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-div92" type="section" level="1" n="85">
          <pb o="[38]" file="0108" n="115"/>
          <p>
            <s xml:id="echoid-s2627" xml:space="preserve">The former four are all conſtructed by making B to fall between A and
              <lb/>
            I, C between U and E, and drawing DH parallel to BC; </s>
            <s xml:id="echoid-s2628" xml:space="preserve">and it will ap-
              <lb/>
            pear by reaſonings ſimilar to thoſe uſed for the like purpoſe in Caſe III. </s>
            <s xml:id="echoid-s2629" xml:space="preserve">of
              <lb/>
            Epitagma I. </s>
            <s xml:id="echoid-s2630" xml:space="preserve">that O muſt fall between E and I as was propoſed. </s>
            <s xml:id="echoid-s2631" xml:space="preserve">See Fig.
              <lb/>
            </s>
            <s xml:id="echoid-s2632" xml:space="preserve">54, 55, 56 and 57.</s>
            <s xml:id="echoid-s2633" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2634" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s2635" xml:space="preserve">In theſe four Caſes, the given ratio of R to S muſt not
              <lb/>
            be leſs than that which the ſquare on the ſum of two mean proportionals
              <lb/>
            between AE and IU, AI and EU bears to the ſquare on EI. </s>
            <s xml:id="echoid-s2636" xml:space="preserve">For it has
              <lb/>
            been proved (Lem. </s>
            <s xml:id="echoid-s2637" xml:space="preserve">II.) </s>
            <s xml:id="echoid-s2638" xml:space="preserve">that when the ratio of the rectangle contained by
              <lb/>
            AO and UO to that contained by EO and IO; </s>
            <s xml:id="echoid-s2639" xml:space="preserve">or, which is the ſame thing,
              <lb/>
            the given ratio of R to S is the leaſt poſſible, the ſquare on EO will be to
              <lb/>
            the ſquare on IO as the rectangle contained by AE and UE is to that con-
              <lb/>
            tained by AI and UI; </s>
            <s xml:id="echoid-s2640" xml:space="preserve">and (
              <emph style="sc">Lem</emph>
            . </s>
            <s xml:id="echoid-s2641" xml:space="preserve">V. </s>
            <s xml:id="echoid-s2642" xml:space="preserve">Fig. </s>
            <s xml:id="echoid-s2643" xml:space="preserve">31.) </s>
            <s xml:id="echoid-s2644" xml:space="preserve">that FG will then be the
              <lb/>
            fum of two mean proportionals between AE and UI, AI and UE: </s>
            <s xml:id="echoid-s2645" xml:space="preserve">it
              <lb/>
            therefore only remains to prove that the rectangle contained by AO and
              <lb/>
            UO is to that contained by EO and IO as the ſquare on FG is to the
              <lb/>
            fquare on EI. </s>
            <s xml:id="echoid-s2646" xml:space="preserve">Now it has been proved in demonſtrating Lem. </s>
            <s xml:id="echoid-s2647" xml:space="preserve">V. </s>
            <s xml:id="echoid-s2648" xml:space="preserve">that the
              <lb/>
            triangles EOG and IOF are ſimilar, and that the angle at V is right,
              <lb/>
            whence it follows that the triangles AOG and FOU are alſo ſimilar, and
              <lb/>
            conſequently that AO is to OG as OF is to UO; </s>
            <s xml:id="echoid-s2649" xml:space="preserve">therefore the rectangle
              <lb/>
            contained by AO and UO is equal to that contained by GO and OF. </s>
            <s xml:id="echoid-s2650" xml:space="preserve">More-
              <lb/>
            over GO is to OF as EO is to IO, and ſo by compoſition and permutation,
              <lb/>
            FG is to EI as OG is to EO, and as OF is to IO: </s>
            <s xml:id="echoid-s2651" xml:space="preserve">hence by compound
              <lb/>
            ratio the ſquare on FG is to the ſquare on EI as the rectangle contained
              <lb/>
            by (OG and OF) AO and UO is to that contained by EO and IO.</s>
            <s xml:id="echoid-s2652" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2653" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s2654" xml:space="preserve">E. </s>
            <s xml:id="echoid-s2655" xml:space="preserve">D.</s>
            <s xml:id="echoid-s2656" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2657" xml:space="preserve">
              <emph style="sc">Scholium</emph>
            . </s>
            <s xml:id="echoid-s2658" xml:space="preserve">In the four Caſes, wherein the given ratio is of a leſs to a
              <lb/>
            greater, and wherein the point O muſt fall between thoſe given ones which
              <lb/>
            bound the antecedent rectangle, the limiting ratio will be a maximum, and
              <lb/>
            the ſame with that which the ſquare on AU bears to the ſquare on FG.</s>
            <s xml:id="echoid-s2659" xml:space="preserve"/>
          </p>
        </div>
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