Gravesande, Willem Jacob 's, An essay on perspective

List of thumbnails

< >
91
91 		92
92
93
93 		94
94 (43)
95
95 (44) 		96
96 (45)
97
97 (46) 		98
98
99
99 		100
100
< >
page |< < (45) of 237 > >|
    <echo version="1.0RC">
      <text xml:lang="en" type="free">
        <div xml:id="echoid-div159" type="section" level="1" n="86">
          <p style="it">
            <s xml:id="echoid-s1141" xml:space="preserve">
              <pb o="45" file="0085" n="96" rhead="on PERSPECTIVE."/>
            and conſequently to V P. </s>
            <s xml:id="echoid-s1142" xml:space="preserve">Therefore if the before-
              <lb/>
            mentioned Plane be ſuppoſed to revolve upon the Line
              <lb/>
            V I, as an Axis, until it coincides with the Per-
              <lb/>
            ſpective Plane, the Center of the Sphere will meet
              <lb/>
            the Perſpective Plane in Q, and the Eye in F;
              <lb/>
            </s>
            <s xml:id="echoid-s1143" xml:space="preserve">whence the Part G E of the Line I V is the tranſ-
              <lb/>
            verſe Diameter of the Ellipſis.</s>
            <s xml:id="echoid-s1144" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s1145" xml:space="preserve">Again let G D E in Figure 30, and g e f, in
              <lb/>
              <note position="right" xlink:label="note-0085-01" xlink:href="note-0085-01a" xml:space="preserve">Fig. 30,
                <lb/>
              31.</note>
            Figure 31 repreſent the Points denoted with the ſame
              <lb/>
            Letters in the foregoing Figure. </s>
            <s xml:id="echoid-s1146" xml:space="preserve">Now if the Cone,
              <lb/>
            whoſe Profile is denoted by the Lines f g and fe be ſup-
              <lb/>
            poſed to be compleated, and to be cut by a Plane paſ-
              <lb/>
            ſing through the Line g e perpendicular to the Plane
              <lb/>
            of the Figure; </s>
            <s xml:id="echoid-s1147" xml:space="preserve">we ſhall have an Ellipſis g 4 e 3
              <lb/>
            ſimilar to that which is the ſought Repreſentation
              <lb/>
            of the Sphere. </s>
            <s xml:id="echoid-s1148" xml:space="preserve">Further if the ſaid Cone be conceived
              <lb/>
            to be cut by a Plane 14 m 3 parallel to its Baſe,
              <lb/>
            and biſecting g e in n, it is manifeſt, that 3 4, the
              <lb/>
            common Section of the Circle 14 m 3, and the Ellip-
              <lb/>
            ſis g 4 e 3, is the conjugate Axis of the Ellip-
              <lb/>
            ſis. </s>
            <s xml:id="echoid-s1149" xml:space="preserve">And therefore this conjugate Axis is equal
              <lb/>
            to the Line 3 4, Perpendicular in the Point n to the
              <lb/>
            Diameter 1 m of the Circle 14 m 3. </s>
            <s xml:id="echoid-s1150" xml:space="preserve">Now draw
              <lb/>
            the Lines E O and G Y in Figure 30, parallel to
              <lb/>
            L M, then the Triangles E G Y and E N M are
              <lb/>
            ſimilar, whence</s>
          </p>
        </div>
        <div xml:id="echoid-div161" type="section" level="1" n="87">
          <head xml:id="echoid-head93" xml:space="preserve">EG: EN:: GY: NM.</head>
          <p>
            <s xml:id="echoid-s1151" xml:space="preserve">But E G is twice E N; </s>
            <s xml:id="echoid-s1152" xml:space="preserve">wherefore G Y is alſo the
              <lb/>
            double of N M, and ſo N M equal to G Z. </s>
            <s xml:id="echoid-s1153" xml:space="preserve">After
              <lb/>
            the ſame manner we demonſtrate, that L N is equal
              <lb/>
            to X E; </s>
            <s xml:id="echoid-s1154" xml:space="preserve">whence it follows, that G D is equal to
              <lb/>
            L M, and is ſo cut in z as L M is in N; </s>
            <s xml:id="echoid-s1155" xml:space="preserve">and there-
              <lb/>
            fore R L or G T of Figure 29, is equal to 34 in
              <lb/>
            Figure 31; </s>
            <s xml:id="echoid-s1156" xml:space="preserve">and conſequently equal to the conjugate
              <lb/>
            Axis of the Ellipſis to be drawn. </s>
            <s xml:id="echoid-s1157" xml:space="preserve">On the other
              <lb/>
            Hand, it is manifeſt by Conſtruction, that ſome one
              <lb/>
            of the Perpendiculars m n, Figure 29, viz. </s>
            <s xml:id="echoid-s1158" xml:space="preserve">that
              <lb/>
            which paſſes through the Center of the </s>
          </p>
        </div>
      </text>
    </echo>
Impressum