Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/029.jpg" pagenum="356"/>
              Section of the Portion be A P O L, the Section of a Rightangled
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              Cone; and let the Axis of the Portion and Diameter of the Section
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              be N O, and the Section of the Surface of the Liquid I S. </s>
              <s>If now
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              the Portion be not erect, then N O ſhall not be at equall Angles with
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              I S. </s>
              <s>Draw R
                <foreign lang="grc">ω</foreign>
              touching the Section of the Rightangled Conoid
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              in P, and parallel to I S: and from the Point P and parall to O N
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              draw
                <emph type="italics"/>
              P
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              F: and take the Centers of Gravity; and of the Solid A
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                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              O L let the Centre be R; and of that which lyeth within the
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              Liquid let the Centre be B; and draw a Line from B to R pro­
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              longing it to G, that G may be the Centre of Gravity of the Solid
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              that is above the Liquid. </s>
              <s>And becauſe N O is ſeſquialter of R
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              O, and is greater than ſeſquialter of the Semi-Parameter; it is ma­
                <lb/>
                <arrow.to.target n="marg1188"/>
                <lb/>
              nifeſt that
                <emph type="italics"/>
              (a)
                <emph.end type="italics"/>
              R O is greater than the
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                <figure id="id.073.01.029.1.jpg" xlink:href="073/01/029/1.jpg" number="26"/>
                <lb/>
              Semi-parameter. ^{*}Let therefore R
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                <arrow.to.target n="marg1189"/>
                <lb/>
              H be equall to the Semi-Parameter,
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                <arrow.to.target n="marg1190"/>
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              ^{*} and O
                <emph type="italics"/>
              H
                <emph.end type="italics"/>
              double to H M. </s>
              <s>Foraſ­
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              much therefore as N O is ſeſquialter
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                <arrow.to.target n="marg1191"/>
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              of R O, and M O of O H,
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              (b)
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              the
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              Remainder N M ſhall be ſeſquialter
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              of the Remainder R H: Therefore
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              the Axis is greater than ſeſquialter
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              of the Semi parameter by the quan­
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              tity of the Line M O. </s>
              <s>And let it be
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              ſuppoſed that the Portion hath not leſſe proportion in Gravity unto
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              the Liquid of equall Maſſe, than the Square that is made of the
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              Exceſſe by which the Axis is greater than ſeſquialter of the Semi­
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              parameter hath to the Square made of the Axis: It is therefore ma­
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              nifeſt that the Portion hath not leſſe proportion in Gravity to the
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              Liquid than the Square of the Line M O hath to the Square of N
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              O: But look what proportion the
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion hath to the Liquid in
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              Gravity, the ſame hath the
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion ſubmerged to the whole Solid:
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              for this hath been demonſtrated
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              (c)
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              above: ^{*}And look what pro­
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                <arrow.to.target n="marg1192"/>
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              portion the ſubmerged Portion hath to the whole
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion, the
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                <arrow.to.target n="marg1193"/>
                <lb/>
              ſame hath the Square of
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              F unto the Square of N O: For it hath
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              been demonſtrated in
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              (d) Lib. de Conoidibus,
                <emph.end type="italics"/>
              that if from a Right­
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                <arrow.to.target n="marg1194"/>
                <lb/>
              angled Conoid two
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortions be cut by Planes in any faſhion pro­
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              duced, theſe
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortions ſhall have the ſame Proportion to each
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              other as the Squares of their Axes: The Square of P F, therefore,
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              hath not leſſe proportion to the Square of N O than the Square of
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              M O hath to the Square of N O: ^{*}Wherefore P F is not leſſe than
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                <arrow.to.target n="marg1195"/>
                <lb/>
              M O, ^{*}nor B P than H O. ^{*}If therefore, a Right Line be drawn
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                <arrow.to.target n="marg1196"/>
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              from H at Right Angles unto N O, it ſhall meet with B
                <emph type="italics"/>
              P,
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              and ſhall
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                <arrow.to.target n="marg1197"/>
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              fall betwixt B and P; let it fall in T:
                <emph type="italics"/>
              (e)
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              And becauſe
                <emph type="italics"/>
              P
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              F is
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                <arrow.to.target n="marg1198"/>
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              parallel to the Diameter, and H T is perpendicular unto the ſame
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              Diameter, and R H equall to the Semi-parameter; a Line drawn
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              from R to T and prolonged, maketh Right Angles with the Line </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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