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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        <div xml:id="echoid-div93" type="section" level="1" n="86">
          <head xml:id="echoid-head101" xml:space="preserve">PROBLEM III.</head>
          <p>
            <s xml:id="echoid-s2660" xml:space="preserve">In this, the point O is ſought without all the given ones, and the three
              <lb/>
            Epitagmas are as in Problem I.</s>
            <s xml:id="echoid-s2661" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2662" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            I. </s>
            <s xml:id="echoid-s2663" xml:space="preserve">There are here eight Caſes, viz. </s>
            <s xml:id="echoid-s2664" xml:space="preserve">four when the order of
              <lb/>
            the given points is the ſame as ſpecified in Epitagma I. </s>
            <s xml:id="echoid-s2665" xml:space="preserve">of Problem I, and
              <lb/>
            O ſought beyond the given point which bounds the antecedent rectangle;
              <lb/>
            </s>
            <s xml:id="echoid-s2666" xml:space="preserve">and four others when O is ſought beyond that which bounds the conſe-
              <lb/>
            quent one: </s>
            <s xml:id="echoid-s2667" xml:space="preserve">the Conſtructions of the four firſt are ſhewn by the ſmall
              <lb/>
            letters b and o in Fig. </s>
            <s xml:id="echoid-s2668" xml:space="preserve">32, 34, 36 and 38; </s>
            <s xml:id="echoid-s2669" xml:space="preserve">and the four latter ones by
              <lb/>
            the ſame letters in Fig. </s>
            <s xml:id="echoid-s2670" xml:space="preserve">33, 35, 37 and 39; </s>
            <s xml:id="echoid-s2671" xml:space="preserve">and the demonſtrations that
              <lb/>
            o will fall as required by the Problem are exactly the ſame as thoſe made
              <lb/>
            uſe of in the laſt mentioned Epitagma. </s>
            <s xml:id="echoid-s2672" xml:space="preserve">It is farther obſervable, that the
              <lb/>
            four firſt Caſes are not poſſible, unleſs the given ratio be of a leſs to a
              <lb/>
            greater; </s>
            <s xml:id="echoid-s2673" xml:space="preserve">nor the four latter, unleſs it be of a greater to a leſs, as is ma-
              <lb/>
            nifeſt without farther illuſtration.</s>
            <s xml:id="echoid-s2674" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2675" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            II. </s>
            <s xml:id="echoid-s2676" xml:space="preserve">Here, as in the ſecond Epitagma of Problem I, the points
              <lb/>
            A and U are one an extreme, and the other an adjacent mean, and there
              <lb/>
            are eight Caſes; </s>
            <s xml:id="echoid-s2677" xml:space="preserve">but it will be ſufficient to exhibit the conſtructions of
              <lb/>
            four of them, the others being not eſſentially different; </s>
            <s xml:id="echoid-s2678" xml:space="preserve">and theſe are ſhewn
              <lb/>
            by the ſmall b and o in Fig. </s>
            <s xml:id="echoid-s2679" xml:space="preserve">40, 41, 42 and 43; </s>
            <s xml:id="echoid-s2680" xml:space="preserve">the demonſtrations that o
              <lb/>
            will fall as required need not be pointed out here; </s>
            <s xml:id="echoid-s2681" xml:space="preserve">but it may be neceſſary
              <lb/>
            to remark that the firſt and third are not poſſible unleſs the given ratio be
              <lb/>
            of a leſs to a greater, nor the ſecond and fourth unleſs it be of a greater to
              <lb/>
            a leſs, as is obvious enough.</s>
            <s xml:id="echoid-s2682" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2683" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            III. </s>
            <s xml:id="echoid-s2684" xml:space="preserve">In which the points A and U are both means, or both
              <lb/>
            extremes; </s>
            <s xml:id="echoid-s2685" xml:space="preserve">and there are here eight Caſes, viz. </s>
            <s xml:id="echoid-s2686" xml:space="preserve">four wherem theſe points
              <lb/>
            are extremes, and four others wherein they are means: </s>
            <s xml:id="echoid-s2687" xml:space="preserve">but theſe laſt being
              <lb/>
            reducible to the former by the ſame method that was uſed in the third Epi-
              <lb/>
            tagmas of the two preceding Problems, I ſhall omit them.</s>
            <s xml:id="echoid-s2688" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2689" xml:space="preserve">All the Caſes of this Epitagma are conſtructed by making B fall beyond
              <lb/>
            I, and C beyond E, with reſpect to A and U; </s>
            <s xml:id="echoid-s2690" xml:space="preserve">and drawing DH parallel to
              <lb/>
            BC. </s>
            <s xml:id="echoid-s2691" xml:space="preserve">That O will fall beyond A in Fig. </s>
            <s xml:id="echoid-s2692" xml:space="preserve">58 and 60, and beyond U in Fig.
              <lb/>
            </s>
            <s xml:id="echoid-s2693" xml:space="preserve">59 and 61 appears hence. </s>
            <s xml:id="echoid-s2694" xml:space="preserve">Draw AG perpendicular to BC, meeting the
              <lb/>
            circle on BC in G: </s>
            <s xml:id="echoid-s2695" xml:space="preserve">by Eu. </s>
            <s xml:id="echoid-s2696" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s2697" xml:space="preserve">13. </s>
            <s xml:id="echoid-s2698" xml:space="preserve">17, the ſquare on AG is equal to </s>
          </p>
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