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

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="040/01/889.jpg" pagenum="196"/>
                <emph type="italics"/>
              Mean-proportional is A F, and between A B and B D, that is, R A and
                <lb/>
              A B the Mean is B F, to which F H is equal; Therefore,
                <emph.end type="italics"/>
              exprædemon­
                <lb/>
              ſtratis,
                <emph type="italics"/>
              the exceſſe A H ſhall be the Time along A B
                <emph.end type="italics"/>
              ex quiete
                <emph type="italics"/>
              in R, or
                <lb/>
              after the Fall out of X; ſince the Time along the ſaid A B
                <emph.end type="italics"/>
              ex quiete
                <emph type="italics"/>
              in
                <lb/>
              A, ſhall be A B. </s>
              <s>Therefore the Time along X A is I B; and along A B
                <lb/>
              after R A, or after X A, is A I: Therefore the Time along X A B ſhall
                <lb/>
              be as A B, namely the ſelf-ſame with the Time along the ſole A B
                <emph.end type="italics"/>
              ex qui­
                <lb/>
              ete
                <emph type="italics"/>
              in A. </s>
              <s>Which was the Propoſition.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>PROBL. XIV. PROP. XXXV.</s>
            </p>
            <p type="main">
              <s>An Inflected Line unto a given
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              erpendicular be­
                <lb/>
              ing aſſigned, to take part in the Inflected Line,
                <lb/>
              along which alone
                <emph type="italics"/>
              ex quiete
                <emph.end type="italics"/>
              a Motion may be
                <lb/>
              made in the ſame Time, as it would be along
                <lb/>
              the ſame together with the Perpendicular.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let the Perpendicular be A B, and a Line inflected to it B C. </s>
              <s>It is
                <lb/>
              required in B C to take a part, along which alone out of Reſt a
                <lb/>
              Motion may be made in the ſame Time as it would along the ſame
                <lb/>
              together with the Perpendicular A B. </s>
              <s>Draw the Horizon A D, with
                <lb/>
              which let the Inclined Line C B prolonged meet in E; and ſuppoſe B F
                <lb/>
              equal to B A, and on the Center E at the diſtance E F deſcribe the Circle
                <lb/>
              F I G; and continue out F E unto the Circumference in G; and as G B
                <lb/>
              is to B F, ſo let B H be to H F; and let H I touch the Circle in I. </s>
              <s>Then
                <lb/>
              out of B erect B K
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.889.1.jpg" xlink:href="040/01/889/1.jpg" number="134"/>
                <lb/>
                <emph type="italics"/>
              Perpendicular to
                <lb/>
              F C, with which
                <lb/>
              let the Line E I L
                <lb/>
              meet in L; and laſt
                <lb/>
              of all let fall L M
                <lb/>
              Perpendicular to E
                <lb/>
              L, meeting B C in
                <lb/>
              M. </s>
              <s>I ſay, that along
                <lb/>
              the Line B M from
                <lb/>
              Rest in B a Motion
                <lb/>
              may be made in the
                <lb/>
              ſame Time, as it
                <lb/>
              would be
                <emph.end type="italics"/>
              ex quiete
                <emph type="italics"/>
              in A along both A B and B M. </s>
              <s>Let E N be made
                <lb/>
              equal to E L. </s>
              <s>And becauſe as G B is to B F, ſo is B H to H F; there­
                <lb/>
              fore, by Permutation as G B is to B H, ſo will B F be to F H; and, by
                <lb/>
              Diviſion, G H ſhall be to H B, as B H is to H F: Wherefore the Rect­
                <lb/>
              angle G H F ſhall be equal to the Square H B: But the ſaid Rectangle
                <lb/>
              is alſo equal to the Square H I: Therefore B H is equal to the ſame H I.
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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