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

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="040/01/956.jpg" pagenum="263"/>
            <p type="head">
              <s>PROPOSITION.</s>
            </p>
            <p type="main">
              <s>If to any Cone or portion of a Cone a Eigure con­
                <lb/>
              ſiſting of Cylinders of equal heights be inſcri­
                <lb/>
              bed and another circumſcribed; and if its Axis
                <lb/>
              be ſo divided as that the part which lyeth be­
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              twixt the point of diviſion and the Vertex be
                <lb/>
              triple to the reſt; the Center of Gravity of
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              the inſcribed Figure ſhall be nearer to the Baſe
                <lb/>
              of the Cone than that point of diviſion: and
                <lb/>
              the Center of Gravity of the circumſcribed
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              ſhall be nearer to the Vertex than that ſame
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              point.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Take therefore a Cone, whoſe Axis is N M. </s>
              <s>Let it be divided
                <lb/>
              in S ſo, as that N S be triple to the remainder S M. </s>
              <s>I ſay, that
                <lb/>
              the Center of Gravity of any Figure inſcribed, as was ſaid, in
                <lb/>
              a Cone doth conſiſt in the Axis N M, and approacheth nearer to the Baſe
                <lb/>
              of the Cone than the point S: and that the Center of Gravity of the
                <lb/>
              Circumſcribed is likewiſe in the Axis N M, and nearer to the Vertex
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              than is S. </s>
              <s>Let a Figure therefore be ſuppoſed to be inſcribed by the Cy­
                <lb/>
              linders whoſe Axis M C, C B, B E, E A are equal. </s>
              <s>Firſt therefore
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              the Cylinder whoſe Axis is M C hath
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.956.1.jpg" xlink:href="040/01/956/1.jpg" number="175"/>
                <lb/>
                <emph type="italics"/>
              to the Cylinder whoſe Axis is C B the
                <lb/>
              ſame proportion as its Baſe hath to
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              the Baſe of the other (for their Alti­
                <lb/>
              tudes are equal.) But this propor­
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              tion is the ſame with that which the
                <lb/>
              Square C N hath to the Square N B.
                <lb/>
              </s>
              <s>And ſo we might prove, that the Cy­
                <lb/>
              linder whoſe Axis is C B hath to the
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              Cylinder whoſe Axis is B E the ſame
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              proportion, as the Square B N hath to
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              the Square N E: and the Cylinder
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              whoſe Axis is B E hath to the Cylin­
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              der whoſe Axis is E A the ſame pro­
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              portion that the Square E N hath to
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              the Square N A. </s>
              <s>But the Lines N C,
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              N B, E N, and N A equally exceed one
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              another, and their exceſs equalleth the
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              leaſt, that is N A. </s>
              <s>Therefore they are certain Magnitudes, to wit, in­
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              ſcribed Cylinders having conſequently to one another the ſame proporti­
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              on as the Squares of Lines that equally exceed one another, and the ex-
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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