Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/062.jpg" pagenum="392"/>
              the Angles at X and N are equall. </s>
              <s>And, therefore, if drawing HK,
                <lb/>
              it be prolonged to
                <foreign lang="grc">ω,</foreign>
              the Centre of Gravity of the whole Portion ſhall
                <lb/>
              be K; of the part which is within the Liquid H; and of the part which
                <lb/>
              is above the Liquid in K
                <foreign lang="grc">ὠ</foreign>
              as ſuppoſe in
                <foreign lang="grc">ω;</foreign>
              and H K perpendicular to
                <lb/>
                <figure id="id.073.01.062.1.jpg" xlink:href="073/01/062/1.jpg" number="63"/>
                <lb/>
              the Surface of the Liquid. </s>
              <s>Therfore
                <lb/>
              along the ſame Right Lines ſhall the
                <lb/>
              part which is within the Liquid move
                <lb/>
              upwards, and the part above it down­
                <lb/>
              wards: And therfore the Portion
                <lb/>
              ſhall reſt with one of its Points
                <lb/>
              touching the Surface of the Liquid,
                <lb/>
              and its Axis ſhall make with the
                <lb/>
                <arrow.to.target n="marg1379"/>
                <lb/>
              ſame an Angle equall to X. </s>
              <s>It is
                <lb/>
              to be demonſtrated in the ſame
                <lb/>
              manner that the Portion that hath
                <lb/>
              the ſame proportion in Gravity to the Liquid, that the Square P F hath
                <lb/>
              to the Square B D, being demitted into the Liquid, ſo, as that its
                <lb/>
              Baſe touch not the Liquid, it ſhall ſtand inclined, ſo, as that its Baſe
                <lb/>
              touch the Surface of the Liquid in one Point only; and its Axis ſhall
                <lb/>
              make therwith an Angle equall to the Angle
                <foreign lang="grc">φ.</foreign>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1374"/>
              A</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1375"/>
              B</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1376"/>
              C</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1377"/>
              D</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1378"/>
              E</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1379"/>
              F</s>
            </p>
            <p type="head">
              <s>COMMANDINE.
                <lb/>
                <arrow.to.target n="marg1380"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1380"/>
              A</s>
            </p>
            <p type="main">
              <s>That is the Square T P to the Square B D.]
                <emph type="italics"/>
              By the twenty ſixth of the Book
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1381"/>
                <lb/>
                <emph type="italics"/>
              of
                <emph.end type="italics"/>
              Archimedes, De Conoidibus & Sphæroidibus:
                <emph type="italics"/>
              Therefore, (a) the Square T P
                <lb/>
              ſhall be equall to the Square X O: And for that reaſon, the Line T P equall to the
                <lb/>
              Line X O.
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1382"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1381"/>
              (a)
                <emph type="italics"/>
              By 9 of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1382"/>
              B</s>
            </p>
            <p type="main">
              <s>The Portions ſhall alſo be equall.]
                <emph type="italics"/>
              By the twenty fifth of the ſame Book.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1383"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1383"/>
              C</s>
            </p>
            <p type="main">
              <s>Again, becauſe that in the Equall and Like Portions, A O Q L
                <lb/>
              and A P M L.]
                <emph type="italics"/>
              For, in the Portion A P M L, deſcribe the Portion A O Q equall
                <lb/>
              to the Portion I P M: The Point Q falleth beneath M; for otherwiſe, the Whole would be
                <lb/>
              equall to the Part. </s>
              <s>Then draw I V parallel to A Q, and cutting the Diameter is
                <emph.end type="italics"/>
                <foreign lang="grc">ψ;</foreign>
                <emph type="italics"/>
              and
                <lb/>
              let I M cut the ſame
                <emph.end type="italics"/>
                <foreign lang="grc">ς;</foreign>
                <emph type="italics"/>
              and A Q in
                <emph.end type="italics"/>
                <foreign lang="grc">ς.</foreign>
                <emph type="italics"/>
              I ſay
                <lb/>
              that the Angle A
                <emph.end type="italics"/>
                <foreign lang="grc">υ</foreign>
                <emph type="italics"/>
              D, is leſſer than the Angle
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.073.01.062.2.jpg" xlink:href="073/01/062/2.jpg" number="64"/>
                <lb/>
                <emph type="italics"/>
              I
                <emph.end type="italics"/>
                <foreign lang="grc">σ</foreign>
                <emph type="italics"/>
              D. </s>
              <s>For the Angle I
                <emph.end type="italics"/>
                <foreign lang="grc">ψ</foreign>
                <emph type="italics"/>
              D is equall to the
                <lb/>
              Angle A
                <emph.end type="italics"/>
                <foreign lang="grc">υ</foreign>
                <emph type="italics"/>
              D: (b) But the interiour Angle
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1384"/>
                <lb/>
                <emph type="italics"/>
              I
                <emph.end type="italics"/>
                <foreign lang="grc">ψ</foreign>
                <emph type="italics"/>
              D is leſſer than the exteriour I
                <emph.end type="italics"/>
                <foreign lang="grc">σ</foreign>
                <emph type="italics"/>
              D: There-
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1385"/>
                <lb/>
                <emph type="italics"/>
              fore, (c) A
                <emph.end type="italics"/>
                <foreign lang="grc">υ</foreign>
                <emph type="italics"/>
              D ſhall alſo be lefter than I
                <emph.end type="italics"/>
                <foreign lang="grc">σ</foreign>
                <emph type="italics"/>
              D.
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1386"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1384"/>
              (b)
                <emph type="italics"/>
              By 29 of the
                <lb/>
              firſt.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1385"/>
                <emph type="italics"/>
              (c) By 16 of the
                <lb/>
              firſt.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1386"/>
              D</s>
            </p>
            <p type="main">
              <s>And the Angle at X, being leſſe
                <lb/>
              than the Angle at N.]
                <emph type="italics"/>
              Thorow O draw twe
                <lb/>
              Lines, O C perpendicular to the Diameter B D, and
                <lb/>
              O X touching the Section in the Point O, and cutting
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1387"/>
                <lb/>
                <emph type="italics"/>
              the Diameter in X: (d) O X ſhall be parallel
                <lb/>
              to A
                <expan abbr="q;">que</expan>
              and the
                <emph.end type="italics"/>
              (e)
                <emph type="italics"/>
              Angle at X, ſhall be equall to
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1388"/>
                <lb/>
                <emph type="italics"/>
              that at
                <emph.end type="italics"/>
                <foreign lang="grc">υ</foreign>
              :
                <emph type="italics"/>
              Therefore, the
                <emph.end type="italics"/>
              (f)
                <emph type="italics"/>
              Angle at X,
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1389"/>
                <lb/>
                <emph type="italics"/>
              ſhall be leſſer than the Angle at
                <emph.end type="italics"/>
                <foreign lang="grc">ς;</foreign>
                <emph type="italics"/>
              that is, to
                <lb/>
              that at N: And, conſequently, X ſhall fall beneath N: Therefore, the Line X B is greater than
                <lb/>
              N B. And, ſince B C is equall to X B, and B S equall to N B; B C ſhall be greater than B S.
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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