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="040/01/960.jpg" pagenum="267"/>
                <emph type="italics"/>
              M ſhall be the Center of Gravity of the remaining proportions by which
                <lb/>
              the Cone exceeds the inſcribed Figure. </s>
              <s>Which is impoſſible. </s>
              <s>Therefore
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              the Center of Gravity of the Cone is not below the point C. </s>
              <s>Nor is it
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              above it. </s>
              <s>For if it may be, let it be R. </s>
              <s>And again aſſume L P cut at
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              pleaſure in N: And as both B C and N P together are to N L, ſo let the
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              Cone be to X. </s>
              <s>And let a Figure be, in like manner, circumſcribed about
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              the Cone, which exceeds the ſaid Cone a leſs quantity than the Solid X.
                <lb/>
              </s>
              <s>And let the Line which intercepts bet wixt its Center of Gravity and C,
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              be leſſer than N P. </s>
              <s>Now take the circumſcribed Figure, whoſe Center
                <lb/>
              let be O; the remainder O R ſhall be greater than the ſaid N L. </s>
              <s>And
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              becauſe, as both together B C and P N is to N L, ſo is the Cone to X:
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              And the exceſs by which the circumſcribed exceeds the Cone is leſſer
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              than X: And B O is leſſer than B C and P N together: And O R grea­
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              ter than L N: The Cone therefore ſhall have much greater proportion to
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              the remaining proportions by which it was exceeded by the circumſcribed
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              Figure, than B O to O R. </s>
              <s>Let it be as M O is to O R. </s>
              <s>M O ſhall
                <lb/>
              be greater than B C; and M ſhall be the Center of Gravity of the pro­
                <lb/>
              portions by which the Cone is exceeded by the circumſcribed Figure.
                <lb/>
              </s>
              <s>Which is inconvenient. </s>
              <s>Therefore the Center of Gravity of the Cone is
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              not above the point C. </s>
              <s>But neither is it below it; as hath been proved.
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              </s>
              <s>Therefore it ſhall be C it ſelf. </s>
              <s>And ſo in like manner may it be demon­
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              ſtrated in any Pyramid.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>PROPOSITION.</s>
            </p>
            <p type="main">
              <s>If there were four Lines continual proportionals;
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              and as the leaſt of them were to the exceſs by
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              which the greateſt exceeds the leaſt, ſo a Line
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              taken at pleaſure ſhould be to 3/4 the exceſs by
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              which the greateſt exceeds the ſecond; and as
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              the Line equal to theſe (
                <emph type="italics"/>
              viz.
                <emph.end type="italics"/>
              to the greateſt,
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              double of the ſecond, and triple of the third)
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              is to the Line equal to the quadruple of the
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              fourth, the quadruple of the ſecond, and the
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              quadruple of the third, ſo ſhould another Line
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              taken be to the exceſs of the greateſt above the
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              ſecond: theſe two Lines taken together ſhall
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              be a fourth part of the greateſt of the propor­
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              tionals.</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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