Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/047.jpg" pagenum="377"/>
              and drawing a Line from Z to T pro­
                <lb/>
                <figure id="id.073.01.047.1.jpg" xlink:href="073/01/047/1.jpg" number="46"/>
                <lb/>
              long it unto G. </s>
              <s>The Centre of
                <lb/>
              Gravity of the whole Portion ſhall
                <lb/>
              be T; of that part which is above
                <lb/>
              the Liquid Z; and of the Remain­
                <lb/>
              der which is within the Liquid, the
                <lb/>
              Centre ſhall be in the Line Z T pro­
                <lb/>
              longed; let it be in G: It ſhall be
                <lb/>
              demonſtrated, as before, that T H
                <lb/>
              is perpendicular to the Surface of
                <lb/>
              the Liquid, and that the Lines
                <lb/>
              drawn thorow Z and G parallel to the ſaid T H, are alſo perpen­
                <lb/>
              diculars unto the ſame: Therefore, the Part which is above the
                <lb/>
              Liquid ſhall move downwards, along that which paſseth thorow Z;
                <lb/>
              and that which is within it, ſhall move upwards, along that which
                <lb/>
              paſseth thorow G: And, therefore, the Portion ſhall not remain
                <lb/>
              ſo inclined, nor ſhall ſo turn about, as that its Axis be perpendicular
                <lb/>
                <arrow.to.target n="marg1285"/>
                <lb/>
              unto the Surface of the Liquid; for the parts towards L ſhall move
                <lb/>
              downwards, and thoſe towards
                <emph type="italics"/>
              A
                <emph.end type="italics"/>
              upwards; as may appear by
                <lb/>
              the things already demonſtrated. </s>
              <s>And, if the Axis ſhould make
                <lb/>
              an Angle with the Surface of the Liquid, leſs than the Angle B;
                <lb/>
              it ſhall in like manner be demonſtrated, that the Portion will not
                <lb/>
                <arrow.to.target n="marg1286"/>
                <lb/>
              reſt, but incline untill that its Axis do make an Angle with the
                <lb/>
              Surface of the Liquid, equall to the Angle B.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1280"/>
              A</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1281"/>
              B</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1282"/>
              C</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1283"/>
              D</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1284"/>
              E</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1285"/>
              F</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1286"/>
              G</s>
            </p>
            <p type="head">
              <s>COMMANDINE.</s>
            </p>
            <p type="main">
              <s>And, therefore, the Square B D doth more exceed the Square
                <lb/>
                <arrow.to.target n="marg1287"/>
                <lb/>
              F Q, than doth the Square B C: And, conſequently, the Line
                <lb/>
              F Q, is leſs than B C; and F leſs than B R.]
                <emph type="italics"/>
              Becauſe the Exceſs by
                <lb/>
              which the Square B D exceedeth the Square B C; having leſs proportion unto the Square B D,
                <lb/>
              than the Exceſs by which the Square B D exceedeth the Square F Q, hath to the ſaid Square
                <emph.end type="italics"/>
              ;
                <lb/>
              (a)
                <emph type="italics"/>
              the Exceſs by which the Square B D exceedeth the Square B C ſhall be leſs than the Exceſs
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1288"/>
                <lb/>
                <emph type="italics"/>
              by which it exceedeth the Square F Q: Therefore, the Square F Q is leſs than the Square B C:
                <lb/>
              and, conſquently, the Line F Q leſs than the Line BC: But F Q hath the ſameproportion
                <lb/>
              to F, that B C hath to B R; for the Antecedents are each Seſquialter of their conſequents:
                <lb/>
              And
                <emph.end type="italics"/>
              (b)
                <emph type="italics"/>
              F Q being leſs than B C, F ſhall alſo be leſs than B R.
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1289"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1287"/>
              A</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1288"/>
              (a)
                <emph type="italics"/>
              By 8. of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1289"/>
              (b)
                <emph type="italics"/>
              By 14. of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>And, for that cauſe, K R hath greater proportion to S Y, than
                <lb/>
              the half of K R hath to
                <foreign lang="grc">ψ</foreign>
              B.]
                <emph type="italics"/>
              For K R is to S Y, as the Square P S is to the Square
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1290"/>
                <lb/>
                <emph type="italics"/>
              S Y: and the half of the Line K R is to the Line
                <emph.end type="italics"/>
                <foreign lang="grc">ψ</foreign>
                <emph type="italics"/>
              B, as the Square E
                <emph.end type="italics"/>
                <foreign lang="grc">ψ</foreign>
                <emph type="italics"/>
              is to the Square
                <emph.end type="italics"/>
                <foreign lang="grc">ψ</foreign>
                <emph type="italics"/>
              B.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1290"/>
              B</s>
            </p>
            <p type="main">
              <s>And S O leſs than
                <foreign lang="grc">ψ</foreign>
              B.]
                <emph type="italics"/>
              For S Y is double of S O.
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1291"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1291"/>
              C</s>
            </p>
            <p type="main">
              <s>And P H greater than F.]
                <emph type="italics"/>
              For P H is equall to S
                <emph.end type="italics"/>
                <foreign lang="grc">ω,</foreign>
                <emph type="italics"/>
              and R
                <emph.end type="italics"/>
                <foreign lang="grc">ψ</foreign>
                <emph type="italics"/>
              equall to F.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1292"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1292"/>
              D</s>
            </p>
            <p type="main">
              <s>And, therefore, the whole Portion ſhall have the ſame propor­</s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1293"/>
                <lb/>
              tion to that part which is above the Liquid, that the Square B D
                <lb/>
              hath to the Square F Q]
                <emph type="italics"/>
              Becauſe that the part ſubmerged, being to the whole Portion
                <lb/>
              as the Exceſs by which the Square B D is greater than the Square F Q, is to the Square B D;
                <lb/>
              the whole Portion, Converting, ſhall be to the part thereof ſubmerged, as the Square B D is to
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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