Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/030.jpg" pagenum="360"/>
              contingent unto the Section in the Point P: Wherefore it alſo
                <lb/>
              maketh Right Angles with the Surface of the Liquid: and that
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              part of the Conoidall Solid which is within the Liquid ſhall move
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              upwards according to the Perpendicular drawn thorow B parallel
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              to R T; and that part which is above the Liquid ſhall move down­
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              wards according to that drawn thorow G, parallel to the ſaid R T:
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              And thus it ſhall continue to do ſo long untill that the Conoid be
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              reſtored to uprightneſſe, or to ſtand according to the Perpendicular.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1188"/>
              (a)
                <emph type="italics"/>
              By 10. of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1189"/>
              A</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1190"/>
              B</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1191"/>
              (b)
                <emph type="italics"/>
              By 19. of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1192"/>
              C</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1193"/>
              (c)
                <emph type="italics"/>
              By 1. of this
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              ſecond Book.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1194"/>
              (d)
                <emph type="italics"/>
              By
                <emph.end type="italics"/>
              6. De Co­
                <lb/>
              noilibus &
                <emph type="italics"/>
              S
                <emph.end type="italics"/>
              phæ­
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              roidibus
                <emph type="italics"/>
              of
                <emph.end type="italics"/>
              Archi­
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              medes.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1195"/>
              D</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1196"/>
              E</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1197"/>
              F</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1198"/>
              (e)
                <emph type="italics"/>
              By 2. of this
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              ſecond Book.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>COMMANDINE.
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                <arrow.to.target n="marg1199"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1199"/>
              A</s>
            </p>
            <p type="main">
              <s>Let therefore R H be equall to the Semi-parameter.]
                <emph type="italics"/>
              So it is to be
                <lb/>
              read, and not R M, as
                <emph.end type="italics"/>
              Tartaglia's
                <emph type="italics"/>
              Tranſlation hath is; which may be made appear from
                <lb/>
              that which followeth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1200"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1200"/>
              B</s>
            </p>
            <p type="main">
              <s>And O H double to H M.]
                <emph type="italics"/>
              In the Tranſlation aforenamed it is falſly render­
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              ed,
                <emph.end type="italics"/>
              O N
                <emph type="italics"/>
              double to
                <emph.end type="italics"/>
              R M.</s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1201"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1201"/>
              C</s>
            </p>
            <p type="main">
              <s>And look what proportion the Submerged Portion hath to the whole
                <lb/>
              Portion, the ſame hath the Square of P F unto the Square of N O.]
                <lb/>
                <emph type="italics"/>
              This place we have reſtored in our Tranſlation, at the requeſt of ſome friends: But it is demon­
                <lb/>
              ſtrated by
                <emph.end type="italics"/>
              Archimedes in Libro de Conoidibus & Sphæroidibus, Propo. </s>
              <s>26.</s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1202"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1202"/>
              D</s>
            </p>
            <p type="main">
              <s>Wherefore P F is not leſſe than M O.]
                <emph type="italics"/>
              For by 10 of the fifth it followeth
                <lb/>
              that the Square of P F is not leſſe than the Square of M O: and therefore neither ſhall the
                <lb/>
              Line P F be leße than the Line M O, by 22 of the
                <emph.end type="italics"/>
              </s>
            </p>
            <figure id="id.073.01.030.1.jpg" xlink:href="073/01/030/1.jpg" number="27"/>
            <p type="main">
              <s>
                <arrow.to.target n="marg1203"/>
                <lb/>
                <emph type="italics"/>
              ſixth.
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1204"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1203"/>
              E</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1204"/>
              (a)
                <emph type="italics"/>
              By 14. of the
                <lb/>
              ſixth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Nor B P than H O,]
                <emph type="italics"/>
              For as P F is to
                <lb/>
              P B, ſo is M O to H O: and, by Permutation, as
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1205"/>
                <lb/>
                <emph type="italics"/>
              P F is to M O, ſo is B P to H O; But P F is not
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              leſſe than M O as hath bin proved; (a) Therefore
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              neither ſhall B P be leſſe than H O.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1205"/>
              F</s>
            </p>
            <p type="main">
              <s>If therefore a Right Line be drawn
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              from H at Right Angles unto N O, it
                <lb/>
              ſhall meet with B P, and ſhall fall be­
                <lb/>
              twixt B and P.]
                <emph type="italics"/>
              This Place was corrupt in the
                <lb/>
              Tranſlation of
                <emph.end type="italics"/>
              Tartaglia
                <emph type="italics"/>
              : But it is thus demonstra­
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              ted. </s>
              <s>In regard that P F is not leſſe than O M, nor P B than O H, if we ſuppoſe P F equall to
                <lb/>
              O M, P B ſhall be likewiſe equall to O H: Wherefore the Line drawn thorow O, parallel to A L
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              ſhall fall without the Section, by 17 of the firſt of our Treatiſe of Conicks; And in regard that
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              B P prolonged doth meet it beneath P; Therefore the Perpendicular drawn thorow H doth
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              alſo meet with the ſame beneath B, and it doth of neceſſity fall betwixt B and P: But the
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              ſame is much more to follow, if we ſuppoſe P F to be greater than O M.
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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