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-div90" type="section" level="1" n="83">
          <head xml:id="echoid-head98" xml:space="preserve">PROBLEM VII. (Fig. 32, 33, 34, &c.)</head>
          <p>
            <s xml:id="echoid-s2337" xml:space="preserve">In any indeſinite ſtraight line let there be aſſigned the points A, E, I
              <lb/>
            and U; </s>
            <s xml:id="echoid-s2338" xml:space="preserve">it is required to cut it in another point, O, ſo that the rectangle
              <lb/>
            contained by the ſegments AO, UO may be to that contained by the ſeg-
              <lb/>
            ments EO, IO in the ratio of two given ſtraight lines, R and S.</s>
            <s xml:id="echoid-s2339" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2340" xml:space="preserve">
              <emph style="sc">Analysis</emph>
            . </s>
            <s xml:id="echoid-s2341" xml:space="preserve">Imagine the thing done, and O the point ſought: </s>
            <s xml:id="echoid-s2342" xml:space="preserve">then will
              <lb/>
            the rectangle AO, UO be to the rectangle EO, IO as R is to S. </s>
            <s xml:id="echoid-s2343" xml:space="preserve">Make
              <lb/>
            UC to EC as R is to S; </s>
            <s xml:id="echoid-s2344" xml:space="preserve">and the rectangle AO, UO will be the rectangle
              <lb/>
            EO, IO as UC is to EC. </s>
            <s xml:id="echoid-s2345" xml:space="preserve">Let now OB be taken a fourth proportional to
              <lb/>
            UO, UC and IO: </s>
            <s xml:id="echoid-s2346" xml:space="preserve">then (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2347" xml:space="preserve">V. </s>
            <s xml:id="echoid-s2348" xml:space="preserve">15.) </s>
            <s xml:id="echoid-s2349" xml:space="preserve">the rectangle AO, UO will be to
              <lb/>
            the rectangle EO, IO as the rectangle UC, OB is to the rectangle EC, OB;
              <lb/>
            </s>
            <s xml:id="echoid-s2350" xml:space="preserve">or (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2351" xml:space="preserve">V. </s>
            <s xml:id="echoid-s2352" xml:space="preserve">16.) </s>
            <s xml:id="echoid-s2353" xml:space="preserve">the rectangle AO, UO is to the rectangle UC, OB as the
              <lb/>
            rectangle EO, IO is to the rectangle EC, OB; </s>
            <s xml:id="echoid-s2354" xml:space="preserve">wherefore ſince UO is to UC
              <lb/>
            as IO to OB, by conſtruction, AO will be to BO as EO to EC; </s>
            <s xml:id="echoid-s2355" xml:space="preserve">and ſo by
              <lb/>
            compoſition or diviſion, CO is to CU as IB to BO, and AB is to BO as
              <lb/>
            CO to EC: </s>
            <s xml:id="echoid-s2356" xml:space="preserve">wherefore ex æquo perturb. </s>
            <s xml:id="echoid-s2357" xml:space="preserve">& </s>
            <s xml:id="echoid-s2358" xml:space="preserve">permut. </s>
            <s xml:id="echoid-s2359" xml:space="preserve">AB is to IB as UC to
              <lb/>
            EC, that is, in the given ratio; </s>
            <s xml:id="echoid-s2360" xml:space="preserve">and hence is given BC, the ſum or dif-
              <lb/>
            ference of CO and BO, as alſo the rectangle contained by them, equal to
              <lb/>
            the rectangle CU, IB, whence thoſe lines themſelves are given by the 85th
              <lb/>
            or 86th of the Data.</s>
            <s xml:id="echoid-s2361" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2362" xml:space="preserve">
              <emph style="sc">Synthesis</emph>
            . </s>
            <s xml:id="echoid-s2363" xml:space="preserve">Make AB to IB, and UC to EC in the given ratio, and de-
              <lb/>
            ſcribe on BC a circle; </s>
            <s xml:id="echoid-s2364" xml:space="preserve">erect, at B the indeſinite perpendicular BK, and take
              <lb/>
            therein BD a mean proportional between AB and EC, or between IB and
              <lb/>
            and UC: </s>
            <s xml:id="echoid-s2365" xml:space="preserve">from D, draw DH, parallel to BC, if O be required any where
              <lb/>
            between B and C; </s>
            <s xml:id="echoid-s2366" xml:space="preserve">but through F, the center of the circle on BC, if it be
              <lb/>
            ſought any where without them, cutting the circle on BC in H. </s>
            <s xml:id="echoid-s2367" xml:space="preserve">Laſtly,
              <lb/>
            draw HO perpendicular to DH, which will cut the indeſinite line in O,
              <lb/>
            the point required.</s>
            <s xml:id="echoid-s2368" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2369" xml:space="preserve">For it is plain from the conſtruction that HO and BD are equal, and
              <lb/>
            (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2370" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s2371" xml:space="preserve">17.) </s>
            <s xml:id="echoid-s2372" xml:space="preserve">the rectangle AB, EC, or the rectangle IB, UC is equal to
              <lb/>
            the ſquare on BD, and therefore equal to the ſquare on HO, which (
              <emph style="sc">Eu</emph>
            .
              <lb/>
            </s>
            <s xml:id="echoid-s2373" xml:space="preserve">III. </s>
            <s xml:id="echoid-s2374" xml:space="preserve">35. </s>
            <s xml:id="echoid-s2375" xml:space="preserve">36.) </s>
            <s xml:id="echoid-s2376" xml:space="preserve">is equal to the rectangle BO, OC. </s>
            <s xml:id="echoid-s2377" xml:space="preserve">Hence (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2378" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s2379" xml:space="preserve">16.) </s>
            <s xml:id="echoid-s2380" xml:space="preserve">AB
              <lb/>
            is to BO as CO is to CE, and CO is to CU as IB is to BO; </s>
            <s xml:id="echoid-s2381" xml:space="preserve">whence,
              <lb/>
            by compoſition or diviſion, AO is to BO as EO is to CE, and UO is </s>
          </p>
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