Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/058.jpg" pagenum="388"/>
              Angles at Y and G being equall; therefore the Lines Y B and G B,
                <lb/>
              and B C and B S ſhall alſo be equall: And therefore C R and S R,
                <lb/>
              and M V and P Z, and V N and Z T, ſhall be equall likewiſe.
                <lb/>
                <arrow.to.target n="marg1356"/>
                <lb/>
              Since therefore M V is Leſſer than double of V N, it is manifeſt that
                <lb/>
              P Z is leſſer than double of Z T.
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              et P
                <foreign lang="grc">ω</foreign>
              be double of
                <foreign lang="grc">ω</foreign>
              T; and
                <lb/>
              drawing a
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine from
                <foreign lang="grc">ω</foreign>
              to K, prolong it to E. </s>
              <s>Now the Centre of
                <lb/>
              Gravity of the whole Portion ſhall be the point K; and the Centre
                <lb/>
              of that part which is in the Liquid ſhall be
                <foreign lang="grc">ω,</foreign>
              and of that which is
                <lb/>
              above the Liquid ſhall be in the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine K E, which let be E: But the
                <lb/>
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine K Z ſhall be perpendicular unto the Surface of the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid:
                <lb/>
              And therefore alſo the Lines drawn thorow the Points E and
                <foreign lang="grc">ω</foreign>
              parall­
                <lb/>
                <arrow.to.target n="marg1357"/>
                <lb/>
              lell unto K Z, ſhall be perpendicular sunto the ſame: Therefore the
                <lb/>
              Portion ſhall not abide, but ſhall turn about ſo, as that its
                <emph type="italics"/>
              B
                <emph.end type="italics"/>
              aſe
                <lb/>
              do not in the leaſt touch the Surface of the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquid; in regard that
                <lb/>
              now when it toucheth in but one Point only, it moveth upwards, on
                <lb/>
                <arrow.to.target n="marg1358"/>
                <lb/>
              the part towards A: It is therefore perſpicuous, that the Portion
                <lb/>
              ſhall conſiſt ſo, as that its Axis ſhall make an Angle with the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              iquids
                <lb/>
              Surface greater than the Angle X.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1350"/>
              B</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1351"/>
              C</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1352"/>
              D</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1353"/>
              E F</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1354"/>
              G</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1355"/>
              H</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1356"/>
              K</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1357"/>
              L</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1358"/>
              M</s>
            </p>
            <p type="head">
              <s>COMMANDINE.
                <lb/>
                <arrow.to.target n="marg1359"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1359"/>
              A</s>
            </p>
            <p type="main">
              <s>If the Portion have leſſer proportion in Gravity to the Liquid,
                <lb/>
              than the Square S B hath to the Square B D, but greater than the
                <lb/>
              Square X O hath to the Square B D.]
                <emph type="italics"/>
              This is the ſecond part of the Tenth
                <lb/>
              propoſition; and the other pat is with their Demonſtrations, ſhall hereafter follow in the ſame Order.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <foreign lang="grc">Ψ</foreign>
              ſhall be greater than
                <emph type="italics"/>
              X
                <emph.end type="italics"/>
              O, but leſſer than the Exceſs by </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1360"/>
                <lb/>
              which the Axis is greater than Seſquialter of the Semi-parameter,
                <lb/>
              that is than S B.]
                <emph type="italics"/>
              This followeth from the 10 of the fifth Book of
                <emph.end type="italics"/>
              Euclids
                <emph type="italics"/>
              Elements.
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1361"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1360"/>
              B</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1361"/>
              C</s>
            </p>
            <p type="main">
              <s>It ſhall be demonſtrated, that M H is double to H N, like as it
                <lb/>
              was demonſtrated, that O G is double to G X.]
                <emph type="italics"/>
              As in the firſt Concluſion
                <lb/>
              of this Propoſition, and from what we have but even now written, thereupon appeareth:
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1362"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1362"/>
              D</s>
            </p>
            <p type="main">
              <s>For in regard that in the like Portions A M Q L and A X D, the
                <lb/>
              Lines A Q and A N are drawn from the Baſes unto the Portions,
                <lb/>
              which Lines contain equall Angles with the ſaid Baſes, Q A ſhall
                <lb/>
              have the ſame proportion to A N, that L A hath to A D.]
                <lb/>
                <emph type="italics"/>
              This we have demonstrated above.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1363"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1363"/>
              E</s>
            </p>
            <p type="main">
              <s>Therefore A N is equall to N Q]
                <emph type="italics"/>
              For ſince that Q A is to A N, as L A to
                <lb/>
              A D; Dividing and Converting, A N ſhall be to N Q as A D to D L: But A D
                <lb/>
              is equall to D L; for that D B is ſuppoſed to be the Diameter of the Portion: Therefore
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1364"/>
                <lb/>
                <emph type="italics"/>
              alſo
                <emph.end type="italics"/>
              (a)
                <emph type="italics"/>
              A N is equall to N
                <expan abbr="q.">que</expan>
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1364"/>
              (a)
                <emph type="italics"/>
              By 14 of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>And A Q parallel to M Y.]
                <emph type="italics"/>
              By the fifth of the ſecond Book of
                <emph.end type="italics"/>
              Apollonius
                <emph type="italics"/>
              his Conicks.
                <emph.end type="italics"/>
                <lb/>
              </s>
              <s>
                <arrow.to.target n="marg1365"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1365"/>
              F</s>
            </p>
            <p type="main">
              <s>And let B D be cut in the Points K and R as hath been ſaid.] </s>
            </p>
            <p type="main">
              <s>
                <arrow.to.target n="marg1366"/>
                <lb/>
                <emph type="italics"/>
              In the firſt Conciuſion of this Propoſition: And let it be cut in K, ſo, as that B K be double to
                <lb/>
              K D, and in R ſo, as that K R may be equall to the Semi-parameter.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1366"/>
              G</s>
            </p>
            <p type="main">
              <s>And, ſeeing that in the Equall and Like Portions A P O L and
                <lb/>
                <arrow.to.target n="marg1367"/>
                <lb/>
              A
                <emph type="italics"/>
              M
                <emph.end type="italics"/>
              Q L, the Lines A O and A Q are drawn from the Extremities
                <lb/>
              of their Baſes, ſo, as that the Portions cut off, do make equall Angles </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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