Gravesande, Willem Jacob 's, An essay on perspective

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          <p>
            <s xml:id="echoid-s1194" xml:space="preserve">
              <pb o="48" file="0090" n="102" rhead="An ESSAY"/>
            might be done) the Point on the Semicircle
              <lb/>
            h z m, as far as the Parallels (as 9 m) are uſeful:
              <lb/>
            </s>
            <s xml:id="echoid-s1195" xml:space="preserve">For when theſe Parallels are uſeleſs, the Point
              <lb/>
            q will fall beyond the Point m: </s>
            <s xml:id="echoid-s1196" xml:space="preserve">But then the
              <lb/>
            Perſpective of the Torus is entirely drawn alrea-
              <lb/>
            dy, if thoſe Parallels were firſt begun to be drawn
              <lb/>
            near to 6 3 z, and the others continually going
              <lb/>
            from it.</s>
            <s xml:id="echoid-s1197" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1198" xml:space="preserve">In order to demonſtrate this Problem, the fol-
              <lb/>
            lowing Lemma is neceſſary.</s>
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        <div xml:id="echoid-div168" type="section" level="1" n="90">
          <head xml:id="echoid-head96" xml:space="preserve">
            <emph style="sc">Lemma</emph>
          .</head>
          <p>
            <s xml:id="echoid-s1200" xml:space="preserve">65. </s>
            <s xml:id="echoid-s1201" xml:space="preserve">If two Circles C D H E and D E F L cut
              <lb/>
              <note position="left" xlink:label="note-0090-01" xlink:href="note-0090-01a" xml:space="preserve">Fig. 34.</note>
            each other, thro’ whoſe Centers C and B the Line
              <lb/>
            C L paſſes, and D E joyns their Interſections;
              <lb/>
            </s>
            <s xml:id="echoid-s1202" xml:space="preserve">then, if the Radius A C or A H be called a, and
              <lb/>
            B F or BL, b, and the Diſtance A B between the
              <lb/>
            two Centers c, I ſay A G is equal to {bb—aa/ec}—{1/2}C.</s>
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        <div xml:id="echoid-div170" type="section" level="1" n="91">
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            <emph style="sc">Demonstration</emph>
          .</head>
          <p style="it">
            <s xml:id="echoid-s1204" xml:space="preserve">Let us call A G, x, and G D or G E, y.
              <lb/>
            </s>
            <s xml:id="echoid-s1205" xml:space="preserve">Then by the Property of the Circle, if y be conceiv’d
              <lb/>
            as an Ordinate of the Circle, C D H; </s>
            <s xml:id="echoid-s1206" xml:space="preserve">yy=aa—xx. </s>
            <s xml:id="echoid-s1207" xml:space="preserve">
              <lb/>
            And if it be likewiſe conſider’d as an Ordinate of
              <lb/>
            the Circle F D L, yy=bb—cc—2cx—xx: </s>
            <s xml:id="echoid-s1208" xml:space="preserve">Whence
              <lb/>
            aa—xx=bb—cc—2cx—xx, and ſo 2cx=bb
              <lb/>
            —aa—cc; </s>
            <s xml:id="echoid-s1209" xml:space="preserve">and dividing each Side of this laſt Equa-
              <lb/>
            tion by 2c, we have a x={bb—aa/2c}{1/2} c. </s>
            <s xml:id="echoid-s1210" xml:space="preserve">Which was
              <lb/>
            to be Demonſtrated.</s>
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