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

Table of figures

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      <text>
        <body>
          <chap>
            <pb xlink:href="040/01/949.jpg" pagenum="256"/>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let there be a Conoid whoſe Axis is A B, and the Center of the
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              circumſcribed Figure C, and the Center of the inſcribed O. </s>
              <s>I ſay
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              the Center of the Conoid is betwixt the points C and O. </s>
              <s>For if
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              not, it ſhall be either above them, or below them, or in one of them. </s>
              <s>Let
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              it be below, as in R. </s>
              <s>And becauſe R is the Center of Gravity of the
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              whole Conoid; and the Center of Gravity of the inſcribed Figure is O:
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              Therefore of the remaining proportions by which the Conoid exceeds
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              the inſcribed Figure the Center of Gravity ſhall be in the Line O R ex­
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              tended towards R, and in that point in which it is ſo determined, that,
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              what proportion the ſaid proportions have to the inſcribed Figure, the
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              ſame ſhall O R have to the Line falling betwixt R and that falling point.
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              </s>
              <s>Let this proportion be that of O R to R X. </s>
              <s>Therefore X falleth either
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              without the Conoid or within, or in its
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.949.1.jpg" xlink:href="040/01/949/1.jpg" number="170"/>
                <lb/>
                <emph type="italics"/>
              Baſe. </s>
              <s>That it falleth without, or in its
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              Baſe it is already manifeſt to be an abſur­
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              dity. </s>
              <s>Let it fall within: and becauſe X R
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              is to R O, as the inſcribed Figure is to
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              the exceſs by which the Conoid exceeds
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              it; the ſame proportion that B R hath to
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              R O, the ſame let the inſcribed Figure
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              have to the Solid K: Which neceſſarily
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              ſhall be leſſer than the ſaid exceſs. </s>
              <s>And let
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              another Figure be inſcribed which may be
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              exceeded by the Conoid a leſs quantity
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              than is K, whoſe Center of Gravity falleth betwixt O and C. </s>
              <s>Let it
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              be V. And, becauſe the firſt Figure is to K as B R to R O, and the ſe­
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              cond Figure, whoſe Center V is greater than the firſt, and exceeded
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              by the Conoid a leſs quantity than is K; what proportion the ſecond
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              Figure hath to the exceſs by which the Conoid exceeds it, the ſame
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              ſhall a Line greater than B R have to R V. </s>
              <s>But R is the Center of Gra­
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              vity of the Conoid; and the Center of the ſecond inſcribed Figure V:
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              The Center therefore of the remaining proportions ſhall be without
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              the Conoid beneath B: Which is impoſſible. </s>
              <s>And by the ſame means
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              we might demonſtrate the Center of Gravity of the ſaid Conoid not to
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              be in the Line C A. </s>
              <s>And that it is none of the points betwixt C and
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              O is manifeſt. </s>
              <s>For ſay, that there other Figures deſcribed, greater
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              ſomething than the inſcribed Figure whoſe Center is O, and leſs than
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              that circumſcribed Figure whoſe Center is C, the Center of the Conoid
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              would fall without the Center of theſe Figures: Which but now was
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              concluded to be impoſſible: It reſts therefore that it be betwixt the Cen­
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              ter of the circumſcribed and inſcribed Figure. </s>
              <s>And if ſo, it ſhall ne­
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              ceſſarily be in that point which divideth the Axis, ſo as that the part
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              towards the Vertex is double to the remainder; ſince N may circum­
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              ſcribe and inſcribe Figures, ſo, that thoſe Lines which fall between
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
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