Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/027.jpg" pagenum="356"/>
                <emph type="italics"/>
              For if, the Line B R being prolonged unto G, G R hath the ſame proportion to R B as the Por­
                <lb/>
              tion of the Conoid I P O S hath to the remaining Figure that lyeth above the Surface of the
                <lb/>
              Liquid, the Toine G ſhall be its Centre of Gravity; by the 8 of the ſecond of
                <emph.end type="italics"/>
              Archimedes
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              de Centro Gravitatis Planorum, vel de
                <emph type="italics"/>
              Æ
                <emph.end type="italics"/>
              quiponderantibus.
                <lb/>
                <arrow.to.target n="marg1177"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1176"/>
              D</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1177"/>
              E</s>
            </p>
            <p type="main">
              <s>R O ſhall be leſs than
                <emph type="italics"/>
              quæ uſque ad Axem
                <emph.end type="italics"/>
              (or than the Semi­
                <lb/>
              parameter.]
                <emph type="italics"/>
              By the 10 Propofit. </s>
              <s>of
                <emph.end type="italics"/>
              Euclids
                <emph type="italics"/>
              fifth Book of Elements. </s>
              <s>The Line
                <emph.end type="italics"/>
              quæ
                <lb/>
              uſque ad Axem,
                <emph type="italics"/>
              (or the Semi-parameter) according to
                <emph.end type="italics"/>
              Archimedes,
                <emph type="italics"/>
              is the half of that
                <emph.end type="italics"/>
                <lb/>
              juxta quam poſſunt, quæ á Sectione ducuntur, (
                <emph type="italics"/>
              or of the Parameter;) as appeareth
                <lb/>
              by the 4 Propoſit of his Book
                <emph.end type="italics"/>
              De Conoidibus & Shpæroidibus:
                <emph type="italics"/>
              and for what reaſon it is
                <lb/>
              ſo called, we have declared in the Commentaries upon him by us publiſhed.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1178"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1178"/>
              F</s>
            </p>
            <p type="main">
              <s>Whereupon the Angle R P
                <foreign lang="grc">ω</foreign>
              ſhall be acute.]
                <emph type="italics"/>
              Let the Line N O be
                <lb/>
              continued out to H, that ſo RH may be equall to
                <lb/>
              the Semi-parameter. </s>
              <s>If now from the Point H
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.073.01.027.1.jpg" xlink:href="073/01/027/1.jpg" number="22"/>
                <lb/>
                <emph type="italics"/>
              a Line be drawn at Right Angles to N H, it ſhall
                <lb/>
              meet with FP without the Section; for being
                <lb/>
              drawn thorow O parallel to A L, it ſhall fall
                <lb/>
              without the Section, by the 17 of our ſirst Book of
                <emph.end type="italics"/>
                <lb/>
              Conicks;
                <emph type="italics"/>
              Therefore let it meet in V: and
                <lb/>
              becauſe F P is parallel to the Diameter, and H
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              V perpendicular to the ſame Diameter, and R H
                <lb/>
              equall to the Semi-parameter, the Line drawn
                <lb/>
              from the Point R to V ſhall make Right Angles
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              with that Line which the Section toucheth in the Point P: that is with K
                <emph.end type="italics"/>
                <foreign lang="grc">ω,</foreign>
                <emph type="italics"/>
              as ſhall anon be
                <lb/>
              demonstrated: Wherefore the Perpendidulat R T falleth betwixt A and
                <emph.end type="italics"/>
                <foreign lang="grc">ω;</foreign>
                <emph type="italics"/>
              and the Argle R
                <emph.end type="italics"/>
                <lb/>
              P
                <foreign lang="grc">ω</foreign>
                <emph type="italics"/>
              ſhall be an Acute Angle.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Let A B C be the Section of a Rightangled Cone, or a Parabola,
                <lb/>
              and its Diameter B D; and let the Line E F touch the
                <lb/>
              ſame in the Point G: and in the Diameter B D take the Line
                <lb/>
              H K equall to the Semi-parameter: and thorow G, G L be­
                <lb/>
              ing drawn parallel to the Diameter, draw KM from the
                <lb/>
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              oint K at Right Angles to B D cutting G L in M: I ſay
                <lb/>
              that the Line prolonged thorow Hand Mis perpendicular to
                <lb/>
              E F, which it cutteth in N.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              For from the Point G draw the Line G O at Right Angles to E F cutting the Diameter in
                <lb/>
              O: and again from the ſame Point draw G P perpendicular to the Diameter: and let the
                <lb/>
              ſaid Diameter prolonged cut the Line E F in
                <expan abbr="q.">que</expan>
              P B ſhall be equall to B Q, by the 35 of
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1179"/>
                <lb/>
                <emph type="italics"/>
              our firſt Book of
                <emph.end type="italics"/>
              Conick
                <emph type="italics"/>
              Sections,
                <emph.end type="italics"/>
              (a)
                <emph type="italics"/>
              and G
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.073.01.027.2.jpg" xlink:href="073/01/027/2.jpg" number="23"/>
                <lb/>
                <emph type="italics"/>
              P a Mean-proportion all betmixt Q P and PO
                <emph.end type="italics"/>
              ;
                <lb/>
                <arrow.to.target n="marg1180"/>
                <lb/>
              (b)
                <emph type="italics"/>
              and therefore the Square of G P ſhall be e­
                <lb/>
              quall to the Rectangle of O P Q: But it is alſo
                <lb/>
              equall to the Rectangle comprehended under P B
                <lb/>
              and the Line
                <emph.end type="italics"/>
              juxta quam poſſunt,
                <emph type="italics"/>
              or the Par­
                <lb/>
              ameter, by the 11 of our firſt Book of
                <emph.end type="italics"/>
              Conicks:
                <lb/>
                <arrow.to.target n="marg1181"/>
                <lb/>
              (c)
                <emph type="italics"/>
              Therefore, look what proportion Q P hath to
                <lb/>
              P B, and the ſame hath the Parameter unto P O:
                <lb/>
              But Q P is double unto
                <emph.end type="italics"/>
              P B,
                <emph type="italics"/>
              for that
                <emph.end type="italics"/>
              P B
                <emph type="italics"/>
              and B
                <lb/>
              Q are equall, as hath been ſaid: And therefore
                <lb/>
              the Parameter ſhall be double to the ſaid P O:
                <lb/>
              and by the ſame Reaſon P O is equall to that which we call the Semi-parameter, that is, to K H
                <emph.end type="italics"/>
              :
                <lb/>
                <arrow.to.target n="marg1182"/>
                <lb/>
                <emph type="italics"/>
              But
                <emph.end type="italics"/>
              (d)
                <emph type="italics"/>
              P G is equall to K M, and
                <emph.end type="italics"/>
              (e)
                <emph type="italics"/>
              the Angle O P G to the Angle H K M; for they are both
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1183"/>
                <lb/>
                <emph type="italics"/>
              Right Angles: And therefore O G alſo is equall to H M, and the Angle P O G unto the
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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