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-div75" type="section" level="1" n="71">
          <head xml:id="echoid-head84" xml:space="preserve">DETERMINATE SECTION.</head>
          <head xml:id="echoid-head85" xml:space="preserve">BOOK I.</head>
          <head xml:id="echoid-head86" xml:space="preserve">PROBLEM I. (Fig. 1.)</head>
          <p>
            <s xml:id="echoid-s1710" xml:space="preserve">In any indefinite ſtraight line, let the Point A be aſſigned; </s>
            <s xml:id="echoid-s1711" xml:space="preserve">it is required
              <lb/>
            to cut it in ſome other point O, ſo that the ſquare on the ſegment AO
              <lb/>
            may be to the ſquare on a given line, P, in the ratio of two given ſtraight
              <lb/>
            lines R and S.</s>
            <s xml:id="echoid-s1712" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1713" xml:space="preserve">
              <emph style="sc">Analysis</emph>
            . </s>
            <s xml:id="echoid-s1714" xml:space="preserve">Since, by Hypotheſis, the ſquare on AO muſt be to the
              <lb/>
            ſquare on P as R is to S, the ſquare on AO will be to the Square on P as
              <lb/>
            the ſquare on R is to the rectangle contained by R and S (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1715" xml:space="preserve">V. </s>
            <s xml:id="echoid-s1716" xml:space="preserve">15.)
              <lb/>
            </s>
            <s xml:id="echoid-s1717" xml:space="preserve">Let there be taken AD, a mean proportional between AB (R) and AC
              <lb/>
            (S); </s>
            <s xml:id="echoid-s1718" xml:space="preserve">then the Square on AO is to the ſquare on P as the ſquare on R is
              <lb/>
            to the ſquare on AD, or (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1719" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1720" xml:space="preserve">22) AO is to P as R to AD; </s>
            <s xml:id="echoid-s1721" xml:space="preserve">conſe-
              <lb/>
            quently, AO is given by
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1722" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1723" xml:space="preserve">12.</s>
            <s xml:id="echoid-s1724" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1725" xml:space="preserve">
              <emph style="sc">Synthesis</emph>
            . </s>
            <s xml:id="echoid-s1726" xml:space="preserve">Make AB equal to R, AC equal to S, and deſcribe on
              <lb/>
            BC a ſemi-circle; </s>
            <s xml:id="echoid-s1727" xml:space="preserve">erect at A the indefinite perpendicular AF, meeting the
              <lb/>
            circle in D, and take AF equal to P; </s>
            <s xml:id="echoid-s1728" xml:space="preserve">draw DB, and parallel thereto FO,
              <lb/>
            meeting the indefinite line in O, the point required.</s>
            <s xml:id="echoid-s1729" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1730" xml:space="preserve">For, by reaſon of the ſimilar triangles ADB, AFO, AO is to AF (P) as
              <lb/>
            AB (R) is to AD; </s>
            <s xml:id="echoid-s1731" xml:space="preserve">therefore (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1732" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1733" xml:space="preserve">22.) </s>
            <s xml:id="echoid-s1734" xml:space="preserve">the ſquare on AO is to the
              <lb/>
            ſquare on P as the ſquare on R is to the ſquare on AD; </s>
            <s xml:id="echoid-s1735" xml:space="preserve">but the ſquare on
              <lb/>
            AD is equal to the rectangle contained by AB (R) and AC (S) by
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1736" xml:space="preserve">VI.
              <lb/>
            </s>
            <s xml:id="echoid-s1737" xml:space="preserve">13. </s>
            <s xml:id="echoid-s1738" xml:space="preserve">17; </s>
            <s xml:id="echoid-s1739" xml:space="preserve">and ſo the ſquare on AO is to the ſquare on P as the ſquare on R
              <lb/>
            is to the rectangle contained by R and S; </s>
            <s xml:id="echoid-s1740" xml:space="preserve">that is (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1741" xml:space="preserve">V. </s>
            <s xml:id="echoid-s1742" xml:space="preserve">15.) </s>
            <s xml:id="echoid-s1743" xml:space="preserve">as R is to S.</s>
            <s xml:id="echoid-s1744" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1745" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s1746" xml:space="preserve">E. </s>
            <s xml:id="echoid-s1747" xml:space="preserve">D.</s>
            <s xml:id="echoid-s1748" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1749" xml:space="preserve">
              <emph style="sc">Scholium</emph>
            . </s>
            <s xml:id="echoid-s1750" xml:space="preserve">Here are no limitations, nor any precautions whatever to be
              <lb/>
            obſerved, except that AB (R) muſt be ſet off from A that way which O
              <lb/>
            is required to fall.</s>
            <s xml:id="echoid-s1751" xml:space="preserve"/>
          </p>
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