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

Table of figures

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      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="040/01/958.jpg" pagenum="265"/>
                <emph type="italics"/>
              I ſay now, that the Line E S is leſſer than K. </s>
              <s>For if not, then let C A
                <lb/>
              be ſuppoſed equal to E O. </s>
              <s>Becauſe therefore O E hath to K the ſame
                <lb/>
              proportion that L hath to X; and the inſcribed Figure is not leſs than
                <lb/>
              the Cylinder L; and the exceſs with which the ſaid Figure is exceeded
                <lb/>
              by the circumſcribed is leſs than the Solid X: therefore the inſcribed
                <lb/>
              Figure ſhall have to the ſaid exceſs
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.958.1.jpg" xlink:href="040/01/958/1.jpg" number="176"/>
                <lb/>
                <emph type="italics"/>
              greater proportion than O E hath to
                <lb/>
              K: But the proportion of O E to K is
                <lb/>
              not leſs than that which O E hath to
                <lb/>
              E S with E S. </s>
              <s>Let it not be leſs than
                <lb/>
              K. </s>
              <s>Therefore the inſcribed Figure
                <lb/>
              hath to the exceſs of the circumſcri­
                <lb/>
              bed Figure above it greater propor­
                <lb/>
              tion than O E hath to E S. </s>
              <s>Therefore
                <lb/>
              as the inſcribed is to the ſaid exceſs,
                <lb/>
              ſo ſhall it be to the Line E S. </s>
              <s>Let E R
                <lb/>
              be a Line greater than E O; and the
                <lb/>
              Center of Gravity of the inſcribed
                <lb/>
              Figure is S; and the Center of the cir­
                <lb/>
              cumſcribed is E. </s>
              <s>It is manifeſt there­
                <lb/>
              fore, that the Center of Gravity of
                <lb/>
              the remaining proportions by which
                <lb/>
              the circumſcribed exceedeth the in
                <lb/>
              ſcribed is in the Line R E, and in that point by which it is ſo termina­
                <lb/>
              ted, that as the inſcribed Figure is to the ſaid proportions, ſo is the Line
                <lb/>
              included betwixt E and that point to the Line E S. </s>
              <s>And this propor­
                <lb/>
              tion hath R E to E S. </s>
              <s>Therefore the Center of Gravity of the remain­
                <lb/>
              ing proportions with which the circumſcribed Figure exceeds the in­
                <lb/>
              ſcribed ſhall be R, which is impoſſible. </s>
              <s>For the Plane drawn thorow
                <lb/>
              R equidiſtant to the Baſe of the Cone doth not cut thoſe proportions. </s>
              <s>It
                <lb/>
              is therefore falſe that the Line E S is not leſſer than K. </s>
              <s>It ſhall therefore
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              be leſs. </s>
              <s>The ſame alſo may be done in a manner not unlike this in Pyra­
                <lb/>
              mides, as ne could demonſtrate.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>COROLLARY.</s>
            </p>
            <p type="main">
              <s>Hence it is manifeſt, that a given Cone may circumſcribe one
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              Figure and inſcribe another conſiſting of Cylinders of equal
                <lb/>
              Altitudes ſo, as that the Lines which are intercepted betwixt
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              their Centers of Gravity and the point which ſo divides the
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              Axis of the Cone, as that the part towards the Vertex is tri­
                <lb/>
              ple to the leſt, are leſs than any given Line.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              For, ſince it hath been demonſtrated, that the ſaid point dividing the
                <lb/>
              Axis, as was ſaid, is alwaies found betwixt the Centers of Gravity
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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