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-div76" type="section" level="1" n="72">
          <p>
            <s xml:id="echoid-s1808" xml:space="preserve">
              <pb o="[17]" file="0087" n="94"/>
            and P will be greater than one to R and half AE, and of courſe, AQ
              <lb/>
            (a fourth proportional to S, R and P) greater than a third proportional
              <lb/>
            to P and half AE; </s>
            <s xml:id="echoid-s1809" xml:space="preserve">in which caſe the rectangle AQ, P will be greater
              <lb/>
            than the ſquare on half AE, and ſo AD (a mean proportional between
              <lb/>
            AQ and P) greater than half AE; </s>
            <s xml:id="echoid-s1810" xml:space="preserve">but when this happens, it is plain that
              <lb/>
            DH can neither cut nor touch the circle on AE, and therefore, the
              <lb/>
            problem becomes impoſſible.</s>
            <s xml:id="echoid-s1811" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div77" type="section" level="1" n="73">
          <head xml:id="echoid-head88" xml:space="preserve">PROBLEM III. (Fig. 4. and 5.)</head>
          <p>
            <s xml:id="echoid-s1812" xml:space="preserve">In any indefinite ſtraight line let there be aſſigned the points A and E;
              <lb/>
            </s>
            <s xml:id="echoid-s1813" xml:space="preserve">it is required to cut it in another point O, ſo that the ſquare on the ſegment
              <lb/>
            AO may be to the rectangle contained by the ſegment EO and a given line
              <lb/>
            P, in the ratio of two given ſtraight lines R and S.</s>
            <s xml:id="echoid-s1814" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1815" xml:space="preserve">
              <emph style="sc">Analysis</emph>
            . </s>
            <s xml:id="echoid-s1816" xml:space="preserve">Suppoſe the thing done, and that O is the point ſought:
              <lb/>
            </s>
            <s xml:id="echoid-s1817" xml:space="preserve">then will the ſquare on AO be to the rectangle EO, P as R to S. </s>
            <s xml:id="echoid-s1818" xml:space="preserve">Make
              <lb/>
            AQ to P as R is to S; </s>
            <s xml:id="echoid-s1819" xml:space="preserve">then will the ſquare on AO be to the rectangle EO,
              <lb/>
            P as AQ is to P; </s>
            <s xml:id="echoid-s1820" xml:space="preserve">or (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1821" xml:space="preserve">V. </s>
            <s xml:id="echoid-s1822" xml:space="preserve">15.) </s>
            <s xml:id="echoid-s1823" xml:space="preserve">the ſquare on AO is to the rectangle EO,
              <lb/>
            P as the rectangle AQ, AO is to the rectangle P, AO; </s>
            <s xml:id="echoid-s1824" xml:space="preserve">wherefore AO is to
              <lb/>
            EO as AQ to AO; </s>
            <s xml:id="echoid-s1825" xml:space="preserve">conſequently by compoſition, or diviſion, AO is to AE
              <lb/>
            as AQ is to OQ, and ſo (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1826" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1827" xml:space="preserve">16.) </s>
            <s xml:id="echoid-s1828" xml:space="preserve">the rectangle AO, OQ is equal to
              <lb/>
            the rectangle AE, AQ; </s>
            <s xml:id="echoid-s1829" xml:space="preserve">and hence, as the ſum or difference of AO and OQ
              <lb/>
            is alſo given, theſe lines themſelves are given by the 85th or 86th of the
              <lb/>
            Data.</s>
            <s xml:id="echoid-s1830" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1831" xml:space="preserve">
              <emph style="sc">Synthesis</emph>
            . </s>
            <s xml:id="echoid-s1832" xml:space="preserve">Take AQ a fourth proportional to S, P and R, and
              <lb/>
            deſcribe thereon a circle; </s>
            <s xml:id="echoid-s1833" xml:space="preserve">erect at A, the indefinite perpendicular AK, and
              <lb/>
            take therein AD, a mean proportional between AE and AQ; </s>
            <s xml:id="echoid-s1834" xml:space="preserve">from D,
              <lb/>
            draw DH, parallel to AE, if O be required beyond E; </s>
            <s xml:id="echoid-s1835" xml:space="preserve">but through F the
              <lb/>
            center of the circle on AQ, if it be ſought beyond A, or between A and
              <lb/>
            E, cutting the ſaid circle in H: </s>
            <s xml:id="echoid-s1836" xml:space="preserve">Laſtly, from H draw HO perpendicular
              <lb/>
            to DH, which will cut the indefinite line in O, the point required.</s>
            <s xml:id="echoid-s1837" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1838" xml:space="preserve">For it is plain from the Conſtruction, that AD and HO are equal; </s>
            <s xml:id="echoid-s1839" xml:space="preserve">and
              <lb/>
            (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1840" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1841" xml:space="preserve">17) the rectangle AE, AQ is equal to the ſquare on AD, and
              <lb/>
            therefore equal to the ſquare on HO; </s>
            <s xml:id="echoid-s1842" xml:space="preserve">but the ſquare on HO is equal to the
              <lb/>
            rectangle AO, OQ, (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1843" xml:space="preserve">III. </s>
            <s xml:id="echoid-s1844" xml:space="preserve">35. </s>
            <s xml:id="echoid-s1845" xml:space="preserve">36) conſequently the rectangle AO, </s>
          </p>
        </div>
      </text>
    </echo>
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