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

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="040/01/927.jpg" pagenum="234"/>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let the Altitude G F of the Parabola F H have the ſame proporti­
                <lb/>
              on to the Altitude C B of the Parabola B D, as the Sublimity B A
                <lb/>
              hath to the Sublimity F E. </s>
              <s>I ſay, that the Amplitude H G is equal
                <lb/>
              to the Amplitude D C. </s>
              <s>For ſince the firſt G F hath the ſame propor­
                <lb/>
              tion to the ſecond C B, as the third B A hath to the fourth F E; There­
                <lb/>
              fore, the Rectangle
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.927.1.jpg" xlink:href="040/01/927/1.jpg" number="158"/>
                <lb/>
                <emph type="italics"/>
              G F E of the firſt and
                <lb/>
              fourth, ſhall be equal to
                <lb/>
              the Rectangle C B A
                <lb/>
              of the ſecond and
                <lb/>
              third: Therefore the
                <lb/>
              Squares that are equal
                <lb/>
              to theſe Rectangles ſhall
                <lb/>
              be equal to one another:
                <lb/>
              But the Square of half of G H is equal to the Rectangle G F E; and
                <lb/>
              the Square of half of C D is equal to the Rectangle C B A: There­
                <lb/>
              fore theſe Squares, and their Sides, and the doubles of their Sides ſhall
                <lb/>
              be equal: But theſe are the Amplitudes G H and C D: Therefore the
                <lb/>
              Propoſition is manifeſt.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>LEMMA
                <emph type="italics"/>
              pro ſequenti.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>If a Right Line be cut according to any proportion, the Squares
                <lb/>
              of the Mean-proportionals between the whole and the two
                <lb/>
              parts are equal to the Square of the whole.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              Let A B be cut according to any proportion in C. </s>
              <s>I ſay, that the
                <lb/>
              Squares of the Mean-proportional Lines between the whole A B and
                <lb/>
              the parts A C and C B, being taken together are equal to the Square of
                <lb/>
              the whole A B. </s>
              <s>And this appeareth, a Semi-
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.927.2.jpg" xlink:href="040/01/927/2.jpg" number="159"/>
                <lb/>
                <emph type="italics"/>
              circle being deſcribed upon the whole Line
                <lb/>
              B A, and from C a Perpendicular being ere­
                <lb/>
              cted C D, and Lines being drawn from D to
                <lb/>
              A, and from D to B. </s>
              <s>For D A is the Mean­
                <lb/>
              proportional betwixt A B and A C; and D B is the Mean-proporti­
                <lb/>
              onal between A B and B C: And the Squares of the Lines D A and
                <lb/>
              D B taken together are equal to the Square of the whole Line A B,
                <lb/>
              the Angle A D B in the Semicircle being a Right-Angle: Therefore
                <lb/>
              the Propoſition is manifest.
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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