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

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="040/01/928.jpg" pagenum="235"/>
            <p type="head">
              <s>THEOR. VII. PROP. X.</s>
            </p>
            <p type="main">
              <s>The
                <emph type="italics"/>
              Impetus
                <emph.end type="italics"/>
              or Moment of any Semiparabola is
                <lb/>
              equal to the Moment of any Moveable falling
                <lb/>
              naturally along the Perpendicular to the Ho­
                <lb/>
              rizon that is equal to the Line compounded of
                <lb/>
              the Sublimity and of the Altitude of the Se­
                <lb/>
              miparabola.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let the Semiparabola be A B, its Sublimity D A, and Altitude
                <lb/>
              A C, of which the Perpendicular D C is compounded. </s>
              <s>I ſay, that
                <lb/>
              the
                <emph.end type="italics"/>
              Impetus
                <emph type="italics"/>
              of the Semiparabola in B is equal to the Moment of
                <lb/>
              the Moveable Naturally falling from D to C. </s>
              <s>Suppoſe D C it ſelf to be
                <lb/>
              the Meaſure of the Time and of the
                <emph.end type="italics"/>
              Impetus;
                <emph type="italics"/>
              and take a Mean-pro­
                <lb/>
              portional betwixt C D and D A, to which let
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.928.1.jpg" xlink:href="040/01/928/1.jpg" number="160"/>
                <lb/>
                <emph type="italics"/>
              C F be equal; and withal let C E be a Mean­
                <lb/>
              proportional between D C and C A: Now C F
                <lb/>
              ſhall be the Meaſure of the Time and of the Mo­
                <lb/>
              ment of the Moveable ſalling along D A out of
                <lb/>
              Reſt in D; and C E ſhall be the Time and Mo­
                <lb/>
              ment of the Moveable falling along A C, out of
                <lb/>
              Reſt in A, and the Moment of the Diagonal E F
                <lb/>
              ſhall be that compounded of both the others,
                <emph.end type="italics"/>
              ſcil.
                <lb/>
                <emph type="italics"/>
              that of the Semiparabola in B. </s>
              <s>And becauſe
                <lb/>
              D C is cut according to any proportion in A, and becauſe C F and C E
                <lb/>
              are Mean-Proportionals between C D and the parts D A and A C; the
                <lb/>
              Squares of them taken together ſhall be equal to the Square of the
                <lb/>
              whole; by the Lemma aforegoing: But the Squares of them are alſo
                <lb/>
              equal to the Square of E F: Therefore D F is equal alſo to the Line D C:
                <lb/>
              Whence it is manifeſt that the Moments along D C, and along the Se­
                <lb/>
              miparabola A B, are equal in C and B: Which was required.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>COROLLARY.</s>
            </p>
            <p type="main">
              <s>Hence it is manifeſt, that of all Parabola's whoſe Altitudes and
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              Sublimities being joyned together are equal, the
                <emph type="italics"/>
              Impetus's
                <emph.end type="italics"/>
              are
                <lb/>
              alſo equal.</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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