Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/048.jpg" pagenum="378"/>
                <emph type="italics"/>
              the Exceſs by which it exceedeth the Square F Q: And, therefore, by Converſion of Proportion,
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              the whole Portion is to the part thereof above the Liquid, as the Square B D is to the Square,
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              F
                <expan abbr="q;">que</expan>
              for the Square B D is ſo much greater than the Exceſs by which it exceedeth the Squar,
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              F Q as is the ſaid Square F
                <expan abbr="q.">que</expan>
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1294"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1293"/>
              E</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1294"/>
              F</s>
            </p>
            <p type="main">
              <s>For the parts towards L ſhall move downwards, and thoſe to­
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              wards A upwards.]
                <emph type="italics"/>
              We thus carrect theſe words, for in
                <emph.end type="italics"/>
              Tartaglia's
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              Tranſlation it
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              is falſly, as I conceive, read
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              Quoniam quæ ex parte L ad ſuperiora ferentur,
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              becauſe
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              the Line thàt paſſeth thorow Z falls perpendicularly on the parts towards L, and that thorow
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              G falleth perpendicularly on the parts towards A: Whereupon the Centre Z, together with thoſe
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              parts which are towards L ſhall move downwards; and the Centre G, together with the parts
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              which are towards A upwards.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1295"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1295"/>
              G</s>
            </p>
            <p type="main">
              <s>It ſhall in like manner be demonſtrated that the Portion ſhall not
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              reſt, but incline untill that its Axis do make an Angle with the
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              Surface of the Liquid, equall to the Angle B.]
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              This may be eaſily demon­
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              ſtratred, as nell from what hath been ſaid in the precedent Propoſition, as alſo from the two
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              latter Figures, by us inſerted
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              </s>
            </p>
            <p type="head">
              <s>PROP. X. THEOR. X.</s>
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            <p type="main">
              <s>
                <emph type="italics"/>
              The Right Portion of a Rightangled Conoid, lighter
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              than the Liquid, when it ſhall have its Axis greater
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              than to be unto the Semiparameter, in proportion as
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              fifteen to four, being demitted into the Liquid, ſo as
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1296"/>
                <lb/>
                <emph type="italics"/>
              that its Baſe touch not the ſame, it ſhall ſometimes
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                <lb/>
                <arrow.to.target n="marg1297"/>
                <lb/>
                <emph type="italics"/>
              ſtand perpendicular; ſometimes inclined; and ſome­
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              times ſo inclined, as that its Baſe touch the Surface
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              of the Liquid in one Point only, and that in two Po-
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1298"/>
                <lb/>
                <emph type="italics"/>
              ſitions; ſometimes ſo that its Baſe be more ſubmer-
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1299"/>
                <lb/>
                <emph type="italics"/>
              ged in the Liquid; and ſometimes ſo as that it doth
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              not in the leaſt touch the Surface of the Liquid;
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1300"/>
                <lb/>
                <emph type="italics"/>
              according to the proportion that it hath to the Liquid
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              in Gravity. </s>
              <s>Every one of which Caſes ſhall be anon
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              demonſtrated.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1296"/>
              A</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1297"/>
              B</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1298"/>
              C</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1299"/>
              D</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1300"/>
              E</s>
            </p>
            <p type="main">
              <s>Let there be a Portion, as hath been ſaid; and it being cut
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              thorow its Axis, by a Plane erect unto the Superficies of the
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              Liquid, let the Section be A P O L, the Section of a Right
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              angled Cone; and the Axis of the Portion and Diameter of the
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              Section B D: and let B D be cut in the Point K, ſo as that B K
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              be double of K D; and in C, ſo as that B D may have the ſame
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                <arrow.to.target n="marg1301"/>
                <lb/>
              proportion to K C, as fifteen to four: It is manifeſt, therefore,
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                <arrow.to.target n="marg1302"/>
                <lb/>
              that K C is greater than the Semi-parameter: Let the </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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