Archimedes, Natation of bodies, 1662

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="073/01/063.jpg" pagenum="397"/>
            <p type="margin">
              <s>
                <margin.target id="marg1387"/>
                <emph type="italics"/>
              (d) By 5 of our ſe­
                <lb/>
              cond of
                <emph.end type="italics"/>
              Conicks.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1388"/>
              (e)
                <emph type="italics"/>
              By 29 of the
                <lb/>
              firſt.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1389"/>
              (f)
                <emph type="italics"/>
              By 39 of our
                <lb/>
              firſt of
                <emph.end type="italics"/>
              Conicks.</s>
            </p>
            <p type="main">
              <s>Therefore, A Q and A M do make equall Acute Angles with
                <lb/>
                <arrow.to.target n="marg1390"/>
                <lb/>
              the Diameters of the Portions.]
                <emph type="italics"/>
              We demonſtrate this as in the Commentaries
                <lb/>
              upon the ſecond Concluſion.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1390"/>
              E</s>
            </p>
            <p type="main">
              <s>It is to be demonſtrated in the ſame manner, that the Portion
                <lb/>
                <arrow.to.target n="marg1391"/>
                <lb/>
              that hath the ſame proportion in Gravity to the Liquid, that the
                <lb/>
              Square P F hath to the Square B D,
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              being demitted into the Liquid, ſo,
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                <figure id="id.073.01.063.1.jpg" xlink:href="073/01/063/1.jpg" number="65"/>
                <lb/>
              as that its Baſe touch not the Li­
                <lb/>
              quid, it ſhall ſtand inclined, ſo, as
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              that its Baſe touch the Surface of the
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              Liquid in one point only; and its Axis
                <lb/>
              ſhall make therewith an angle equall
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              to the Angle
                <foreign lang="grc">φ.</foreign>
              ]
                <emph type="italics"/>
              Let the Portion be to the
                <lb/>
              Liquid in Gravity, as the Square P F to the
                <lb/>
              Square B D: and being demitted into the
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              Liquid, ſo inclined, as that its Baſe touch not
                <lb/>
              the Liquid, let it be cut thorow the Axis by a
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              Plane erect to the Surface of the Liquid, that
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              that the Section may be A M O L, the Section
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              of a Rightangled Cone; and, let the Section of the Liquids Surface be I O; and the Axit
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              of the Portion and Diameter of the Section B D; which let be cut into the ſame parts as
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              we ſaid before, and draw M N parallel to I O, that it may touch the Section in the Point
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              M; and M T parallel to B D, and P M S perpe ndicular to the ſame. </s>
              <s>It is to be demon­
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              strated, that the Portion ſhall not reſt, but ſhall incline, ſo, as that it touch the Liquids
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              Surface, in one Point of its Baſe only. </s>
              <s>For,
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.073.01.063.2.jpg" xlink:href="073/01/063/2.jpg" number="66"/>
                <lb/>
                <emph type="italics"/>
              draw P C perpendicular to B D; and drawing
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              a Line from A to F, prolong it till it meet with
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              the Section in
                <expan abbr="q;">que</expan>
              and thorow P draw P
                <emph.end type="italics"/>
                <foreign lang="grc">φ</foreign>
                <emph type="italics"/>
              pa­
                <lb/>
              rallel to A Q: Now, by the things allready de­
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              monſtrated by us, A F and F Q ſhall be equall
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              to one another. </s>
              <s>And being that the Portion hath
                <lb/>
              the ſame proportion in Gravity unto the Liquid,
                <lb/>
              that the Square P F hath to the Square B D; and
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              ſeeing that the part ſubmerged, hath the ſame pro-
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1392"/>
                <lb/>
                <emph type="italics"/>
              partion to the whole Portion; that is, the Squàre
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              M T to the Square B D; (g) the Square M T
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              ſhall be equall to the Square P F; and, by the
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              ſame reaſon, the Line M T equall to the Line
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              P F. </s>
              <s>So that there being drawn in the equall & like
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              portions A P Q Land A M O L, the Lines A Q and I O which cut off equall Portions, the
                <lb/>
              firſt from the Extreme term of the Baſe, the laſt not from the Extremity; it followeth, that
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              A Q drawn from the Extremity, containeth a leſſer Acute Angle with the Diameter of the
                <lb/>
              Portion, than I O: But the Line P
                <emph.end type="italics"/>
                <foreign lang="grc">φ</foreign>
                <emph type="italics"/>
              is parallel to the Line A Q, and M N to I O: There­
                <lb/>
              fore, the Angle at
                <emph.end type="italics"/>
                <foreign lang="grc">φ</foreign>
                <emph type="italics"/>
              ſhall be leſſer than the Angle at N; but the Line B C greater than B S;
                <lb/>
              and S R, that is, M X, greater than C R, that is, than P Y: and, by the ſame reaſon, X T
                <lb/>
              leſſer than Y F. And, ſince P Y is double to Y F, M X ſhall be greater than double to
                <lb/>
              Y F, and much greater than double of X T. </s>
              <s>Let M H be double to H T, and draw a
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              Line from H to K, prolonging it. </s>
              <s>Now, the Centre of Gravity of the whole Portion
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              ſhall be the Point K; of the part within the Liquid H; and of the Remaining part above
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              the Liquid in the Line H K produced, as ſuppoſe in
                <emph.end type="italics"/>
                <foreign lang="grc">ω</foreign>
                <emph type="italics"/>
              It ſhall be demonſtrated in the ſame
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              manner, as before, that both the Line K H and thoſe that are drawn thorow the Points H
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              and
                <emph.end type="italics"/>
                <foreign lang="grc">ω</foreign>
                <emph type="italics"/>
              parallel to the ſaid K H, are perpendicular to the Surface of the Liquid: The
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              Portion therefore, ſhall not reſt; but when it ſhall be enclined ſo far as to touch the Sur­
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              face of the Liquid in one Point and no more, then it ſhall ſtay. </s>
              <s>For the Angle at N
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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