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-s2503" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            III. </s>
            <s xml:id="echoid-s2504" xml:space="preserve">There are here but four Caſes, viz. </s>
            <s xml:id="echoid-s2505" xml:space="preserve">when the order of the
              <lb/>
            given points is A, E, I, U; </s>
            <s xml:id="echoid-s2506" xml:space="preserve">A, I, E, U; </s>
            <s xml:id="echoid-s2507" xml:space="preserve">E, A, U, I; </s>
            <s xml:id="echoid-s2508" xml:space="preserve">or E, U, A, I;
              <lb/>
            </s>
            <s xml:id="echoid-s2509" xml:space="preserve">the two ſirſt of theſe are not poſſible unleſs the given ratio be the ratio of a
              <lb/>
            greater to a leſs; </s>
            <s xml:id="echoid-s2510" xml:space="preserve">nor the two latter, unleſs it be of a leſs to a greater, and
              <lb/>
            as theſe are reduced to the ſirſt two by reading every where E for A, I
              <lb/>
            for U, and the contrary, I ſhall omit ſpecifying them.</s>
            <s xml:id="echoid-s2511" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2512" xml:space="preserve">
              <emph style="sc">Case</emph>
            I. </s>
            <s xml:id="echoid-s2513" xml:space="preserve">If the order of the given points be A, E, I, U, the conſtruc-
              <lb/>
            tion will be effected by Fig. </s>
            <s xml:id="echoid-s2514" xml:space="preserve">44, wherein B is made to fall beyond I, and
              <lb/>
            C beyond E, and DH is drawn parallel to BC. </s>
            <s xml:id="echoid-s2515" xml:space="preserve">That O, when this con-
              <lb/>
            ſtruction is uſed, will fall between E and I, is eaſily made appear by rea-
              <lb/>
            ſoning in a manner ſimilar to what was done in Caſe V. </s>
            <s xml:id="echoid-s2516" xml:space="preserve">of Epitagma I.</s>
            <s xml:id="echoid-s2517" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2518" xml:space="preserve">
              <emph style="sc">Case</emph>
            II. </s>
            <s xml:id="echoid-s2519" xml:space="preserve">The conſtruction of this Caſe, where the order of the points
              <lb/>
            is A, I, E, U, is not materially different from that above exhibited as ap-
              <lb/>
            pears by Fig. </s>
            <s xml:id="echoid-s2520" xml:space="preserve">45, and that O will fall between I and E is manifeſt without
              <lb/>
            farther illuſtration.</s>
            <s xml:id="echoid-s2521" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2522" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s2523" xml:space="preserve">In theſe two Caſes the given ratio of R to S cannot be
              <lb/>
            leſs than that which the ſquare on AU bears to the ſquare on a line which
              <lb/>
            is the difference of two mean proportionals between AI and EU, AE and
              <lb/>
            IU. </s>
            <s xml:id="echoid-s2524" xml:space="preserve">For by Lemma I. </s>
            <s xml:id="echoid-s2525" xml:space="preserve">the leaſt ratio which the rectangle contained by AO
              <lb/>
            and UO can have to the rectangle contained by EO and IO; </s>
            <s xml:id="echoid-s2526" xml:space="preserve">or, which
              <lb/>
            is the ſame thing, that R can have to S, will be when the point O is
              <lb/>
            the interſection of the diameter AU, of a circle AYUV, with a ſtraight
              <lb/>
            line YV. </s>
            <s xml:id="echoid-s2527" xml:space="preserve">joining the tops of two perpendiculars EV, IY to the indeſi-
              <lb/>
            nite line, on contrary ſides thereof, and terminating in the periphery of
              <lb/>
            the circle. </s>
            <s xml:id="echoid-s2528" xml:space="preserve">Produce VE (Fig. </s>
            <s xml:id="echoid-s2529" xml:space="preserve">27.) </s>
            <s xml:id="echoid-s2530" xml:space="preserve">to meet the circle again in K, and
              <lb/>
            draw the diameter KL; </s>
            <s xml:id="echoid-s2531" xml:space="preserve">join LY and KY, on which, produced, let fall
              <lb/>
            the perpendicular UF. </s>
            <s xml:id="echoid-s2532" xml:space="preserve">Now, ſince by Lemma III. </s>
            <s xml:id="echoid-s2533" xml:space="preserve">KF is a mean propor-
              <lb/>
            tional between AI and EU, and YF a mean proportional between AE
              <lb/>
            and IU: </s>
            <s xml:id="echoid-s2534" xml:space="preserve">it remains only to prove that the ratio of the rectangle con-
              <lb/>
            tained by AO and OU to the rectangle contained by EO and OI is the
              <lb/>
            ſame with the ratio which the ſquare on AU bears to the ſquare on KY,
              <lb/>
            which is the diſſerence between KF and YF. </s>
            <s xml:id="echoid-s2535" xml:space="preserve">Becauſe the angles E and
              <lb/>
            KYL are both right, and the angles EVO and KYL equal (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2536" xml:space="preserve">III. </s>
            <s xml:id="echoid-s2537" xml:space="preserve">21.)
              <lb/>
            </s>
            <s xml:id="echoid-s2538" xml:space="preserve">the triangles EVO and YLK are ſimilar; </s>
            <s xml:id="echoid-s2539" xml:space="preserve">and ſo VO is to EO as AU (LK)
              <lb/>
            is to KY; </s>
            <s xml:id="echoid-s2540" xml:space="preserve">or the ſquare on VO is to the ſquare on EO as the ſquare on AU
              <lb/>
            is to the ſquare on KY. </s>
            <s xml:id="echoid-s2541" xml:space="preserve">Now the triangles EVO, IYO being alſo </s>
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