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

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="040/01/945.jpg" pagenum="252"/>
            <p type="head">
              <s>COROLLARY.</s>
            </p>
            <p type="main">
              <s>Hence it is manifeſt, that a Conoid may be inſcribed in a Para­
                <lb/>
              bolical Figure, and another circumſcribed, ſo, as that the
                <lb/>
              Centers of their Gravities may be diſtant from the point N
                <lb/>
              leſs than any Line given.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              For if we aſſume a Line ſexcuple of the propoſed Line, and make the
                <lb/>
              Axis of the Cylinders, of which the Figures are compounded given
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              leſſer than this aſſumed Line, there ſhall fall Lines between the Centers
                <lb/>
              of Gravities of theſe Figures and the mark N that are leſs than the
                <lb/>
              Line propoſed.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>The former Propoſition another way.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let the Axis of the Conoid (which let be C D) be divided in
                <lb/>
              O, ſo, as that C O be double to O D. </s>
              <s>It is to be proved that the
                <lb/>
              Center of Gravity of the inſcribed Figure is in the Line O D;
                <lb/>
              and the Center of the circumſcribed in C O. </s>
              <s>Let the Plane of the Fi­
                <lb/>
              gures be cut through the Axis and C, as hath been ſaid. </s>
              <s>Becauſe there­
                <lb/>
              fore the Cylinders S N, T M, V I,
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.945.1.jpg" xlink:href="040/01/945/1.jpg" number="168"/>
                <lb/>
                <emph type="italics"/>
              X E are to one another as the Squares
                <lb/>
              of the Lines S D, T N, V M, X I;
                <lb/>
              and theſe are to one another as the
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              Lines N C, C M, C I, C E: but
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              theſe do exceed one another equally;
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              and the exceſs is equal to the leaſt, to
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              wit, C E: And the Cylinder T M is
                <lb/>
              equal to the Cylinder Q N; and the
                <lb/>
              Cylinder V I equal to P N; and X E
                <lb/>
              is equal to L N: Therefore the Cylin­
                <lb/>
              ders S N, Q N, P N, and L N do
                <lb/>
              equally exceed one another, and the
                <lb/>
              exceſs is equal to the leaſt of them,
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              namely, to the Cylinder L N. </s>
              <s>But
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              the exceſs of the Cylinder S N, above
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              the Cylinder Q N is a Ring whoſe
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              height is Q T; that is, N D; and
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              its breadth S
                <expan abbr="q.">que</expan>
              And the exceſs of the Cylinder Q N above P N, is a
                <lb/>
              Ring, whoſe breadth is Q P. </s>
              <s>And the exceſs of the Cylinder P N above
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              L N is a Ring, whoſe breadth is P L. </s>
              <s>Wherefore the ſaid Rings S Q,
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              Q P, P L, are equal to another, and to the Cylinder L N. </s>
              <s>Therefore the
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              Ring S T equalleth the Cylinder X E: the Ring Q V, which is double
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              to S T, equalleth the Cylinder V I; which likewiſe is double to the
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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