Apollonius <Pergaeus>; Lawson, John, The two books of Apollonius Pergaeus, concerning tangencies, as they have been restored by Franciscus Vieta and Marinus Ghetaldus : with a supplement to which is now added, a second supplement, being Mons. Fermat's Treatise on spherical tangencies

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        <div xml:id="echoid-div78" type="section" level="1" n="74">
          <pb o="[19]" file="0089" n="96"/>
          <p>
            <s xml:id="echoid-s1894" xml:space="preserve">
              <emph style="sc">Synthesis</emph>
            . </s>
            <s xml:id="echoid-s1895" xml:space="preserve">Make EB equal to R, EC equal to S, and deſcribe on
              <lb/>
            BC a circle; </s>
            <s xml:id="echoid-s1896" xml:space="preserve">erect at E the perpendicular ED, meeting the periphery of the
              <lb/>
            circle in D; </s>
            <s xml:id="echoid-s1897" xml:space="preserve">alſo at A erect the perpendicular AF equal to R; </s>
            <s xml:id="echoid-s1898" xml:space="preserve">draw AD,
              <lb/>
            which produce, if neceſſary, to cut the indefinite line, as in O, which will
              <lb/>
            be the point required.</s>
            <s xml:id="echoid-s1899" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1900" xml:space="preserve">For becauſe of the ſimilar triangles AOF, EOD, AO is to EO as AF
              <lb/>
            (R) is to DE; </s>
            <s xml:id="echoid-s1901" xml:space="preserve">therefore the ſquare on AO is to the ſquare on EO as the
              <lb/>
            ſquare on R is to the ſquare on DE (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1902" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1903" xml:space="preserve">22); </s>
            <s xml:id="echoid-s1904" xml:space="preserve">but the ſquare on DE
              <lb/>
            is equal to the rectangle contained by R and S; </s>
            <s xml:id="echoid-s1905" xml:space="preserve">therefore the ſquare on AO is
              <lb/>
            to the ſquare on EO as the ſquare on R is to the rectangle R, S; </s>
            <s xml:id="echoid-s1906" xml:space="preserve">that is as
              <lb/>
            R is to S, by
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1907" xml:space="preserve">V. </s>
            <s xml:id="echoid-s1908" xml:space="preserve">15.</s>
            <s xml:id="echoid-s1909" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1910" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s1911" xml:space="preserve">E. </s>
            <s xml:id="echoid-s1912" xml:space="preserve">D.</s>
            <s xml:id="echoid-s1913" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1914" xml:space="preserve">
              <emph style="sc">Scholium</emph>
            . </s>
            <s xml:id="echoid-s1915" xml:space="preserve">This Problem alſo hath three Epitagmas, which I enumerate
              <lb/>
            as in the laſt. </s>
            <s xml:id="echoid-s1916" xml:space="preserve">The firſt is conſtructed by Fig. </s>
            <s xml:id="echoid-s1917" xml:space="preserve">6, wherein the perpendicu-
              <lb/>
            lars DE and AF are ſet off on the ſame ſide of the indefinite line; </s>
            <s xml:id="echoid-s1918" xml:space="preserve">the ſecond
              <lb/>
            by Fig. </s>
            <s xml:id="echoid-s1919" xml:space="preserve">7, where they are ſet off on contrary ſides, and the third by Fig. </s>
            <s xml:id="echoid-s1920" xml:space="preserve">8,
              <lb/>
            in which they are again ſet off on the ſame ſide. </s>
            <s xml:id="echoid-s1921" xml:space="preserve">The ſecond has no limits;
              <lb/>
            </s>
            <s xml:id="echoid-s1922" xml:space="preserve">but in the firſt R muſt be leſs, and in the third greater than S, for reaſons
              <lb/>
            too obvious to be inſiſted on; </s>
            <s xml:id="echoid-s1923" xml:space="preserve">and hence, both theſe caſes are impoſſible
              <lb/>
            when the given ratio is that of equality.</s>
            <s xml:id="echoid-s1924" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div79" type="section" level="1" n="75">
          <head xml:id="echoid-head90" xml:space="preserve">PROBLEM V. (Fig. 9. 10. 11. 12. 13. 14. 15. 16.)</head>
          <p>
            <s xml:id="echoid-s1925" xml:space="preserve">In any indefinite ſtraight line let there be aſſigned the points A, E and I;
              <lb/>
            </s>
            <s xml:id="echoid-s1926" xml:space="preserve">it is required to cut it in another point, O, ſo that the rectangle contained
              <lb/>
            by the ſegment AO and a given ſtraight line P may be to the rectangle
              <lb/>
            contained by the ſegments EO, IO in the ratio of two given ſtraight lines
              <lb/>
            R and S.</s>
            <s xml:id="echoid-s1927" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1928" xml:space="preserve">
              <emph style="sc">Analysis</emph>
            . </s>
            <s xml:id="echoid-s1929" xml:space="preserve">Conceive the thing done, and O the point ſought: </s>
            <s xml:id="echoid-s1930" xml:space="preserve">then
              <lb/>
            would the rectangle AO, P be to the rectangle EO, IO as R to S. </s>
            <s xml:id="echoid-s1931" xml:space="preserve">Make
              <lb/>
            IQ to P as S is to R; </s>
            <s xml:id="echoid-s1932" xml:space="preserve">then will the rectangle AO, P be to the rectangle
              <lb/>
            EO, IO as P is to IQ; </s>
            <s xml:id="echoid-s1933" xml:space="preserve">or (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1934" xml:space="preserve">V. </s>
            <s xml:id="echoid-s1935" xml:space="preserve">15.) </s>
            <s xml:id="echoid-s1936" xml:space="preserve">the rectangle AO, P be to the
              <lb/>
            rectangle EO, IO as the rectangle IO, P is to the rectangle IO, IQ; </s>
            <s xml:id="echoid-s1937" xml:space="preserve">and
              <lb/>
            hence (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1938" xml:space="preserve">V. </s>
            <s xml:id="echoid-s1939" xml:space="preserve">15. </s>
            <s xml:id="echoid-s1940" xml:space="preserve">16) AO is to EO as IO is to IQ; </s>
            <s xml:id="echoid-s1941" xml:space="preserve">whence, by compoſition
              <lb/>
            or diviſion, AE is to EO as OQ is to IQ: </s>
            <s xml:id="echoid-s1942" xml:space="preserve">therefore (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1943" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1944" xml:space="preserve">16.) </s>
            <s xml:id="echoid-s1945" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>
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