Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/046.jpg" pagenum="376"/>
              and F leſs than B R. </s>
              <s>Let R
                <foreign lang="grc">Ψ</foreign>
              be equall to F; and draw
                <foreign lang="grc">Ψ</foreign>
              E
                <lb/>
              perpendicular to B D; which let be in power the half of that
                <lb/>
              which the Lines K R and
                <foreign lang="grc">Ψ</foreign>
              B containeth; and draw a Line from
                <lb/>
              B to E: I ſay that the Portion demitted into the Liquid, ſo as that
                <lb/>
              its Baſe be wholly within the Liquid, ſhall ſo ſtand, as that its Axis
                <lb/>
              do make an Angle with the Liquids Surface, equall to the Angle B.
                <lb/>
              </s>
              <s>For let the
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion be demitted into the Liquid, as hath been ſaid;
                <lb/>
              and let the Axis not make an Angle with the Liquids Surface, equall
                <lb/>
              to B, but firſt a greater: and the ſame being cut thorow the Axis
                <lb/>
              by a Plane erect unto the Surface of the Liquid, let the Section of
                <lb/>
              the Portion be A P O L, the Section of a Rightangled Cone; the
                <lb/>
              Section of the Surface of the Liquid
                <foreign lang="grc">Γ</foreign>
              I; and the Axis of the
                <lb/>
              Portion and Diameter of the Section N O; which let be cut in
                <lb/>
              the Points
                <foreign lang="grc">ω</foreign>
              and T, as before: and draw Y P, parallelto
                <foreign lang="grc">Γ</foreign>
              I, and
                <lb/>
              touching the Section in P, and MP parallel to N O, and P S perpen­
                <lb/>
              dicular to the Axis. </s>
              <s>And becauſe now that the Axis of the Portion
                <lb/>
              maketh an
                <emph type="italics"/>
              A
                <emph.end type="italics"/>
              ngle with the Liquids Surface greater than the Angle
                <lb/>
              B, the Angle S Y P ſhall alſo be greater than the Angle B: And,
                <lb/>
              therefore, the Square P S hath greater proportion to the Square
                <lb/>
                <arrow.to.target n="marg1281"/>
                <lb/>
              S Y, than the Square
                <foreign lang="grc">Ψ</foreign>
              E hath to the Square
                <foreign lang="grc">Ψ</foreign>
              B: And, for that
                <lb/>
              cauſe, K R hath greater proportion to S Y, than the half of K R
                <lb/>
              hath to
                <foreign lang="grc">Ψ</foreign>
              B: Therefore, S Y is leſs than the double of
                <foreign lang="grc">Ψ</foreign>
              B; and
                <lb/>
                <arrow.to.target n="marg1282"/>
                <lb/>
              S O leſs than
                <foreign lang="grc">ψ</foreign>
              B:
                <emph type="italics"/>
              A
                <emph.end type="italics"/>
              nd, therefore, S
                <foreign lang="grc">ω</foreign>
              is greater than R
                <foreign lang="grc">ψ</foreign>
              ; and
                <lb/>
                <arrow.to.target n="marg1283"/>
                <lb/>
              P H greater than F.
                <emph type="italics"/>
              A
                <emph.end type="italics"/>
              nd, becauſe that the
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion hath the
                <lb/>
              ſame proportion in Gravity unto the Liquid, that the Exceſs by
                <lb/>
              which the Square B D, is greater than the Square F Q, hath unto
                <lb/>
              the Square B D; and that as the Portion is in proportion to the
                <lb/>
              Liquid in Gravity, ſo is the part thereof ſubmerged unto the whole
                <lb/>
              Portion; It followeth that the part ſubmerged, hath the ſame
                <lb/>
              proportion to the whole
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion, that the Exceſs by which the
                <lb/>
              Square B D is greater than the Square F Q hath unto the Square
                <lb/>
              B D:
                <emph type="italics"/>
              A
                <emph.end type="italics"/>
              nd, therefore, the whole
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion ſhall have the ſame propor­
                <lb/>
                <arrow.to.target n="marg1284"/>
                <lb/>
              tion to that part which is above the
                <lb/>
                <figure id="id.073.01.046.1.jpg" xlink:href="073/01/046/1.jpg" number="45"/>
                <lb/>
              Liquid, that the Square B D hath to
                <lb/>
              the Square F Q: But as the whole
                <lb/>
              Portion is to that part which is above
                <lb/>
              the Liquid, ſo is the Square N O unto
                <lb/>
              the Square P M: Therefore, P M
                <lb/>
              ſhall be equall to F Q: But it
                <lb/>
              hath been demonſtrated, that P H is
                <lb/>
              greater than F. And, therefore,
                <lb/>
              MH ſhall be leſs than
                <expan abbr="q;">que</expan>
              and P H
                <lb/>
              greater than double of H M. </s>
              <s>Let
                <lb/>
              therefore, P Z be double to Z M: </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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