Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/064.jpg" pagenum="398"/>
                <figure id="id.073.01.064.1.jpg" xlink:href="073/01/064/1.jpg" number="67"/>
                <lb/>
                <emph type="italics"/>
              ſhall be equall to the Angle at
                <emph.end type="italics"/>
                <foreign lang="grc">φ;</foreign>
                <emph type="italics"/>
              and the Line B S
                <lb/>
              equall to the Line B C; and S R to C R: Where­
                <lb/>
              fore, M H ſhall be likewiſe equall to P Y. There­
                <lb/>
              fore, having drawn HK and prolonged it; the
                <lb/>
              Centre of Gravity of the whole Portion ſhall be
                <lb/>
              K; of that which is in the Liquid H; and of
                <lb/>
              that which is above it, the Centre ſhall be in
                <lb/>
              the Line prolonged: let it be in
                <emph.end type="italics"/>
                <foreign lang="grc">ω.</foreign>
                <emph type="italics"/>
              There­
                <lb/>
              fore, along that ſame Line K H, which is per­
                <lb/>
              pendicular to the Surface of the Liquid, ſhall
                <lb/>
              the part which is within the Liquid move up­
                <lb/>
              wards, and that which is above the Liquld
                <lb/>
              downwards: And, for this cauſe, the Portion,
                <lb/>
              ſhall be no longer moved, but ſhall ſtay, and
                <lb/>
              reſt, ſo, as that its Baſe do touch the Liquids Surface in but one Point; and its Axis
                <lb/>
              maketh an Angle therewith equall to the Angle
                <emph.end type="italics"/>
                <foreign lang="grc">φ</foreign>
                <emph type="italics"/>
              ; And, this is that which we were to
                <lb/>
              demonſtrate.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1391"/>
              F</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1392"/>
              (g)
                <emph type="italics"/>
              By 9 of t
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>CONCLVSION IV.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              If the Portion have greater proportion in Gravity
                <lb/>
              to the Liquid, than the Square F P to the Square
                <lb/>
              B D, but leſſer than that of the Square X O to the
                <lb/>
              Square B D, being demitted into the Liquid,
                <lb/>
              and inclined, ſo, as that its Baſe touch not the
                <lb/>
              Liquid, it ſhall ſtand and reſt, ſo, as that its Baſe
                <lb/>
              ſhall be more ſubmerged in the Liquid.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Again, let the Portion have greater proportion in
                <lb/>
              Gravity to the Liquid, than the Square F P to the
                <lb/>
              Square B D, but leſſer than that of the Square X O to
                <lb/>
              the Square B D; and as the Portion is in Gravity to the Liquid,
                <lb/>
              ſo let the Square made of the Line
                <foreign lang="grc">ψ</foreign>
              be to the Square B D.
                <foreign lang="grc">Ψ</foreign>
                <lb/>
              ſhall be greater than F P, and leſſer than X O. Apply, therefore,
                <lb/>
              the right Line I V to fall betwixt the Portions A V Q L and A X D;
                <lb/>
              and let it be equall to
                <foreign lang="grc">ψ,</foreign>
              and parallel to B D; and let it meet
                <lb/>
              the Remaining Section in Y: V Y ſhall alſo be proved double
                <lb/>
              to Y I, like as it hath been demonſtrated, that O G is double off
                <lb/>
              G X. And, draw from V, the Line V
                <foreign lang="grc">ω,</foreign>
              touching the Section
                <lb/>
              A V Q L in V; and drawing a Line from A to I, prolong it unto
                <lb/>
                <expan abbr="q.">que</expan>
              We prove in the ſame manner, that the Line A I is equall
                <lb/>
              to I
                <expan abbr="q;">que</expan>
              and that A Q is parallel to V
                <foreign lang="grc">ω.</foreign>
              It is to be demonſtrated,
                <lb/>
              that the Portion being demitted into the Liquid, and ſo inclined,
                <lb/>
              as that its Baſe touch not the Liquid, ſhall ſtand, ſo, that its Baſe
                <lb/>
              ſhall be more ſubmerged in the Liquid, than to touch it Surface in </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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