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

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            <p type="main">
              <s>
                <pb xlink:href="040/01/959.jpg" pagenum="266"/>
                <emph type="italics"/>
              of the Circumſcribed and inſcribed Figures: and that it's poſſible, that
                <lb/>
              there be a Line in the middle betwixt thoſe Centers that is leſs than any
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              Line aſſigned; it followeth that the ſame given Line be much leſs that
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              lyeth betwixt one of the ſaid Centers and the ſaid point that divides
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              the Axis.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>PROPOSITION.</s>
            </p>
            <p type="main">
              <s>The Center of Gravity divideth the Axis of any
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              Cone or Pyramid ſo, that the part next the
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              Vertex is triple to the remainder.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let there be a Cone whoſe Axis is A B. </s>
              <s>And in C let it be divided,
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              ſo that A C be triple to the remaining part C B. </s>
              <s>It is to be proved,
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              that C is the Center of Gravity of the Cone. </s>
              <s>For if it be not, the
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              Cone's Center ſhall be either above or below the point C. </s>
              <s>Let it be firſt
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              beneath, and let it be E. </s>
              <s>And draw the Line L P, by it ſelf, equal to
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              C E; which divided at pleaſure in N. </s>
              <s>And as both B E and P N to­
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              gether are to P N, ſo let the Cone be to the Solid X: and inſcribe in the
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              Cone a Solid Figure of Cylinders that have equal Baſes, whoſe Center
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              of Gravity is leſs diſtant from the point C than is the Line L N, and
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              the exceſs of the Cone above it leſs than the Solid X. </s>
              <s>And that this
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              may be done is manifeſt from what hath been already demonſtrated.
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              </s>
              <s>Now let the inſcribed Figure be ſuch as
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.959.1.jpg" xlink:href="040/01/959/1.jpg" number="177"/>
                <lb/>
                <emph type="italics"/>
              was required, whoſe Center of Gravity
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              let be I. </s>
              <s>The Line I E therefore ſhall be
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              greater than N P together with L P. </s>
              <s>Let
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              C E and I C leſs L N be equal: And be­
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              cauſe both together B E and N P is to N P
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              as the Cone to X: and the exceſs by which
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              the Cone exceeds the inſcribed Figure is
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              leſs than the Solid X: Therefore the Cone
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              ſhall have greater proportion to the ſaid
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              X S than both B E and N P to N P: and, by
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              Diviſion, the inſcribed Figure ſhall have
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              greater proportion to the exceſs by which
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              the Cone exceeds it, than B E to N P: But B E hath leſs proportion to
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              E I than to N P with I E. </s>
              <s>Let N P be greater. </s>
              <s>Then the inſcribed Fi­
                <lb/>
              gure hath to the exceſs of the Cone above it much greater proportion
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              than B E to E I. </s>
              <s>Therefore as the inſcribed Figure is to the ſaid exceſs,
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              ſo ſhall a Line bigger than B E be to E I. </s>
              <s>Let that Line be M E. Becauſe,
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              therefore, M E is to E I as the inſcribed Figure is to the exceſs of the
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              Cone above the ſaid Figure, and D is the Center of Gravity of the
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              Cone, and I the Center of Gravity of the inſcribed Figure: Therefore
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
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