Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/068.jpg" pagenum="402"/>
              make equall Angles; and that in the Triangles N F S and G
                <foreign lang="grc">ω</foreign>
              C
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              the Angles at F and
                <foreign lang="grc">ω</foreign>
              are equall; as alſo, that S B and B C, and
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              S R and C R are equall to one another: And, therefore, N X and
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              G Y are alſo equall; and X T and Y I. </s>
              <s>And ſince G H is double
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              to H I, N X ſhall be leſſer than double of X T. </s>
              <s>Let N M therefore
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              be double to M T; and drawing a Line from M to K, prolong it
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              unto E. </s>
              <s>Now the Centre of Gravity of the whole ſhall be the
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              Point K; of the part which is in the Liquid the Point M; and
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              that of the part which is above the Liquid in the Line prolonged
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              as ſuppoſe in E. Therefore, by what was even now demonſtrated
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              it is manifeſt that the Portion ſhall not ſtay thus, but ſhall incline, ſo
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              as that its Baſe do in no wiſe touch the Surface of the Liquid
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              And that the Portion will ſtand, ſo, as to make an Angle with the
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              Surface of the Liquid leſſer than
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                <figure id="id.073.01.068.1.jpg" xlink:href="073/01/068/1.jpg" number="75"/>
                <lb/>
              the Angle
                <foreign lang="grc">φ,</foreign>
              ſhall thus be demon
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              ſtrated. </s>
              <s>Let it, if poſſible, ſtand,
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              ſo, as that it do not make an Angle
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              leſſer than the Angle
                <foreign lang="grc">φ;</foreign>
              and diſpoſe
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              all things elſe in the ſame manner a
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              before; as is done in the preſet
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              Figure. </s>
              <s>We are to demonſtrat
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              in the ſame method, that N T is e­
                <lb/>
              quall to
                <foreign lang="grc">ψ;</foreign>
              and by the ſame reaſor
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              equall alſo to G I. </s>
              <s>And ſince that in
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              the Triangles P
                <foreign lang="grc">φ</foreign>
              C and N F S, the Angle F is not leſſer than the
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              Angle
                <foreign lang="grc">φ,</foreign>
              B F ſhall not be greater than B C: And, therefore, neither
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              ſhall S R be leſſer than C R; nor N X than P Y: But ſince P F is
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              greater than N T, let P F be Seſquialter of P Y: N T ſhall be leſſer
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              than Seſquialter of N X: And, therefore, N X ſhall be greate
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              than double of X T. </s>
              <s>Let N M be double of M T; and drawing
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              Line from M to K prolong it. </s>
              <s>It is manifeſt, now, by what hath
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              been ſaid, that the Portion ſhall not continue in this poſition, but ſhall
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              turn about, ſo, as that its Axis do make an Angle with the Surface
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              of the Liquid, leſſer than the Angle
                <foreign lang="grc">φ.</foreign>
              </s>
            </p>
          </chap>
        </body>
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    </archimedes>
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