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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      <text xml:lang="en" type="free">
        <div xml:id="echoid-div79" type="section" level="1" n="75">
          <p>
            <s xml:id="echoid-s1945" xml:space="preserve">
              <pb o="[20]" file="0090" n="97"/>
            rectangle EO, OQ is equal to the rectangle AE, IQ; </s>
            <s xml:id="echoid-s1946" xml:space="preserve">conſequently, as the
              <lb/>
            ſum or difference of EO and OQ is alſo given, thoſe lines themſelves are
              <lb/>
            given by the 85th or 86th of the Data.</s>
            <s xml:id="echoid-s1947" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1948" xml:space="preserve">
              <emph style="sc">Synthesis</emph>
            . </s>
            <s xml:id="echoid-s1949" xml:space="preserve">Take IQ a fourth proportional to R, S and P, and
              <lb/>
            deſcribe on EQ a circle; </s>
            <s xml:id="echoid-s1950" xml:space="preserve">erect at E the indefinite perpendicular EK, and
              <lb/>
            take therein ED a mean proportional between AE and IQ; </s>
            <s xml:id="echoid-s1951" xml:space="preserve">from D draw
              <lb/>
            DH, parallel to EQ, if O muſt lie any where between the points E and Q;
              <lb/>
            </s>
            <s xml:id="echoid-s1952" xml:space="preserve">but through F, the center of the circle on EQ if it muſt fall without them,
              <lb/>
            cutting the ſaid circle in H: </s>
            <s xml:id="echoid-s1953" xml:space="preserve">Laſtly, draw HO perpendicular to DH, which
              <lb/>
            will meet the indeſinite line in O, the point required.</s>
            <s xml:id="echoid-s1954" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1955" xml:space="preserve">For it is manifeſt from the conſtruction that ED and HO are equal; </s>
            <s xml:id="echoid-s1956" xml:space="preserve">and
              <lb/>
            (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1957" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1958" xml:space="preserve">17.) </s>
            <s xml:id="echoid-s1959" xml:space="preserve">the rectangle AE, IQ is equal to the ſquare on ED, and
              <lb/>
            therefore equal to the ſquare on HO; </s>
            <s xml:id="echoid-s1960" xml:space="preserve">but the ſquare on HO is equal to the
              <lb/>
            rectangle EO, OQ (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1961" xml:space="preserve">III. </s>
            <s xml:id="echoid-s1962" xml:space="preserve">35. </s>
            <s xml:id="echoid-s1963" xml:space="preserve">36.)</s>
            <s xml:id="echoid-s1964" xml:space="preserve">: </s>
            <s xml:id="echoid-s1965" xml:space="preserve">therefore the rectangle AE, IQ is
              <lb/>
            equal to the rectangle EO, OQ; </s>
            <s xml:id="echoid-s1966" xml:space="preserve">and hence (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1967" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1968" xml:space="preserve">16.) </s>
            <s xml:id="echoid-s1969" xml:space="preserve">AE is to OE as OQ
              <lb/>
            to IQ, whence, by compoſition or diviſion, AO is to EO as OI to IQ; </s>
            <s xml:id="echoid-s1970" xml:space="preserve">but
              <lb/>
            IQ is to P as S to R, or inverſely, P is to IQ as R to S; </s>
            <s xml:id="echoid-s1971" xml:space="preserve">and ſo, by compound
              <lb/>
            ratio, the rectangle AO, P is to the rectangle EO, IQ as the rectangle IO,
              <lb/>
            R is to the rectangle IQ, S; </s>
            <s xml:id="echoid-s1972" xml:space="preserve">that is (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1973" xml:space="preserve">V. </s>
            <s xml:id="echoid-s1974" xml:space="preserve">15 and 16.) </s>
            <s xml:id="echoid-s1975" xml:space="preserve">the rectangle AO,
              <lb/>
            P is to the rectangle IO, R as EO is to S; </s>
            <s xml:id="echoid-s1976" xml:space="preserve">or the rectangle AO, P is to the
              <lb/>
            rectangle IO, R as the rectangle IO, EO is to the rectangle IO, S
              <lb/>
            (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1977" xml:space="preserve">V. </s>
            <s xml:id="echoid-s1978" xml:space="preserve">16.) </s>
            <s xml:id="echoid-s1979" xml:space="preserve">the rectangle AO, P is to the rectangle EO, IO as the rectangle
              <lb/>
            IO, R is to the rectangle IO, S; </s>
            <s xml:id="echoid-s1980" xml:space="preserve">that is (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1981" xml:space="preserve">V. </s>
            <s xml:id="echoid-s1982" xml:space="preserve">15.) </s>
            <s xml:id="echoid-s1983" xml:space="preserve">as R is to S. </s>
            <s xml:id="echoid-s1984" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s1985" xml:space="preserve">E. </s>
            <s xml:id="echoid-s1986" xml:space="preserve">D.</s>
            <s xml:id="echoid-s1987" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1988" xml:space="preserve">
              <emph style="sc">Scholium</emph>
            . </s>
            <s xml:id="echoid-s1989" xml:space="preserve">This Problem may be conſidered as having three Epitagmas,
              <lb/>
            or general Caſes, viz. </s>
            <s xml:id="echoid-s1990" xml:space="preserve">when A, the point which bounds the ſegment aſſigned
              <lb/>
            for the co efficient of the given line P being an extreme, O is ſought be-
              <lb/>
            tween it and the next thereto, or beyond all the points with reſpect to A;
              <lb/>
            </s>
            <s xml:id="echoid-s1991" xml:space="preserve">ſecondly, where A is the middle point; </s>
            <s xml:id="echoid-s1992" xml:space="preserve">and thirdly, when A being again
              <lb/>
            an extreme, O is ſought beyond it, or between the other two points E and
              <lb/>
            I: </s>
            <s xml:id="echoid-s1993" xml:space="preserve">and each of theſe is ſubdiviſible into four more particular ones.</s>
            <s xml:id="echoid-s1994" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1995" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            I. </s>
            <s xml:id="echoid-s1996" xml:space="preserve">Here the four Caſes are when E being the middle point,
              <lb/>
            O is required between A and E, or beyond I; </s>
            <s xml:id="echoid-s1997" xml:space="preserve">and theſe are both con-
              <lb/>
            ſtructed at once by Fig. </s>
            <s xml:id="echoid-s1998" xml:space="preserve">9: </s>
            <s xml:id="echoid-s1999" xml:space="preserve">when I is the middle point and O ſought between
              <lb/>
            A and I or beyond E; </s>
            <s xml:id="echoid-s2000" xml:space="preserve">and theſe are both conſtructed at once by Fig. </s>
            <s xml:id="echoid-s2001" xml:space="preserve">10.
              <lb/>
            </s>
            <s xml:id="echoid-s2002" xml:space="preserve">and in both of theſe IQ is ſet off from I contrary to that direction which </s>
          </p>
        </div>
      </text>
    </echo>
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