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-div72" type="section" level="1" n="68">
          <pb o="[7]" file="0077" n="84"/>
        </div>
        <div xml:id="echoid-div73" type="section" level="1" n="69">
          <head xml:id="echoid-head82" xml:space="preserve">PROBLEM III.</head>
          <p>
            <s xml:id="echoid-s1354" xml:space="preserve">To cut a given indefinite right line in one point, ſo that of the three ſeg-
              <lb/>
            ments intercepted between the ſame, and three points given, the rectangle
              <lb/>
            under two of them may be to the ſquare of the remaining one in a given ratio.</s>
            <s xml:id="echoid-s1355" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1356" xml:space="preserve">In the indefinite line let the three points be A, E, I. </s>
            <s xml:id="echoid-s1357" xml:space="preserve">it is then required to be
              <lb/>
            cut again in O, ſo that OA x OE may be to
              <emph style="ol">OI</emph>
              <emph style="sub">2</emph>
            (let the ſituation of I be
              <lb/>
            what it may) in a given ratio, which ratio let be expreſſed by EL to LI.
              <lb/>
            </s>
            <s xml:id="echoid-s1358" xml:space="preserve">[And here I cannot but obſerve with
              <emph style="sc">Hugo</emph>
            D'
              <emph style="sc">Omerique</emph>
            , page 113. </s>
            <s xml:id="echoid-s1359" xml:space="preserve">that
              <lb/>
            this Problem, viz. </s>
            <s xml:id="echoid-s1360" xml:space="preserve">‘To exhibit two lines in a given ratio whoſe ſum, or whoſe
              <lb/>
            difference is given,’ ought to have had a place in the Elements as a Propoſition; </s>
            <s xml:id="echoid-s1361" xml:space="preserve">
              <lb/>
            or at leaſt to have been annext as a Scholium to the 9th or 10th of the VIth
              <lb/>
            Book.</s>
            <s xml:id="echoid-s1362" xml:space="preserve">] And be the ſituation of L alſo what it may, either between A and E,
              <lb/>
            or between A and I, or between E and I, or beyond either extreme. </s>
            <s xml:id="echoid-s1363" xml:space="preserve">To the three
              <lb/>
            points E, L, I, and the right line AI, let be found, by
              <emph style="sc">Problem</emph>
            II, a fourth
              <lb/>
            point O ſuch, that AI x OE: </s>
            <s xml:id="echoid-s1364" xml:space="preserve">OI x OL:</s>
            <s xml:id="echoid-s1365" xml:space="preserve">: EI: </s>
            <s xml:id="echoid-s1366" xml:space="preserve">IL. </s>
            <s xml:id="echoid-s1367" xml:space="preserve">And let ſuch a Caſe be
              <lb/>
            choſen of
              <emph style="sc">Problem</emph>
            II, that, according as AO is greater or leſs than AI, ſo of
              <lb/>
            the three rectangles, deſcribed in
              <emph style="sc">Lemma</emph>
            V, made by the four points E, O,
              <lb/>
            I, L, that of IO x EL may accordingly be greater or leſs than that of EI x OL.</s>
            <s xml:id="echoid-s1368" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1369" xml:space="preserve">D
              <emph style="sc">EMONSTRATION</emph>
            .</s>
            <s xml:id="echoid-s1370" xml:space="preserve">On ſuppoſition then that ſuch a Caſe of
              <emph style="sc">Problem</emph>
            II. </s>
            <s xml:id="echoid-s1371" xml:space="preserve">is
              <lb/>
            made uſe of, we have
              <lb/>
            AI x OE: </s>
            <s xml:id="echoid-s1372" xml:space="preserve">OI x OL:</s>
            <s xml:id="echoid-s1373" xml:space="preserve">: EI: </s>
            <s xml:id="echoid-s1374" xml:space="preserve">IL</s>
          </p>
          <p>
            <s xml:id="echoid-s1375" xml:space="preserve">And by
              <emph style="sc">Lemma</emph>
            IV, OL x EI: </s>
            <s xml:id="echoid-s1376" xml:space="preserve">OE x IL:</s>
            <s xml:id="echoid-s1377" xml:space="preserve">: AI: </s>
            <s xml:id="echoid-s1378" xml:space="preserve">OI</s>
          </p>
          <p>
            <s xml:id="echoid-s1379" xml:space="preserve">And by Diviſion or Compoſition EL x OI: </s>
            <s xml:id="echoid-s1380" xml:space="preserve">OE x IL:</s>
            <s xml:id="echoid-s1381" xml:space="preserve">: AO: </s>
            <s xml:id="echoid-s1382" xml:space="preserve">OI</s>
          </p>
          <p>
            <s xml:id="echoid-s1383" xml:space="preserve">This appears from
              <emph style="sc">Lemma</emph>
            V.</s>
            <s xml:id="echoid-s1384" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1385" xml:space="preserve">Then again by
              <emph style="sc">Lemma</emph>
            IV, AO x OE: </s>
            <s xml:id="echoid-s1386" xml:space="preserve">
              <emph style="ol">OI</emph>
              <emph style="sub">2</emph>
            :</s>
            <s xml:id="echoid-s1387" xml:space="preserve">: EL: </s>
            <s xml:id="echoid-s1388" xml:space="preserve">IL.</s>
            <s xml:id="echoid-s1389" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1390" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s1391" xml:space="preserve">E. </s>
            <s xml:id="echoid-s1392" xml:space="preserve">D.</s>
            <s xml:id="echoid-s1393" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1394" xml:space="preserve">This Problem has two
              <emph style="sc">Epitagmas</emph>
            . </s>
            <s xml:id="echoid-s1395" xml:space="preserve">The firſt wherein OI, whoſe ſquare is
              <lb/>
            ſought, is bounded by I an extreme point of the three given ones. </s>
            <s xml:id="echoid-s1396" xml:space="preserve">And this
              <lb/>
            again admits of three Caſes. </s>
            <s xml:id="echoid-s1397" xml:space="preserve">The ſecond is when the point I is the middle
              <lb/>
            point. </s>
            <s xml:id="echoid-s1398" xml:space="preserve">And this again has three caſes. </s>
            <s xml:id="echoid-s1399" xml:space="preserve">And there remain two Anomalous
              <lb/>
            Caſes, wherein Problem II. </s>
            <s xml:id="echoid-s1400" xml:space="preserve">is of no uſe, which muſt therefore be conſtructed
              <lb/>
            by themſelves.</s>
            <s xml:id="echoid-s1401" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1402" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            I. </s>
            <s xml:id="echoid-s1403" xml:space="preserve">
              <emph style="sc">Case</emph>
            I. </s>
            <s xml:id="echoid-s1404" xml:space="preserve">Let the ratio given, EL to LI, be inequalitatis
              <lb/>
            majoris, i. </s>
            <s xml:id="echoid-s1405" xml:space="preserve">e. </s>
            <s xml:id="echoid-s1406" xml:space="preserve">of a greater to a leſs; </s>
            <s xml:id="echoid-s1407" xml:space="preserve">and the point O ſought be required to lie
              <lb/>
            between I and the next point to it E, or elſe to lie beyond I the other way;
              <lb/>
            </s>
            <s xml:id="echoid-s1408" xml:space="preserve">for the ſame conſtruction ſerves for both. </s>
            <s xml:id="echoid-s1409" xml:space="preserve">Here
              <emph style="sc">Case</emph>
            I. </s>
            <s xml:id="echoid-s1410" xml:space="preserve">of
              <emph style="sc">Problem</emph>
            II. </s>
            <s xml:id="echoid-s1411" xml:space="preserve">is </s>
          </p>
        </div>
      </text>
    </echo>
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