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-div80" type="section" level="1" n="76">
          <pb o="[22]" file="0092" n="99"/>
          <p>
            <s xml:id="echoid-s2033" xml:space="preserve">
              <emph style="sc">Analysis</emph>
            . </s>
            <s xml:id="echoid-s2034" xml:space="preserve">Let us conceive the thing effected, and that O is really the
              <lb/>
            point ſought. </s>
            <s xml:id="echoid-s2035" xml:space="preserve">Then, by ſuppoſition, the rectangle AO, EO is to the
              <lb/>
            ſquare on IO as R to S. </s>
            <s xml:id="echoid-s2036" xml:space="preserve">Make EC to IC as R is to S; </s>
            <s xml:id="echoid-s2037" xml:space="preserve">and the rectangle
              <lb/>
            AO, EO is to the ſquare on IO as EC to IC. </s>
            <s xml:id="echoid-s2038" xml:space="preserve">Let now OB be taken a
              <lb/>
            fourth proportional to EO, EC and IO; </s>
            <s xml:id="echoid-s2039" xml:space="preserve">then (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2040" xml:space="preserve">V. </s>
            <s xml:id="echoid-s2041" xml:space="preserve">15.) </s>
            <s xml:id="echoid-s2042" xml:space="preserve">the rectangle
              <lb/>
            AO, EO is to the ſquare on IO as the rectangle EC, OB is to the rectangle
              <lb/>
            IC, OB; </s>
            <s xml:id="echoid-s2043" xml:space="preserve">and ſo by permutation, the rectangle AO, EO is to the rectangle
              <lb/>
            EC, OB as the ſquare on IO is to the rectangle IC, OB; </s>
            <s xml:id="echoid-s2044" xml:space="preserve">and becauſe EO is
              <lb/>
            to EC as IO to BO, AO will be to OB as IO to IC, and ſo by compoſition,
              <lb/>
            or diviſion CO is to EC as IB to OB, and AB is to OB as CO to IC;
              <lb/>
            </s>
            <s xml:id="echoid-s2045" xml:space="preserve">whence, ex æquo perturb. </s>
            <s xml:id="echoid-s2046" xml:space="preserve">et permut. </s>
            <s xml:id="echoid-s2047" xml:space="preserve">AB is to IB as EC to IC; </s>
            <s xml:id="echoid-s2048" xml:space="preserve">that is in the
              <lb/>
            given ratio, and hence is given BC, the ſum or difference of CO and BO,
              <lb/>
            as alſo the rectangle contained by them, equal to the rectangle AB, IC,
              <lb/>
            wherefore theſe lines themſelves are given by the 85th or 86th of the Data.</s>
            <s xml:id="echoid-s2049" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2050" xml:space="preserve">
              <emph style="sc">Synthesis</emph>
            . </s>
            <s xml:id="echoid-s2051" xml:space="preserve">Make AB to IB and EC to IC in the given ratio, and
              <lb/>
            deſcribe on BC a circle; </s>
            <s xml:id="echoid-s2052" xml:space="preserve">erect, at B, the indefinite perpendicular BK, and
              <lb/>
            take therein BD a mean proportional between AB and IC, or between IB
              <lb/>
            and EC: </s>
            <s xml:id="echoid-s2053" xml:space="preserve">from D draw DH parallel to CB, if O muſt fall between B and C;
              <lb/>
            </s>
            <s xml:id="echoid-s2054" xml:space="preserve">but through F, the center of the circle on BC, if it muſt fall without them,
              <lb/>
            cutting the ſ@id circle in H; </s>
            <s xml:id="echoid-s2055" xml:space="preserve">then draw HO perpendicular to DH, which
              <lb/>
            will cut the indefinite line in O, the point required.</s>
            <s xml:id="echoid-s2056" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2057" xml:space="preserve">For it is plain from the conſtruction that BD and HO are equal, and
              <lb/>
            (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2058" xml:space="preserve">IV. </s>
            <s xml:id="echoid-s2059" xml:space="preserve">17.) </s>
            <s xml:id="echoid-s2060" xml:space="preserve">the rectangle AB, IC, or the rectangle IB, EC is equal to the
              <lb/>
            ſquare on BD, and therefore equal to the ſquare on HO, which (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2061" xml:space="preserve">III.
              <lb/>
            </s>
            <s xml:id="echoid-s2062" xml:space="preserve">35. </s>
            <s xml:id="echoid-s2063" xml:space="preserve">36.) </s>
            <s xml:id="echoid-s2064" xml:space="preserve">is equal to the rectangle BO, CO: </s>
            <s xml:id="echoid-s2065" xml:space="preserve">conſequently (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2066" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s2067" xml:space="preserve">16.) </s>
            <s xml:id="echoid-s2068" xml:space="preserve">AB
              <lb/>
            is to BO as CO to IC; </s>
            <s xml:id="echoid-s2069" xml:space="preserve">alſo EC is to CO as BO is to IB; </s>
            <s xml:id="echoid-s2070" xml:space="preserve">wherefore, by
              <lb/>
            compoſition or diviſion, AO is to BO as IO to IC, and EO to EC as IO
              <lb/>
            to BO: </s>
            <s xml:id="echoid-s2071" xml:space="preserve">conſequently by compound ratio, the rectangle contained by AO and
              <lb/>
            EO is to the rectangle contained by BO and EC, as the ſquare on IO is to
              <lb/>
            the rectangle contained by BO and IC; </s>
            <s xml:id="echoid-s2072" xml:space="preserve">by permutation, the rectangle
              <lb/>
            contained by AO and EO is to the ſquare on IO as the rectangle contained
              <lb/>
            by BO and EC is to the rectangle contained by BO and IC, that is (Euc. </s>
            <s xml:id="echoid-s2073" xml:space="preserve">
              <lb/>
            v. </s>
            <s xml:id="echoid-s2074" xml:space="preserve">15.) </s>
            <s xml:id="echoid-s2075" xml:space="preserve">as EC is to IC, or as R to S.</s>
            <s xml:id="echoid-s2076" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2077" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s2078" xml:space="preserve">E. </s>
            <s xml:id="echoid-s2079" xml:space="preserve">D.</s>
            <s xml:id="echoid-s2080" xml:space="preserve"/>
          </p>
        </div>
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    </echo>
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