Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="073/01/039.jpg" pagenum="369"/>
            <p type="head">
              <s>PROP. VII. THE OR. VII.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              The Right Portion of a Rightangled Conoid lighter
                <lb/>
              than the Liquid, when it ſhall have its Axis greater
                <lb/>
              than Seſquialter of the Semi-parameter, but leſſe
                <lb/>
              than to be unto the ſaid Semi-parameter in proportion
                <lb/>
              as fiſteen to fower, being demitted into the Liquid ſo
                <lb/>
              as that its Baſe be wholly within the Liquid, it ſhall
                <lb/>
              never ſtand ſo as that its Baſe do touch the Surface
                <lb/>
              of the Liquid, but ſo, that it be wholly within the
                <lb/>
              Liquid, and ſhall not in the leaſt touch its Surface.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Let there be a Portion as hath been ſaid; and let it be de­
                <lb/>
              mitted into the Liquid, as we have ſuppoſed, ſo as that its
                <lb/>
              Baſe do touch the Surface in one Point only: It is to be de­
                <lb/>
              monſtrated that the ſame ſhall not ſo
                <lb/>
                <figure id="id.073.01.039.1.jpg" xlink:href="073/01/039/1.jpg" number="37"/>
                <lb/>
              continue, but ſhall turn about in
                <lb/>
              ſuch manner as that its Baſe do in no
                <lb/>
              wiſe touch the Surface of the Liquid.
                <lb/>
              </s>
              <s>For let it be cut thorow its Axis by
                <lb/>
              a Plane erect upon the Liquids Sur­
                <lb/>
              face: and let the Section be A P O L,
                <lb/>
              the Section of a Rightangled
                <lb/>
              Cone; the Section of the Liquids
                <lb/>
              Surface S L; and the Axis of the
                <lb/>
              Portion and Diameter of the Section P F: and let P F be cut in
                <lb/>
              R, ſo, as that R P may be double to R F, and in
                <foreign lang="grc">ω</foreign>
              ſo as that P F
                <lb/>
              may be to R
                <foreign lang="grc">ω</foreign>
              as fifteen to fower: and draw
                <foreign lang="grc">ω</foreign>
              K at Right Angles </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1237"/>
                <lb/>
              to P F:
                <emph type="italics"/>
              (a)
                <emph.end type="italics"/>
              R
                <foreign lang="grc">ω</foreign>
              ſhall be leſſe than the Semi-parameter. </s>
              <s>There­
                <lb/>
              fore let R H be ſuppoſed equall to the Semi-parameter: and
                <lb/>
              draw C O touching the Section in O and parallel unto S L; and
                <lb/>
              let N O be parallel unto P F; and firſt let N O cut K
                <foreign lang="grc">ω</foreign>
              in the Point
                <lb/>
              I, as in the former Schemes: It ſhall be demonſtrated that N O is
                <lb/>
              to O I either ſeſquialter, or greater than ſeſquialter. </s>
              <s>Let O I be
                <lb/>
              leſſe than double to I N; and let O B be double to B N: and let
                <lb/>
              them be diſpoſed like as before. </s>
              <s>We might likewiſe demonſtrate
                <lb/>
              that if a Line be drawn thorow R and T it will make Right Angles
                <lb/>
              with the Line C O, and with the Surface of the Liquid: Where­
                <lb/>
              fore Lines being drawn from the Points B and G parallels unto
                <lb/>
              R T, they alſo ſhall be Perpendiculars to the Surface of the Liquid:
                <lb/>
              The Portion therefore which is above the Liquid ſhall move </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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