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

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="040/01/850.jpg" pagenum="157"/>
            <p type="head">
              <s>THEOR. V. PROP. V.</s>
            </p>
            <p type="main">
              <s>The proportion of the Times of the Deſcents
                <lb/>
              along Planes that have different Inclinations
                <lb/>
              and Lengths, and the Elivations unequal, is
                <lb/>
              compounded of the proportion of the Lengths
                <lb/>
              of thoſe Planes, and of the ſubduple proporti­
                <lb/>
              on of their Elevations Reciprocally taken.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let A B and A C be Planes inclined after different manners,
                <lb/>
              whoſe Lengths are unequal, as alſo their Elevations. </s>
              <s>I ſay,
                <lb/>
              the proportion of the Time of the Deſcent along A C to the
                <lb/>
              Time along A B, is compounded of the proportion of the ſaid A C
                <lb/>
              to A B, and of the ſubduple proportion of their Elevation Recipro­
                <lb/>
              cally taken. </s>
              <s>For let the Perpendicular A D be drawn, with which
                <lb/>
              let the Horizontal Lines B G and C D interſect, and let A L be a
                <lb/>
              Mean-proportional between C A and A E; and from the point L let
                <lb/>
              a Parallel be drawn to the Horizon interſecting
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.850.1.jpg" xlink:href="040/01/850/1.jpg" number="92"/>
                <lb/>
                <emph type="italics"/>
              the Plane A C in F; and A F ſhall be a Mean
                <lb/>
              proportional between C A and A E. </s>
              <s>And becauſe
                <lb/>
              the Time along A C is to the Time along A E, as
                <lb/>
              the Line F A to A E; and the Time along A E is
                <lb/>
              to the Time along A B, as the ſaid A E to the ſaid
                <lb/>
              A B: It is manifeſt that the Time along A C is to
                <lb/>
              the Time along A B, as A F to A B. </s>
              <s>It remaineth,
                <lb/>
              therefore, to be demonſtrated, that the proportion
                <lb/>
              of A F to A B is compounded of the proportion of
                <lb/>
              C A to A B, and of the proportion of G A to A L;
                <lb/>
              which is the ſubduple proportion of the Elevati­
                <lb/>
              ons D A and A G Reciprocally taken. </s>
              <s>But that is manifeſt, C A
                <lb/>
              being put between F A and A B: For the proportion of F A to A C
                <lb/>
              is the ſame as that of L A to A D, or of G A to A L; which is ſub­
                <lb/>
              duple of the proportion of the Elevations G A and A D; and the
                <lb/>
              proportion of C A to A B is the proportion of the Lengths: Therefore
                <lb/>
              the Propoſition is manifeſt.
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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