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-div74" type="section" level="1" n="70">
          <pb o="[10]" file="0080" n="87"/>
          <p>
            <s xml:id="echoid-s1492" xml:space="preserve">In the indefinite line let the four points be A, E, I, U. </s>
            <s xml:id="echoid-s1493" xml:space="preserve">It is then required
              <lb/>
            to be cut again in O ſo that OA x OU may be to OE x OI (be the Poſition
              <lb/>
            of the four given points what they may) in the ratio of AL to LE, let the
              <lb/>
            point L fall alſo as it may.</s>
            <s xml:id="echoid-s1494" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1495" xml:space="preserve">
              <emph style="sc">Construction</emph>
            . </s>
            <s xml:id="echoid-s1496" xml:space="preserve">To the three points E, L, U, and the right line UI, let
              <lb/>
            be found by the IId
              <emph style="sc">Problem</emph>
            a fourth point O, ſo that UI x OE may be to
              <lb/>
            OU x OL as AE to AL. </s>
            <s xml:id="echoid-s1497" xml:space="preserve">And let ſuch a
              <emph style="sc">Case</emph>
            be choſen of the IId
              <emph style="sc">Problem</emph>
              <lb/>
            that, according as UO is required greater or leſs than UI, or their ſum ſhall
              <lb/>
            conſtitute OI, ſo of the three rectangles deſcribed in the Vth
              <emph style="sc">Lemma</emph>
            made
              <lb/>
            by the four points E, O, A, L, that of OE x AL may accordingly be greater
              <lb/>
            or leſs than OL x AE, or their ſum conſtitute that of OA x EL.</s>
            <s xml:id="echoid-s1498" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1499" xml:space="preserve">N. </s>
            <s xml:id="echoid-s1500" xml:space="preserve">B. </s>
            <s xml:id="echoid-s1501" xml:space="preserve">U being now uſed to repreſent one of the given points, in all the
              <lb/>
            following Diagrams I have ſubſtituted V in the place where U was uſed
              <lb/>
            before.</s>
            <s xml:id="echoid-s1502" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1503" xml:space="preserve">
              <emph style="sc">Demonstration</emph>
            . </s>
            <s xml:id="echoid-s1504" xml:space="preserve">On ſuppoſition therefore that ſuch a
              <emph style="sc">Case</emph>
            of the
              <lb/>
            IId
              <emph style="sc">Problem</emph>
            is made uſe of,
              <lb/>
            We have UI x OE: </s>
            <s xml:id="echoid-s1505" xml:space="preserve">OU x OL:</s>
            <s xml:id="echoid-s1506" xml:space="preserve">: AE: </s>
            <s xml:id="echoid-s1507" xml:space="preserve">AL</s>
          </p>
          <p>
            <s xml:id="echoid-s1508" xml:space="preserve">And by Inverſion OU x OL: </s>
            <s xml:id="echoid-s1509" xml:space="preserve">UI x OE:</s>
            <s xml:id="echoid-s1510" xml:space="preserve">: AL: </s>
            <s xml:id="echoid-s1511" xml:space="preserve">AE</s>
          </p>
          <p>
            <s xml:id="echoid-s1512" xml:space="preserve">And by
              <emph style="sc">Lemma</emph>
            IV. </s>
            <s xml:id="echoid-s1513" xml:space="preserve">AL x OE: </s>
            <s xml:id="echoid-s1514" xml:space="preserve">AE x OL:</s>
            <s xml:id="echoid-s1515" xml:space="preserve">: OU: </s>
            <s xml:id="echoid-s1516" xml:space="preserve">UI</s>
          </p>
          <p>
            <s xml:id="echoid-s1517" xml:space="preserve">Hence by Compoſition or Diviſion, &</s>
            <s xml:id="echoid-s1518" xml:space="preserve">c. </s>
            <s xml:id="echoid-s1519" xml:space="preserve">AL x OE: </s>
            <s xml:id="echoid-s1520" xml:space="preserve">OA x LE:</s>
            <s xml:id="echoid-s1521" xml:space="preserve">: OU
              <lb/>
            : </s>
            <s xml:id="echoid-s1522" xml:space="preserve">OI as appears by
              <emph style="sc">Lemma</emph>
            V.</s>
            <s xml:id="echoid-s1523" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1524" xml:space="preserve">Then again by
              <emph style="sc">Lemma</emph>
            IV. </s>
            <s xml:id="echoid-s1525" xml:space="preserve">OU x OA: </s>
            <s xml:id="echoid-s1526" xml:space="preserve">OI x OE:</s>
            <s xml:id="echoid-s1527" xml:space="preserve">: AL: </s>
            <s xml:id="echoid-s1528" xml:space="preserve">LE</s>
          </p>
          <p>
            <s xml:id="echoid-s1529" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s1530" xml:space="preserve">E. </s>
            <s xml:id="echoid-s1531" xml:space="preserve">D.</s>
            <s xml:id="echoid-s1532" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1533" xml:space="preserve">This
              <emph style="sc">Problem</emph>
            has three
              <emph style="sc">Epitagmas</emph>
            . </s>
            <s xml:id="echoid-s1534" xml:space="preserve">The Iſt whereof is when of the
              <lb/>
            two aſſigned points A and U, the one of them is an extreme, and the other an
              <lb/>
            alternate mean; </s>
            <s xml:id="echoid-s1535" xml:space="preserve">and this admits of three
              <emph style="sc">Cases</emph>
            . </s>
            <s xml:id="echoid-s1536" xml:space="preserve">The IId is when A and U
              <lb/>
            are both of them extremes; </s>
            <s xml:id="echoid-s1537" xml:space="preserve">and this has four
              <emph style="sc">Cases</emph>
            . </s>
            <s xml:id="echoid-s1538" xml:space="preserve">The IIId is when of
              <lb/>
            A and U one of them is an extreme, and the other is the point next to it; </s>
            <s xml:id="echoid-s1539" xml:space="preserve">and
              <lb/>
            this has three Caſes. </s>
            <s xml:id="echoid-s1540" xml:space="preserve">And there remain three more Anomalous Caſes, wherein
              <lb/>
            the IId
              <emph style="sc">Problem</emph>
            is of no uſe, but which may be reduced to one, as ſhall be
              <lb/>
            ſhewn in it's proper place.</s>
            <s xml:id="echoid-s1541" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1542" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            I. </s>
            <s xml:id="echoid-s1543" xml:space="preserve">
              <emph style="sc">Case</emph>
            I. </s>
            <s xml:id="echoid-s1544" xml:space="preserve">Let A the firſt aſſigned point be an extreme, and
              <lb/>
            U the ſecond aſſigned point be an alternate mean; </s>
            <s xml:id="echoid-s1545" xml:space="preserve">and let the point O be
              <lb/>
            ſought between the firſt aſſigned A and the next point to it E; </s>
            <s xml:id="echoid-s1546" xml:space="preserve">or between
              <lb/>
            the ſecond aſſigned U and the laſt I. </s>
            <s xml:id="echoid-s1547" xml:space="preserve">For the ſame Conſtruction ſerves
              <lb/>
            for both.</s>
            <s xml:id="echoid-s1548" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>
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