Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/025.jpg" pagenum="354"/>
              Therefore B and R are equall. </s>
              <s>And becauſe that of the Magni­
                <lb/>
              tude FA the
                <emph type="italics"/>
              G
                <emph.end type="italics"/>
              ravity is B: Therefore of the Liquid Body
                <emph type="italics"/>
              N
                <emph.end type="italics"/>
              I the
                <lb/>
              Gravity is O R. </s>
              <s>As F A is to N I, ſo is B to O R, or, ſo is R to
                <lb/>
              O R: But as R is to O R, ſo is I to N I, and A to F A: Therefore
                <lb/>
                <arrow.to.target n="marg1164"/>
                <lb/>
              I is to N I, as F A to N I: And as I to N I ſo is
                <emph type="italics"/>
              (b)
                <emph.end type="italics"/>
              A to F A.
                <lb/>
              </s>
              <s>Therefore F A is to N I, as A is to F A: Which was to be demon­
                <lb/>
              ſtrated.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1163"/>
              (a)
                <emph type="italics"/>
              By 5. of the
                <lb/>
              firſt of this.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1164"/>
              (b)
                <emph type="italics"/>
              By 11. of the
                <lb/>
              fifth of
                <emph.end type="italics"/>
              Eucl.</s>
            </p>
            <p type="head">
              <s>PROP. II. THEOR. II.
                <lb/>
                <arrow.to.target n="marg1165"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1165"/>
              A</s>
            </p>
            <p type="main">
              <s>^{*}
                <emph type="italics"/>
              The Right Portion of a Right angled Conoide, when it
                <lb/>
              ſhall have its Axis leſſe than
                <emph.end type="italics"/>
              ſeſquialter ejus quæ ad
                <lb/>
              Axem (
                <emph type="italics"/>
              or of its
                <emph.end type="italics"/>
              Semi-parameter)
                <emph type="italics"/>
              having any what
                <lb/>
              ever proportion to the Liquid in Gravity, being de­
                <lb/>
              mitted into the Liquid ſo as that its Baſe touch not
                <lb/>
              the ſaid Liquid, and being ſet ſtooping, it ſhall not
                <lb/>
              remain ſtooping, but ſhall be restored to uprightneſſe.
                <lb/>
              </s>
              <s>I ſay that the ſaid Portion ſhall ſtand upright when
                <lb/>
              the Plane that cuts it ſhall be parallel unto the Sur­
                <lb/>
              face of the Liquid.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Let there be a Portion of a Rightangled Conoid, as hath been
                <lb/>
              ſaid; and let it lye ſtooping or inclining: It is to be demon­
                <lb/>
              ſtrated that it will not ſo continue but ſhall be reſtored to re­
                <lb/>
              ctitude. </s>
              <s>For let it be cut through the Axis by a plane erect upon
                <lb/>
              the Surface of the Liquid, and let the Section of the Portion be
                <lb/>
              A PO L, the Section of a Rightangled Cone, and let the Axis
                <lb/>
                <figure id="id.073.01.025.1.jpg" xlink:href="073/01/025/1.jpg" number="20"/>
                <lb/>
              of the Portion and Diameter of the
                <lb/>
              Section be N O: And let the Sect­
                <lb/>
              ion of the Surface of the Liquid be
                <lb/>
              I S. </s>
              <s>If now the Portion be not
                <lb/>
              erect, then neither ſhall A L be Pa­
                <lb/>
              rallel to I S: Wherefore N O will
                <lb/>
              not be at Right Angles with I S. </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1166"/>
                <lb/>
              Draw therefore K
                <foreign lang="grc">ω,</foreign>
              touching the Section of the Cone I, in the
                <lb/>
              Point P [that is parallel to I S: and from the Point P unto I S
                <lb/>
                <arrow.to.target n="marg1167"/>
                <lb/>
              draw P F parallel unto O N, ^{*} which ſhall be the Diameter of the
                <lb/>
              Section I P O S, and the Axis of the Portion demerged in the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              i­
                <lb/>
                <arrow.to.target n="marg1168"/>
                <lb/>
              quid. </s>
              <s>In the next place take the Centres of Gravity: ^{*} and of
                <lb/>
              the Solid Magnitude A P O L, let the Centre of Gravity be R; and
                <lb/>
                <arrow.to.target n="marg1169"/>
                <lb/>
              of I P O S let the Centre be B: ^{*} and draw a Line from B to R
                <lb/>
              prolonged unto G; which let be the Centre of Gravity of the </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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