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="040/01/924.jpg" pagenum="231"/>
                <emph type="italics"/>
              Parabola G D. </s>
              <s>Let G M be a Mean-proportional betwixt A B and
                <lb/>
              G L; G M ſhall be the Time, and the Moment or
                <emph.end type="italics"/>
              Impetus
                <emph type="italics"/>
              in G of the
                <lb/>
              Project falling from L, (for it hath been ſuppoſed that A B is the Mea­
                <lb/>
              ſure of the Time and
                <emph.end type="italics"/>
              Impetus.)
                <emph type="italics"/>
              Again, let G N be a Mean-propor­
                <lb/>
              tional betwixt B C and C G: this G N ſhall be the Meaſure of the
                <lb/>
              Time and the
                <emph.end type="italics"/>
                <lb/>
              Impetus
                <emph type="italics"/>
              of the
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.040.01.924.1.jpg" xlink:href="040/01/924/1.jpg" number="156"/>
                <lb/>
                <emph type="italics"/>
              Project falling
                <lb/>
              from G to C.
                <lb/>
              </s>
              <s>If therefore a
                <lb/>
              Line be drawn
                <lb/>
              from M to N
                <lb/>
              it ſhall be the
                <lb/>
              the Meaſure of
                <lb/>
              the
                <emph.end type="italics"/>
              Impetus
                <emph type="italics"/>
              of
                <lb/>
              the Project a­
                <lb/>
              long the Para­
                <lb/>
              bola B D, ſcri­
                <lb/>
              king in the
                <lb/>
              term D. Which
                <emph.end type="italics"/>
                <lb/>
              Impetus,
                <emph type="italics"/>
              I ſay,
                <lb/>
              is greater than the
                <emph.end type="italics"/>
              Impetus
                <emph type="italics"/>
              of the Project along the Parabola B D,
                <lb/>
              whoſe quantity was A E. </s>
              <s>For becauſe G N is ſuppoſed the Mean-pro­
                <lb/>
              portional betwixt B C and C G, and B C is equal to B E, that is to H G;
                <lb/>
              (for they are each of them ſubduple to D C:) Therefore as C G is to
                <lb/>
              G N, ſo ſhall N G be to G K: and, as C G or H G is to G K, ſo ſhall the
                <lb/>
              Square N G be to the Square of G K: But as H G is to G K, ſo was
                <lb/>
              K G ſuppoſed to be to G L: Therefore as N G is to the Square G K, ſo
                <lb/>
              is K G to G L: But as K G is to G L, ſo is the Square K G unto the
                <lb/>
              Square G M, (for G M is the Mean between K G and G L:) Therefore
                <lb/>
              the three Squares N G, K G, and G M are continual proportionals: And
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              the two extream ones N G and G M taken together, that is the Square
                <lb/>
              M N is greater than double the Square K G, to which the Square A E
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              is double: Therefore the Square M N is greater than the Square A E:
                <lb/>
              and the Line M N greater than the Line A E: Which was to be de­
                <lb/>
              monſtrated.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>CORROLLARY I.</s>
            </p>
            <p type="main">
              <s>Hence it appeareth, that on the contrary, in the Project out of D
                <lb/>
              along the Semiparabola D B, leſs
                <emph type="italics"/>
              Impetus
                <emph.end type="italics"/>
              is required than
                <lb/>
              along any other according to the greater or leſſer Elevation
                <lb/>
              of the Semiparabola B D, which is according to the Tan­
                <lb/>
              gent A D, containing half a Right-Angle upon the Hori­
                <lb/>
              zon.</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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