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

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      <text>
        <body>
          <chap>
            <pb xlink:href="040/01/849.jpg" pagenum="156"/>
            <p type="head">
              <s>THEOR. IV. PROP. IV.</s>
            </p>
            <p type="main">
              <s>The Times of the Motions along equal Planes,
                <lb/>
              but unequally inclined, are to each other in
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              ſubduple proportion of the Elevations of thoſe
                <lb/>
              Planes Reciprocally taken.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let there proceed from the term B two equal Planes, but une­
                <lb/>
              qually inclined, B A and B C, and let A E and C D be Hori­
                <lb/>
              zontal Lines, drawn as far as the Perpendicular B D: Let the
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              Elevation of the Plane B A be B E; and let the Elevation of the
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              Plane B C be B D: And let B I be a Mean Proportional between the
                <lb/>
              Elevations D B and B E: It is manifeſt
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.849.1.jpg" xlink:href="040/01/849/1.jpg" number="91"/>
                <lb/>
                <emph type="italics"/>
              that the proportion of D B to B I, is ſub­
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              duple the proportion of D B to B E. </s>
              <s>Now
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              I ſay, that the proportion of the Times
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              of the Deſcents or Motions along the
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              Planes B A and B C, are the ſame with
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              the proportion of D B to B I Reciprocal­
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              ly taken: So that to the Time B A the
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              Elevation of the other Plane B C, that is
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              B D be Homologal; and to the Time along
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              B C, B I be Homologal: Therefore it is
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              to be demonſtrated, That the Time along B A is to the Time along
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              B C, as D B is to B I. </s>
              <s>Let I S be drawn equidiſtant from D C. </s>
              <s>And
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              becauſe it hath been demonſtrated that the Time of the Deſcent
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              along B A, is to the Time of the Deſcent along the Perpendicular
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              B E, as the ſaid B A is to B E; and the Time along B E is to the
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              Time along B D, as B E is to B I; and the Time along B D is to the
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              Time along B C, as B D to B C, or as B I to B S: Therefore,
                <emph.end type="italics"/>
              ex æqua­
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              li,
                <emph type="italics"/>
              the Time along B A ſhall be to the Time along B C as B A to B S,
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              or as C B to BS: But C B is to B S, as D B to B I: Therefore the
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              Propoſition is manifeſt:
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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