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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          <pb o="[33]" file="0103" n="110"/>
          <p>
            <s xml:id="echoid-s2417" xml:space="preserve">
              <emph style="sc">Case</emph>
            II. </s>
            <s xml:id="echoid-s2418" xml:space="preserve">If the order oſ the given points be retained, but the ratio be of
              <lb/>
            a greater to a leſs, B muſt ſall beyond I with reſpect to A, and C beyond
              <lb/>
            E, as in Fig. </s>
            <s xml:id="echoid-s2419" xml:space="preserve">33, and DH muſt be drawn as before: </s>
            <s xml:id="echoid-s2420" xml:space="preserve">and it may be proved
              <lb/>
            by reaſonings ſimilar to thoſe uſed in the ſirſt Caſe that O will fall between
              <lb/>
            I and U, as was required.</s>
            <s xml:id="echoid-s2421" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2422" xml:space="preserve">
              <emph style="sc">Cases</emph>
            III and IV. </s>
            <s xml:id="echoid-s2423" xml:space="preserve">Theſe caſes are conſtructed exactly in the ſame man-
              <lb/>
            ner as Caſes I and II reſpectively; </s>
            <s xml:id="echoid-s2424" xml:space="preserve">and the reaſonings to prove that O will
              <lb/>
            fall as it ought arethe ſame with thoſe made uſe of in Caſe I, as will appear
              <lb/>
            by inſpecting Fig. </s>
            <s xml:id="echoid-s2425" xml:space="preserve">34. </s>
            <s xml:id="echoid-s2426" xml:space="preserve">and 35.</s>
            <s xml:id="echoid-s2427" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2428" xml:space="preserve">
              <emph style="sc">Case</emph>
            V. </s>
            <s xml:id="echoid-s2429" xml:space="preserve">Let now the order of the given points be A, E, U, I, and the
              <lb/>
            given ratio of a leſs to a greater. </s>
            <s xml:id="echoid-s2430" xml:space="preserve">Then muſt B be made to fall beyond A
              <lb/>
            and C beyond U, as in Fig. </s>
            <s xml:id="echoid-s2431" xml:space="preserve">36, and DH is here to be drawn parallel to
              <lb/>
            to BC: </s>
            <s xml:id="echoid-s2432" xml:space="preserve">and that O will fall as required may be made to appear thus. </s>
            <s xml:id="echoid-s2433" xml:space="preserve">Draw
              <lb/>
            EF and UG perpendicular to BC. </s>
            <s xml:id="echoid-s2434" xml:space="preserve">Now (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2435" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s2436" xml:space="preserve">13. </s>
            <s xml:id="echoid-s2437" xml:space="preserve">17.) </s>
            <s xml:id="echoid-s2438" xml:space="preserve">the ſquare on
              <lb/>
            EF is equal to the rectangle contained by EC and EB, the ſquare on UG
              <lb/>
            to the rectangle contained by UC and UB; </s>
            <s xml:id="echoid-s2439" xml:space="preserve">and, by conſtruction, the
              <lb/>
            ſquare on HO to the rectangle contained by EC and AB, or to the rect-
              <lb/>
            angle contained by UC and IB; </s>
            <s xml:id="echoid-s2440" xml:space="preserve">and ſince, by ſuppoſition, EB is greater
              <lb/>
            than AB, and UB leſs than IB, the rectangle EC, AB, or its equal, the
              <lb/>
            rectangle UC, IB will be leſs than the rectangle EC, EB, and greater than
              <lb/>
            the rectangle UC, UB; </s>
            <s xml:id="echoid-s2441" xml:space="preserve">and conſequently the ſquare on HO will be leſs
              <lb/>
            than the ſquare on EF and greater than the ſquare on UG; </s>
            <s xml:id="echoid-s2442" xml:space="preserve">and therefore
              <lb/>
            HO leſs than EF and greater than UG: </s>
            <s xml:id="echoid-s2443" xml:space="preserve">but this could not be the Caſe
              <lb/>
            unleſs O fell between E and U, as was was required.</s>
            <s xml:id="echoid-s2444" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2445" xml:space="preserve">
              <emph style="sc">Case</emph>
            VI. </s>
            <s xml:id="echoid-s2446" xml:space="preserve">If the order of the points be retained; </s>
            <s xml:id="echoid-s2447" xml:space="preserve">but the given ratio be
              <lb/>
            of a greater to a leſs, B muſt then fall beyond I, and C beyond E; </s>
            <s xml:id="echoid-s2448" xml:space="preserve">and
              <lb/>
            DH is drawn as in the preceding Caſe (See Fig. </s>
            <s xml:id="echoid-s2449" xml:space="preserve">37.) </s>
            <s xml:id="echoid-s2450" xml:space="preserve">moreover, that O
              <lb/>
            will fall between E and U may be made to appear, by reaſonings ſimilar to
              <lb/>
            thoſe there made uſe on the like occaſion.</s>
            <s xml:id="echoid-s2451" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2452" xml:space="preserve">
              <emph style="sc">Case</emph>
            VII. </s>
            <s xml:id="echoid-s2453" xml:space="preserve">Is conſtructed exactly in the ſame manner as Caſe V. </s>
            <s xml:id="echoid-s2454" xml:space="preserve">and,</s>
          </p>
          <p>
            <s xml:id="echoid-s2455" xml:space="preserve">
              <emph style="sc">Case</emph>
            VIII. </s>
            <s xml:id="echoid-s2456" xml:space="preserve">As Caſe VI: </s>
            <s xml:id="echoid-s2457" xml:space="preserve">the truth of which will appear by. </s>
            <s xml:id="echoid-s2458" xml:space="preserve">barely in-
              <lb/>
            ſpecting Fig. </s>
            <s xml:id="echoid-s2459" xml:space="preserve">38 and 39.</s>
            <s xml:id="echoid-s2460" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2461" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            II. </s>
            <s xml:id="echoid-s2462" xml:space="preserve">In this Epitagma there are alſo eight Caſes, viz. </s>
            <s xml:id="echoid-s2463" xml:space="preserve">when the
              <lb/>
            order of the given points is A, U, E, I; </s>
            <s xml:id="echoid-s2464" xml:space="preserve">A, U, I, E; </s>
            <s xml:id="echoid-s2465" xml:space="preserve">U, A, E, I; </s>
            <s xml:id="echoid-s2466" xml:space="preserve">or
              <lb/>
            U, A, I, E; </s>
            <s xml:id="echoid-s2467" xml:space="preserve">and the given ratio of a leſs to a greater: </s>
            <s xml:id="echoid-s2468" xml:space="preserve">and there are
              <lb/>
            four others wherein the order of the points are the ſame as in thoſe, </s>
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