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

Table of Notes

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            <s xml:id="echoid-s2583" xml:space="preserve">
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            at I, the perpendicular IG: </s>
            <s xml:id="echoid-s2584" xml:space="preserve">by Eu. </s>
            <s xml:id="echoid-s2585" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s2586" xml:space="preserve">13. </s>
            <s xml:id="echoid-s2587" xml:space="preserve">17. </s>
            <s xml:id="echoid-s2588" xml:space="preserve">the ſquare on IG is equal
              <lb/>
            to the rectangle contained by IB and IC; </s>
            <s xml:id="echoid-s2589" xml:space="preserve">and the ſquare on HO is equal
              <lb/>
            to the rectangle contained by IB and UC. </s>
            <s xml:id="echoid-s2590" xml:space="preserve">Now IC is by ſuppoſition
              <lb/>
            greater than UC, and therefore the rectangle IB, IC is greater than the
              <lb/>
            rectangle IB, UC: </s>
            <s xml:id="echoid-s2591" xml:space="preserve">conſequently the ſquare on IG is greater than the
              <lb/>
            ſquare on HO, and IG than HO; </s>
            <s xml:id="echoid-s2592" xml:space="preserve">whence O muſt fall between I and B,
              <lb/>
            much more between I and A. </s>
            <s xml:id="echoid-s2593" xml:space="preserve">And in the ſame manner it may be proved
              <lb/>
            that the point o falls between U and E.</s>
            <s xml:id="echoid-s2594" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2595" xml:space="preserve">
              <emph style="sc">Case</emph>
            IV. </s>
            <s xml:id="echoid-s2596" xml:space="preserve">In which the order of the given points is U, E, A, I; </s>
            <s xml:id="echoid-s2597" xml:space="preserve">it is
              <lb/>
            conſtructed exactly in the ſame manner as Caſe III, and is exhibited by
              <lb/>
            Fig. </s>
            <s xml:id="echoid-s2598" xml:space="preserve">49.</s>
            <s xml:id="echoid-s2599" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2600" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            II. </s>
            <s xml:id="echoid-s2601" xml:space="preserve">There are here only four Caſes, becauſe, as in Epi-
              <lb/>
            tagma I. </s>
            <s xml:id="echoid-s2602" xml:space="preserve">it is indifferent whether the given ratio be of a leſs to a greater,
              <lb/>
            or of a greater to a leſs; </s>
            <s xml:id="echoid-s2603" xml:space="preserve">and the two laſt of thoſe, viz. </s>
            <s xml:id="echoid-s2604" xml:space="preserve">where the order
              <lb/>
            of the given points is E, A, U, I; </s>
            <s xml:id="echoid-s2605" xml:space="preserve">or E, U, A, I, being reducible to the
              <lb/>
            two former by reading every where I for A, E for U, and the contrary,
              <lb/>
            I ſhall omit ſaying any thing of their conſtructions, except that they are
              <lb/>
            exhibited by Fig. </s>
            <s xml:id="echoid-s2606" xml:space="preserve">52 and 53.</s>
            <s xml:id="echoid-s2607" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2608" xml:space="preserve">
              <emph style="sc">Case</emph>
            I. </s>
            <s xml:id="echoid-s2609" xml:space="preserve">The order of the given points, being A,E,I,U, make B to
              <lb/>
            fall between A and I, C between E and U, and draw DH through the
              <lb/>
            center of the circle on BC, as is done in Fig. </s>
            <s xml:id="echoid-s2610" xml:space="preserve">50; </s>
            <s xml:id="echoid-s2611" xml:space="preserve">and O will fall as re-
              <lb/>
            quired for reaſons ſimilar to thoſe urged in Caſe I. </s>
            <s xml:id="echoid-s2612" xml:space="preserve">of the firſt Epitagma
              <lb/>
            of this Problem.</s>
            <s xml:id="echoid-s2613" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2614" xml:space="preserve">
              <emph style="sc">Case</emph>
            II. </s>
            <s xml:id="echoid-s2615" xml:space="preserve">If the order of the given points be A, I, E, U, the conſtruc-
              <lb/>
            tion will be as in Fig. </s>
            <s xml:id="echoid-s2616" xml:space="preserve">51, where B and C are made to fall, and DH is
              <lb/>
            drawn as in Caſe I.</s>
            <s xml:id="echoid-s2617" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2618" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            III. </s>
            <s xml:id="echoid-s2619" xml:space="preserve">Here there are eight Caſes, viz. </s>
            <s xml:id="echoid-s2620" xml:space="preserve">four where in the
              <lb/>
            order of the given points is A, U, E, I; </s>
            <s xml:id="echoid-s2621" xml:space="preserve">A, U, I, E; </s>
            <s xml:id="echoid-s2622" xml:space="preserve">U, A, E, I; </s>
            <s xml:id="echoid-s2623" xml:space="preserve">and
              <lb/>
            U, A, I, E, and the given ratio of a greater to a leſs, when O will fall
              <lb/>
            between the two given points, which bound the conſequent rectangle;
              <lb/>
            </s>
            <s xml:id="echoid-s2624" xml:space="preserve">and four others@ wherein the order of the given points is the ſame as
              <lb/>
            here, but the given ratio of a leſs to a greater, and in which the point
              <lb/>
            O will fall between the points that bound the antecedent rectangle; </s>
            <s xml:id="echoid-s2625" xml:space="preserve">but
              <lb/>
            as theſe laſt are reducible to the former by the ſame means which have been
              <lb/>
            uſed on former ſimilar occaſions, I ſhall not ſtop to ſpecify them.</s>
            <s xml:id="echoid-s2626" xml:space="preserve"/>
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