in regard that
                <lb/>
              M P is equall to F Q, it is manifeſt that P M is greater than ſeſqui­
                <lb/>
              alter of P H; and that P H is leſſe than double of
                <emph type="italics"/>
              H
                <emph.end type="italics"/>
              M.
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              et P Z
                <lb/>
              be double to Z M. </s>
              <s>The Centre of Gravity of the whole Solid ſhall
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              again be T; that of the part which is within the Liquid Z; and
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              drawing a Line from Z to T, the Centre of Gravity of that which
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              is above the Liquid ſhall be found in that Line portracted, that is
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              in G: Therefore, Perpendiculars being drawn thorow Z and G
                <lb/>
                <arrow.to.target n="marg1252"/>
                <lb/>
              unto the Surface of the Liquid that are parallel to T H, it followeth
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              that the ſaid Portion ſhall not ſtay, but ſhall turn about till
                <lb/>
              that its Axis do make an Angle with the Waters Surface greater than
                <lb/>
              that which it now maketh. </s>
              <s>And becauſe that when before we </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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