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-div77" type="section" level="1" n="73">
          <p>
            <s xml:id="echoid-s1845" xml:space="preserve">
              <pb o="[18]" file="0088" n="95"/>
            is equal to the rectangle AE, AQ; </s>
            <s xml:id="echoid-s1846" xml:space="preserve">and hence (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1847" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1848" xml:space="preserve">16.) </s>
            <s xml:id="echoid-s1849" xml:space="preserve">OQ is to AQ
              <lb/>
            as AE is to AO; </s>
            <s xml:id="echoid-s1850" xml:space="preserve">therefore, by compoſition or diviſion, AO is to AQ as
              <lb/>
            EO is to AO; </s>
            <s xml:id="echoid-s1851" xml:space="preserve">but by conſtruction, AQ is to R as P is to S, and ſo, by
              <lb/>
            compound ratio, the rectangle AO, AQ is to the rectangle R, AQ as the
              <lb/>
            rectangle EO, P is to the rectangle AO, S; </s>
            <s xml:id="echoid-s1852" xml:space="preserve">or (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1853" xml:space="preserve">V. </s>
            <s xml:id="echoid-s1854" xml:space="preserve">15. </s>
            <s xml:id="echoid-s1855" xml:space="preserve">and 16.) </s>
            <s xml:id="echoid-s1856" xml:space="preserve">the ſquare
              <lb/>
            on AO is to the rectangle EO, P as the rectangle AO, R is to the rectangle
              <lb/>
            AO, S; </s>
            <s xml:id="echoid-s1857" xml:space="preserve">that is, as R to S.</s>
            <s xml:id="echoid-s1858" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1859" xml:space="preserve">
              <emph style="sc">Scholium</emph>
            . </s>
            <s xml:id="echoid-s1860" xml:space="preserve">This Problem hath three Epitagmas alſo, which I ſtill enu-
              <lb/>
            merate as before. </s>
            <s xml:id="echoid-s1861" xml:space="preserve">The firſt and ſecond are conſtructed by Fig. </s>
            <s xml:id="echoid-s1862" xml:space="preserve">4, where DH
              <lb/>
            is drawn through F, the center of the circle on AQ: </s>
            <s xml:id="echoid-s1863" xml:space="preserve">and theſe have no limi-
              <lb/>
            tations. </s>
            <s xml:id="echoid-s1864" xml:space="preserve">The third is conſtructed as in Fig. </s>
            <s xml:id="echoid-s1865" xml:space="preserve">5, where DH is drawn parallel
              <lb/>
            to AQ; </s>
            <s xml:id="echoid-s1866" xml:space="preserve">and here the given ratio of R to S muſt not be leſs than the ratio
              <lb/>
            which four times AE bears to P: </s>
            <s xml:id="echoid-s1867" xml:space="preserve">for if it be, AE will be greater than
              <lb/>
            one-fourth Part of AQ (a fourth proportional to S, R and P) in which
              <lb/>
            caſe the rectangle contained by AE and AQ will be greater than the ſquare
              <lb/>
            on half AQ, and conſequently AD (a mean proportional between AE and
              <lb/>
            AQ) greater than half AQ; </s>
            <s xml:id="echoid-s1868" xml:space="preserve">but it is plain when this is the caſe, that DH
              <lb/>
            will neither cut nor touch the circle on AQ, and therefore the problem is
              <lb/>
            impoſſible.</s>
            <s xml:id="echoid-s1869" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div78" type="section" level="1" n="74">
          <head xml:id="echoid-head89" xml:space="preserve">PROBLEM IV. (Fig. 6. 7. and 8.)</head>
          <p>
            <s xml:id="echoid-s1870" xml:space="preserve">In any indefinite ſtraight line, let there be aſſigned the points A and E;
              <lb/>
            </s>
            <s xml:id="echoid-s1871" xml:space="preserve">it is required to cut it in another point O, ſo that the two ſquares on the
              <lb/>
            ſegments AO, EO, may obtain the Ratio of two given ſtraight lines,
              <lb/>
            R and S.</s>
            <s xml:id="echoid-s1872" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1873" xml:space="preserve">
              <emph style="sc">Analysis</emph>
            . </s>
            <s xml:id="echoid-s1874" xml:space="preserve">Imagine the thing to be effected, and that O is really the
              <lb/>
            point required: </s>
            <s xml:id="echoid-s1875" xml:space="preserve">then will the ſquare on AO be to the ſquare on EO as
              <lb/>
            R to S; </s>
            <s xml:id="echoid-s1876" xml:space="preserve">or (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1877" xml:space="preserve">V. </s>
            <s xml:id="echoid-s1878" xml:space="preserve">15.) </s>
            <s xml:id="echoid-s1879" xml:space="preserve">the ſquare on AO will be to the ſquare on EO
              <lb/>
            as the ſquare on R is to the rectangle contained by R and S. </s>
            <s xml:id="echoid-s1880" xml:space="preserve">Let DE be
              <lb/>
            made a mean proportional between EB (R) and EC (S). </s>
            <s xml:id="echoid-s1881" xml:space="preserve">Then
              <lb/>
            (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1882" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1883" xml:space="preserve">17.) </s>
            <s xml:id="echoid-s1884" xml:space="preserve">the ſquare on AO will be to the ſquare on EO as the
              <lb/>
            ſquare on R to the ſquare on DE; </s>
            <s xml:id="echoid-s1885" xml:space="preserve">and ſo (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1886" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1887" xml:space="preserve">22.) </s>
            <s xml:id="echoid-s1888" xml:space="preserve">AO to EO as R to
              <lb/>
            DE; </s>
            <s xml:id="echoid-s1889" xml:space="preserve">and hence both AO and EO will be given by the converſe of Prop. </s>
            <s xml:id="echoid-s1890" xml:space="preserve">38.
              <lb/>
            </s>
            <s xml:id="echoid-s1891" xml:space="preserve">of Eu. </s>
            <s xml:id="echoid-s1892" xml:space="preserve">Data.</s>
            <s xml:id="echoid-s1893" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>
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