Salusbury, Thomas, Mathematical collections and translations (Tome I), 1667

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="040/01/1065.jpg" pagenum="371"/>
              Portion demitted into the Liquid, like as hath been ſaid, ſhall ſtand
                <lb/>
              enclined ſo as that its Axis do make an Angle with the Surface of
                <lb/>
              the Liquid equall unto the Angle E B
                <foreign lang="grc">Ψ.</foreign>
              For demit any Portion
                <lb/>
              into the Liquid ſo as that its Baſe
                <lb/>
                <figure id="id.040.01.1065.1.jpg" xlink:href="040/01/1065/1.jpg" number="265"/>
                <lb/>
              touch not the Liquids Surface;
                <lb/>
              and, if it can be done, let the
                <lb/>
              Axis not make an Angle with the
                <lb/>
              Liquids Surface equall to the
                <lb/>
              Angle E B
                <foreign lang="grc">Ψ</foreign>
              ; but firſt, let it be
                <lb/>
              greater: and the Portion being
                <lb/>
              cut thorow the Axis by a Plane e­
                <lb/>
              rect unto [
                <emph type="italics"/>
              or upon
                <emph.end type="italics"/>
              ] the Surface of
                <lb/>
              the Liquid, let the Section be A P
                <lb/>
              O L the Section of a Rightangled
                <lb/>
              Cone; the Section of the Surface of the Liquid X S; and let the
                <lb/>
              Axis of the Portion and Diameter of the Section be N O: and
                <lb/>
              draw P Y parallel to X S, and touching the Section A P O L in P;
                <lb/>
              and P M parallel to N O; and P I perpendicular to N O: and
                <lb/>
              moreover, let B R be equall to O
                <foreign lang="grc">ω,</foreign>
              and R K to T
                <foreign lang="grc">ω;</foreign>
              and let
                <foreign lang="grc">ω</foreign>
              H
                <lb/>
              be perpendicular to the Axis. </s>
              <s>Now becauſe it hath been ſuppoſed
                <lb/>
                <arrow.to.target n="marg1241"/>
                <lb/>
              that the Axis of the Portion doth make an Angle with the Surface
                <lb/>
              of the Liquid greater than the Angle B, the Angle P Y I ſhall be
                <lb/>
              greater than the Angle B: Therefore the Square P I hath greater
                <lb/>
                <arrow.to.target n="marg1242"/>
                <lb/>
              proportion to the Square Y I, than the Square E
                <foreign lang="grc">Ψ</foreign>
              hath to the
                <lb/>
              Square
                <foreign lang="grc">Ψ</foreign>
              B: But as the Square P I is to the Square Y I, ſo is the
                <lb/>
                <arrow.to.target n="marg1243"/>
                <lb/>
              Line K R unto the Line I Y; and as the Square E
                <foreign lang="grc">Ψ</foreign>
              is to the Square
                <lb/>
                <arrow.to.target n="marg1244"/>
                <lb/>
                <foreign lang="grc">Ψ</foreign>
              B, ſo is half of the Line K R unto the Line
                <foreign lang="grc">Ψ</foreign>
              B: Wherefore
                <lb/>
                <emph type="italics"/>
              (a)
                <emph.end type="italics"/>
              K R hath greater proportion to I Y, than the half of K R hath
                <lb/>
                <arrow.to.target n="marg1245"/>
                <lb/>
              to
                <foreign lang="grc">Ψ</foreign>
              B: And, conſequently, I Y isleſſe than the double of
                <foreign lang="grc">Ψ</foreign>
              B,
                <lb/>
              and is the double of O I: Therefore O I is leſſe than
                <foreign lang="grc">Ψ</foreign>
              B; and I
                <foreign lang="grc">ω</foreign>
                <lb/>
                <arrow.to.target n="marg1246"/>
                <lb/>
              greater than
                <foreign lang="grc">Ψ</foreign>
              R: but
                <foreign lang="grc">Ψ</foreign>
              R is equall to F: Therefore I
                <foreign lang="grc">ω</foreign>
              is greater
                <lb/>
                <arrow.to.target n="marg1247"/>
                <lb/>
              than F. </s>
              <s>And becauſe that the Portion is ſuppoſed to be in Gra­
                <lb/>
              vity unto the Liquid, as the Square F Q is to the Square B D; and
                <lb/>
              ſince that as the Portion is to the Liquid in Gravity, ſo is the part
                <lb/>
              thereof ſubmerged unto the whole Portion; and in regard that as
                <lb/>
              the part thereof ſubmerged is to the whole, ſo is the Square P M to
                <lb/>
              the Square O N; It followeth, that the Square P M is to the Square
                <lb/>
              N O, as the Square F Q is to the Square B D: And therefore F
                <lb/>
                <arrow.to.target n="marg1248"/>
                <lb/>
              Q is equall to P M: But it hath been demonſtrated that P H is
                <lb/>
                <arrow.to.target n="marg1249"/>
                <lb/>
              greater than F: It is manifeſt, therefore, that P M is leſſe than
                <lb/>
              ſeſquialter of P H: And conſequently that P H is greater than
                <lb/>
              the double of H M. </s>
              <s>Let P Z be double to Z M: T ſhall be the Cen­
                <lb/>
              tre of Gravity of the whole Solid; the Centre of that part of it
                <lb/>
              which is within the Liquid, the Point Z; and of the remaining
                <lb/>
                <arrow.to.target n="marg1250"/>
                <lb/>
              part the Centre ſhall be in the Line Z T prolonged unto G. </s>
              <s>In </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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