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-s2468" xml:space="preserve">
              <pb o="[34]" file="0104" n="111"/>
            the ratio of a greater to a leſs; </s>
            <s xml:id="echoid-s2469" xml:space="preserve">but theſe, I ſhall ſhew, may be reduced
              <lb/>
            to four.</s>
            <s xml:id="echoid-s2470" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2471" xml:space="preserve">
              <emph style="sc">Case</emph>
            I. </s>
            <s xml:id="echoid-s2472" xml:space="preserve">The order of the given points being A, U, E, I; </s>
            <s xml:id="echoid-s2473" xml:space="preserve">and the given
              <lb/>
            ratio a leſs to a greater, the conſtruction will be eſſected by Fig. </s>
            <s xml:id="echoid-s2474" xml:space="preserve">40,
              <lb/>
            wherein B is made to fall beyond A with reſpect to I and C beyond U, and
              <lb/>
            DH is drawn through the center of the circle on BC. </s>
            <s xml:id="echoid-s2475" xml:space="preserve">And O will fall
              <lb/>
            between U and E for reaſons ſimilar to thoſe urged in the ſirſt Caſe of Epi-
              <lb/>
            tagma I. </s>
            <s xml:id="echoid-s2476" xml:space="preserve">It is moreover obvious that the conſtruction will not be eſſentially
              <lb/>
            different ſhould the points E and I change places, and therefore need not
              <lb/>
            here be made a new Caſe.</s>
            <s xml:id="echoid-s2477" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2478" xml:space="preserve">
              <emph style="sc">Case</emph>
            II. </s>
            <s xml:id="echoid-s2479" xml:space="preserve">The order of the points being the ſame as in the laſt Caſe, let
              <lb/>
            the given ratio be of a greater to a leſs; </s>
            <s xml:id="echoid-s2480" xml:space="preserve">then, as in Fig. </s>
            <s xml:id="echoid-s2481" xml:space="preserve">41, B muſt fall
              <lb/>
            beyond I, and C beyond E; </s>
            <s xml:id="echoid-s2482" xml:space="preserve">but DC muſt ſtill be drawn through the
              <lb/>
            center of the circle on BC. </s>
            <s xml:id="echoid-s2483" xml:space="preserve">It is manifeſt that this conſtruction will ſerve
              <lb/>
            for that Caſe wherein the points A and U change ſituations, if the ratio be,
              <lb/>
            as here, of a greater to a leſs.</s>
            <s xml:id="echoid-s2484" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2485" xml:space="preserve">
              <emph style="sc">Case</emph>
            III. </s>
            <s xml:id="echoid-s2486" xml:space="preserve">Here, let the order of the points be U, A, I, E, and the
              <lb/>
            given ratio of a leſs to a greater, and the Conſtruction will be aſſected by
              <lb/>
            Fig. </s>
            <s xml:id="echoid-s2487" xml:space="preserve">42, in which B falls beyond A, and C beyond U with reſpect to I and
              <lb/>
            E: </s>
            <s xml:id="echoid-s2488" xml:space="preserve">and the ſame conſtruction will ſerve if I and E change places, but the
              <lb/>
            ratio remain the ſame.</s>
            <s xml:id="echoid-s2489" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2490" xml:space="preserve">
              <emph style="sc">Case</emph>
            IV. </s>
            <s xml:id="echoid-s2491" xml:space="preserve">If the poſition of the points be retained, but the ratio be
              <lb/>
            made of a leſs to a greater; </s>
            <s xml:id="echoid-s2492" xml:space="preserve">then muſt B fall beyond I (Fig. </s>
            <s xml:id="echoid-s2493" xml:space="preserve">43.) </s>
            <s xml:id="echoid-s2494" xml:space="preserve">and C
              <lb/>
            beyond E; </s>
            <s xml:id="echoid-s2495" xml:space="preserve">but DH drawn as before. </s>
            <s xml:id="echoid-s2496" xml:space="preserve">That O muſt fall as was required,
              <lb/>
            in theſe three laſt caſes, is obvious enough from what has been ſaid be-
              <lb/>
            fore on the like occaſion: </s>
            <s xml:id="echoid-s2497" xml:space="preserve">and it is alſo plain that the conſtruction will
              <lb/>
            not be materially diſſerent though A and U change places.</s>
            <s xml:id="echoid-s2498" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2499" xml:space="preserve">
              <emph style="sc">Scholium</emph>
            . </s>
            <s xml:id="echoid-s2500" xml:space="preserve">That none of the Caſes of theſe two Epitagmas are ſubject
              <lb/>
            to Limitations, might be proved with the utmoſt rigour of geometrical
              <lb/>
            reaſoning was it not ſuſſiciently manifeſt from conſidering that as the point
              <lb/>
            O approaches points A, or U, the ratio of the rectangle AO, OU to the
              <lb/>
            rectangle EO, OI will become very ſmall, and as it approaches the points
              <lb/>
            E, or I the ſaid ratio will become very great: </s>
            <s xml:id="echoid-s2501" xml:space="preserve">and nothing hinders that
              <lb/>
            the ſaid point may ſall any where between thoſe.</s>
            <s xml:id="echoid-s2502" xml:space="preserve"/>
          </p>
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