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

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="040/01/882.jpg" pagenum="189"/>
            <p type="head">
              <s>THEOR.
                <emph type="italics"/>
              XIX.
                <emph.end type="italics"/>
              PROP.
                <emph type="italics"/>
              XXX.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>If a Perpendicular be let fall from any point of the
                <lb/>
              Horizontal Line, and out of another point in
                <lb/>
              the ſame Horizontal Line a Plane be drawn
                <lb/>
              forth untill it meet the Perpendicular, along
                <lb/>
              which a Moveable deſcendeth in the ſhorteſt
                <lb/>
              time unto the ſaid Perpendicular, this Plane
                <lb/>
              ſhall be that which cutteth off a part equall to
                <lb/>
              the diſtance of the aſſigned point from the end
                <lb/>
              of the Perpendicular.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let the Perpendicular B D be let fall from the point B of the Ho­
                <lb/>
              rizontal Line A C, in which let there be any point C, and in the
                <lb/>
              Perpendicular let the Diſtance B E be ſuppoſed equal to the Di­
                <lb/>
              ſtance B C, and draw C E. </s>
              <s>I ſay, that of all Planes inclined out of
                <lb/>
              the point C till they meet the Perpendicular C E is that, along which
                <lb/>
              in the ſhorteſt of all Times the Deſcent
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.882.1.jpg" xlink:href="040/01/882/1.jpg" number="128"/>
                <lb/>
                <emph type="italics"/>
              is made unto the Perpendicular. </s>
              <s>For
                <lb/>
              let the Planes C F and C G be inclined
                <lb/>
              above and below, and draw I K a Tan­
                <lb/>
              gent unto the Semidiameter B C of the
                <lb/>
              deſcribed Circle in C, which ſhall be
                <lb/>
              equidiſtant from the Perpendicular;
                <lb/>
              and unto the ſaid C F let E K be Paral­
                <lb/>
              lel cutting the Circumference of the Cir­
                <lb/>
              cle in L: It is manifeſt that the Time of
                <lb/>
              the Deſcent along L E is equal to the
                <lb/>
              Time of the Deſcent along C E: But
                <lb/>
              the Time along K E is longer than along
                <lb/>
              L E: Therefore the Time along K E is
                <lb/>
              longer than that along C E: But the
                <lb/>
              Time along K E is equal to the Time a­
                <lb/>
              long C F, they being equal, and drawn
                <lb/>
              according to the ſame Inclination: Likewiſe ſince C G, and I E are
                <lb/>
              equal, and inclined according to the ſame Inclination, the Times of the
                <lb/>
              Motions along them ſhall be equal: But H E being ſhorter than I E, the
                <lb/>
              Time along it is alſo ſhorter than I E: Therefore the Time alſo along
                <lb/>
              C E, (which is equal to the Time along H E) ſhall be ſhorter than the
                <lb/>
              Time along I E: The Propoſition, therefore, is manifeſt.
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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