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-div79" type="section" level="1" n="75">
          <p>
            <s xml:id="echoid-s2002" xml:space="preserve">
              <pb o="[21]" file="0091" n="98"/>
            bears therefrom, and DH drawn through F, the center of the circle on EQ:
              <lb/>
            </s>
            <s xml:id="echoid-s2003" xml:space="preserve">none of theſe Caſes are ſubject to any Limitations.</s>
            <s xml:id="echoid-s2004" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2005" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            II. </s>
            <s xml:id="echoid-s2006" xml:space="preserve">Wherein A is the middle point, and the Caſes, when O
              <lb/>
            is ſought beyond E, between E and A, between A and I or beyond I. </s>
            <s xml:id="echoid-s2007" xml:space="preserve">The
              <lb/>
            firſt and third of which are conſtructed at once by Fig. </s>
            <s xml:id="echoid-s2008" xml:space="preserve">11, wherein IQ is
              <lb/>
            ſet off from I towards A and DH drawn through F, the center of the circle
              <lb/>
            on EQ. </s>
            <s xml:id="echoid-s2009" xml:space="preserve">The ſecond and fourth are conſtructed at once, alſo, by Fig. </s>
            <s xml:id="echoid-s2010" xml:space="preserve">12.
              <lb/>
            </s>
            <s xml:id="echoid-s2011" xml:space="preserve">where IQ is ſet off from I the contrary way to that which A lies, and DH
              <lb/>
            drawn parallel to EQ. </s>
            <s xml:id="echoid-s2012" xml:space="preserve">There are no Limitations to any of theſe Caſes.</s>
            <s xml:id="echoid-s2013" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2014" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            III. </s>
            <s xml:id="echoid-s2015" xml:space="preserve">Here, E being the middle point, the Caſes are, when O
              <lb/>
            muſt lie beyond A, or between E and I; </s>
            <s xml:id="echoid-s2016" xml:space="preserve">and the ſame Caſes occur when
              <lb/>
            I is made the middle point. </s>
            <s xml:id="echoid-s2017" xml:space="preserve">The firſt is conſtructed by Fig. </s>
            <s xml:id="echoid-s2018" xml:space="preserve">13, the ſecond
              <lb/>
            by Fig. </s>
            <s xml:id="echoid-s2019" xml:space="preserve">14, the third by Fig. </s>
            <s xml:id="echoid-s2020" xml:space="preserve">15, and the fourth by Fig. </s>
            <s xml:id="echoid-s2021" xml:space="preserve">16: </s>
            <s xml:id="echoid-s2022" xml:space="preserve">in every one
              <lb/>
            of which IQ is ſet off from I towards A, and DH drawn parallel to EQ.
              <lb/>
            </s>
            <s xml:id="echoid-s2023" xml:space="preserve">The Limits are that the given ratio of R to S, muſt not be leſs than the ratio
              <lb/>
            which the rectangle AE, P bears to the ſquare on half the Sum, or half the
              <lb/>
            difference of AE, and a fourth propor tional to R, S and P; </s>
            <s xml:id="echoid-s2024" xml:space="preserve">that is, to the
              <lb/>
            ſquare on half EQ: </s>
            <s xml:id="echoid-s2025" xml:space="preserve">ſince if it ſhould, the rectangle contained by AE and
              <lb/>
            the ſaid fourth proportional will be greater than the ſquare on half EQ; </s>
            <s xml:id="echoid-s2026" xml:space="preserve">
              <lb/>
            and of courſe ED (a mean proportional between them) greater than half
              <lb/>
            EQ, in which Caſe DH can neither cut nor touch the circle on EQ, and
              <lb/>
            ſo the problem be impoſſible. </s>
            <s xml:id="echoid-s2027" xml:space="preserve">It is farther obſervable in the two laſt caſes,
              <lb/>
            that to have the former of them poſſible, AE muſt be leſs, and to have
              <lb/>
            the latter poſſible, EI muſt be greater than the above-mentioned half
              <lb/>
            ſum; </s>
            <s xml:id="echoid-s2028" xml:space="preserve">for if this latter part of the Limitation be not obſerved, theſe caſes
              <lb/>
            are changed into one another.</s>
            <s xml:id="echoid-s2029" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div80" type="section" level="1" n="76">
          <head xml:id="echoid-head91" xml:space="preserve">PROBLEM VI.
            <lb/>
          (Fig. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.)</head>
          <p>
            <s xml:id="echoid-s2030" xml:space="preserve">In any indefinite ſtraight line let there be aſſigned the points A, E and I;
              <lb/>
            </s>
            <s xml:id="echoid-s2031" xml:space="preserve">it is required to cut it in another point O, ſo that the rectangle contained
              <lb/>
            by the ſegments AO, EO may be to the ſquare on IO in the ratio of two
              <lb/>
            given ſtraight lines, R and S.</s>
            <s xml:id="echoid-s2032" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>
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