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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          <p>
            <s xml:id="echoid-s2541" xml:space="preserve">
              <pb o="[36]" file="0106" n="113"/>
            OY will be to OV as OI is to OE, and (Eu. </s>
            <s xml:id="echoid-s2542" xml:space="preserve">V. </s>
            <s xml:id="echoid-s2543" xml:space="preserve">15. </s>
            <s xml:id="echoid-s2544" xml:space="preserve">16.) </s>
            <s xml:id="echoid-s2545" xml:space="preserve">the rectangle
              <lb/>
            contained by VO and YO or its equal, the rectangle contained by AO and
              <lb/>
            OU, is to the rectangle contained by EO and OI as the ſquare on OV is
              <lb/>
            to the ſquare on EO, as the ſquaree on AU is to the ſquare on KY.</s>
            <s xml:id="echoid-s2546" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2547" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s2548" xml:space="preserve">E. </s>
            <s xml:id="echoid-s2549" xml:space="preserve">D.</s>
            <s xml:id="echoid-s2550" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2551" xml:space="preserve">
              <emph style="sc">Scholium</emph>
            . </s>
            <s xml:id="echoid-s2552" xml:space="preserve">It might be obſerved that in the two Caſes of this Epitagma
              <lb/>
            where the points A and U are means, the limiting ratio will be a maxi-
              <lb/>
            mum inſtead of a minimum; </s>
            <s xml:id="echoid-s2553" xml:space="preserve">and that ratio will be the ſame with that which
              <lb/>
            the ſquare on KY bears to the ſquare on EI, as is plain from what hath
              <lb/>
            been advanced above.</s>
            <s xml:id="echoid-s2554" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div92" type="section" level="1" n="85">
          <head xml:id="echoid-head100" xml:space="preserve">PROBLEM II. (Fig. 46 to 57.)</head>
          <p>
            <s xml:id="echoid-s2555" xml:space="preserve">Where O is ſought between a mean and an extream point: </s>
            <s xml:id="echoid-s2556" xml:space="preserve">and here,
              <lb/>
            as in the firſt Problem, there are three Epitagmas. </s>
            <s xml:id="echoid-s2557" xml:space="preserve">Firſt when the points
              <lb/>
            A and U, which bound the ſegments containing the antecedent rectangle,
              <lb/>
            are one an extreme, and the other an alternate mean; </s>
            <s xml:id="echoid-s2558" xml:space="preserve">ſecondly when they
              <lb/>
            are both means, or both extremes; </s>
            <s xml:id="echoid-s2559" xml:space="preserve">thirdly when they are one an extreme,
              <lb/>
            and the other an adjacent mean.</s>
            <s xml:id="echoid-s2560" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2561" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            I. </s>
            <s xml:id="echoid-s2562" xml:space="preserve">There are here eight Caſes, but they are conſtructed at
              <lb/>
            four times, becauſe it is indifferent whether the given ratio be of a leſs to a
              <lb/>
            greater, or of a greater to a leſs.</s>
            <s xml:id="echoid-s2563" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2564" xml:space="preserve">
              <emph style="sc">Case</emph>
            I. </s>
            <s xml:id="echoid-s2565" xml:space="preserve">The order of the given points being A, E, U, I, as in Fig. </s>
            <s xml:id="echoid-s2566" xml:space="preserve">46,
              <lb/>
            make B to fall between A and I, C between U and E, and draw DH
              <lb/>
            through the center of the circle on BC; </s>
            <s xml:id="echoid-s2567" xml:space="preserve">and O will fall between A and
              <lb/>
            E, becauſe AB is to BO as CO is to CE, and therefore, if AB be greater
              <lb/>
            than BO, CO muſt be greater than CE, and if leſs, leſs; </s>
            <s xml:id="echoid-s2568" xml:space="preserve">but this cannot
              <lb/>
            be the caſe if O falls either beyond A or E: </s>
            <s xml:id="echoid-s2569" xml:space="preserve">and the like abſurdity fol.
              <lb/>
            </s>
            <s xml:id="echoid-s2570" xml:space="preserve">lows if o be ſuppoſed to fall otherwiſe than between I and U.</s>
            <s xml:id="echoid-s2571" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2572" xml:space="preserve">
              <emph style="sc">Case</emph>
            II. </s>
            <s xml:id="echoid-s2573" xml:space="preserve">Wherein the order of the points is U, I, A, E; </s>
            <s xml:id="echoid-s2574" xml:space="preserve">and it is con-
              <lb/>
            ſtructed in the very ſame manner that Caſe I. </s>
            <s xml:id="echoid-s2575" xml:space="preserve">is, as appears by barely
              <lb/>
            inſpecting Fig. </s>
            <s xml:id="echoid-s2576" xml:space="preserve">47.</s>
            <s xml:id="echoid-s2577" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2578" xml:space="preserve">
              <emph style="sc">Case</emph>
            III. </s>
            <s xml:id="echoid-s2579" xml:space="preserve">If the order of the given points be A, I, U, E (Fig. </s>
            <s xml:id="echoid-s2580" xml:space="preserve">48.) </s>
            <s xml:id="echoid-s2581" xml:space="preserve">the
              <lb/>
            points B and C muſt be made to fall as in the two preceding Caſes; </s>
            <s xml:id="echoid-s2582" xml:space="preserve">but
              <lb/>
            DH muſt be drawn parallel to BC, and O will fall as required. </s>
            <s xml:id="echoid-s2583" xml:space="preserve">For </s>
          </p>
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