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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            <s xml:id="echoid-s2119" xml:space="preserve">
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            and ſecondly when it is ſought beyond both the other given points: </s>
            <s xml:id="echoid-s2120" xml:space="preserve">in the
              <lb/>
            firſt, the ratio of R to S muſt be of a greater to a leſs, and in the latter of
              <lb/>
            a leſs to a greater. </s>
            <s xml:id="echoid-s2121" xml:space="preserve">The former is conſtructed in Fig. </s>
            <s xml:id="echoid-s2122" xml:space="preserve">17. </s>
            <s xml:id="echoid-s2123" xml:space="preserve">at the ſame time
              <lb/>
            with Caſe I of Epitagma I. </s>
            <s xml:id="echoid-s2124" xml:space="preserve">and is repreſented by the ſmall letters h and o:
              <lb/>
            </s>
            <s xml:id="echoid-s2125" xml:space="preserve">the latter in Fig. </s>
            <s xml:id="echoid-s2126" xml:space="preserve">18, and pointed out by the ſame letters. </s>
            <s xml:id="echoid-s2127" xml:space="preserve">That O will
              <lb/>
            fall in both as is required needs not inſiſting on.</s>
            <s xml:id="echoid-s2128" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2129" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            III. </s>
            <s xml:id="echoid-s2130" xml:space="preserve">In which there are ſix Caſes, viz. </s>
            <s xml:id="echoid-s2131" xml:space="preserve">I being the middle
              <lb/>
            point, when O is ſought beyond A, or beyond E; </s>
            <s xml:id="echoid-s2132" xml:space="preserve">and that whether the
              <lb/>
            given ratio be of a leſs to a greater, or of a greater to a leſs; </s>
            <s xml:id="echoid-s2133" xml:space="preserve">and again, I
              <lb/>
            being an extreme point, when O is ſought between A and E, and that let
              <lb/>
            the order of the points A and E be what it will.</s>
            <s xml:id="echoid-s2134" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2135" xml:space="preserve">
              <emph style="sc">Cases</emph>
            I and II. </s>
            <s xml:id="echoid-s2136" xml:space="preserve">Are when I is a mean point and the given ratio of a
              <lb/>
            leſs to a greater; </s>
            <s xml:id="echoid-s2137" xml:space="preserve">and theſe are both conſtructed at once by Fig. </s>
            <s xml:id="echoid-s2138" xml:space="preserve">20,
              <lb/>
            wherein B is made to fall beyond A, and C beyond E with reſpect to the
              <lb/>
            middle point I, and DH is drawn through the center of the circle on BC.</s>
            <s xml:id="echoid-s2139" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2140" xml:space="preserve">
              <emph style="sc">Cases</emph>
            III and IV. </s>
            <s xml:id="echoid-s2141" xml:space="preserve">Here, the points remaining as before, the given
              <lb/>
            ratio is of a greater to a leſs; </s>
            <s xml:id="echoid-s2142" xml:space="preserve">and the conſtruction will be effected by
              <lb/>
            making B fall beyond E, and C beyond A, and drawing DH parallel to
              <lb/>
            BC, as in Fig. </s>
            <s xml:id="echoid-s2143" xml:space="preserve">21 and 22.</s>
            <s xml:id="echoid-s2144" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2145" xml:space="preserve">
              <emph style="sc">Case</emph>
            V. </s>
            <s xml:id="echoid-s2146" xml:space="preserve">Wherein I is one extreme point and A the other, and O is
              <lb/>
            ſought between A and E: </s>
            <s xml:id="echoid-s2147" xml:space="preserve">in conſtructing this Caſe, B muſt be made to
              <lb/>
            fall between A and I, C between E and I, and DH drawn parallel to BC,
              <lb/>
            as is done in Fig. </s>
            <s xml:id="echoid-s2148" xml:space="preserve">23. </s>
            <s xml:id="echoid-s2149" xml:space="preserve">The directions for Conſtructing Caſe VI. </s>
            <s xml:id="echoid-s2150" xml:space="preserve">are exactly
              <lb/>
            the ſame, as will appear by barely inſpecting Fig. </s>
            <s xml:id="echoid-s2151" xml:space="preserve">24.</s>
            <s xml:id="echoid-s2152" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2153" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s2154" xml:space="preserve">It is plain that in the four laſt Caſes, the ratio which the
              <lb/>
            rectangle contained by AO and EO bears to the ſquare on IO, or which is
              <lb/>
            the ſame thing, the given ratio of R to S cannot exceed a certain limit;
              <lb/>
            </s>
            <s xml:id="echoid-s2155" xml:space="preserve">and it is farther obvious that the ſaid limit will be when the ſtraight line
              <lb/>
            DH becomes a tangent to the circle on BC, as in Fig. </s>
            <s xml:id="echoid-s2156" xml:space="preserve">25. </s>
            <s xml:id="echoid-s2157" xml:space="preserve">26, for after
              <lb/>
            that the problem is manifeſtly impoſſible. </s>
            <s xml:id="echoid-s2158" xml:space="preserve">Now when DH is a tangent to
              <lb/>
            the circle on BC, HO will be equal to half BC; </s>
            <s xml:id="echoid-s2159" xml:space="preserve">but the ſquare on HO
              <lb/>
            is equal to the rectangle contained by IB and EC, wherefore the ſquare on
              <lb/>
            half BC will then be equal to the rectangle contained by IB and EC. </s>
            <s xml:id="echoid-s2160" xml:space="preserve">
              <lb/>
            Moreover, by the conſtruction, R is to S as AB is to IB, and as EC is to
              <lb/>
            IC; </s>
            <s xml:id="echoid-s2161" xml:space="preserve">therefore by compoſition or diviſion, the ſum or difference of R and
              <lb/>
            S is to R as EI to EC, and the ſaid ſum or difference is to S as AI is </s>
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