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

Table of contents

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[51.] PROBLEM X.
Page: 39 ((27))
[52.] PROBLEM XI.
Page: 39 ((27))
[53.] PROBLEM XII.
Page: 40 ((28))
[54.] PROBLEM XIII.
Page: 40 ((28))
[55.] PROBLEM XIV.
Page: 40 ((28))
[56.] PROBLEM XV.
Page: 40 ((28))
[57.] Synopſis of the PROBLEMS.
Page: 41 ((29))
[58.] THE TWO BOOKS OF APOLLONIUS PERGÆUS, CONCERNING DETERMINATE SECTION, As they have been Reſtored by WILLEBRORDUS SNELLIUS. By JOHN LAWSON, B. D. Rector of Swanſcombe, Kent. TO WHICH ARE ADDED, THE SAME TWO BOOKS, BY WILLIAM WALES, BEING AN ENTIRE NEW WORK. LONDON: Printed by G. BIGG, Succeſſor to D. LEACH. And ſold by B. White, in Fleet-Street; L. Davis, in Holborne; J. Nourse, in the Strand; and T. Payne, near the Mews-Gate. MDCC LXXII.
Page: 64
[59.] ADVERTISEMENT.
Page: 66 ([i])
[60.] EXTRACT from PAPPUS's Preface to his Seventh Book in Dr. HALLEY's Tranſlation. DE SECTIONE DETERMINATA II.
Page: 68 ([iii])
[61.] THE PREFACE.
Page: 70 ([v])
[62.] PROBLEMS CONCERNING DETERMINATE SECTION. PROBLEM I.
Page: 78
[63.] LEMMA I.
Page: 79 ([2])
[64.] LEMMA II.
Page: 80 ([3])
[65.] LEMMA III.
Page: 80 ([3])
[66.] PROBLEM II.
Page: 80 ([3])
[67.] LEMMA IV.
Page: 83 ([6])
[68.] LEMMA V.
Page: 83 ([6])
[69.] PROBLEM III.
Page: 84 ([7])
[70.] PROBLEM IV.
Page: 86 ([9])
[71.] DETERMINATE SECTION. BOOK I. PROBLEM I. (Fig. 1.)
Page: 92 ([15])
[72.] PROBLEM II. (Fig. 2 and 3.)
Page: 93 ([16])
[73.] PROBLEM III. (Fig. 4. and 5.)
Page: 94 ([17])
[74.] PROBLEM IV. (Fig. 6. 7. and 8.)
Page: 95 ([18])
[75.] PROBLEM V. (Fig. 9. 10. 11. 12. 13. 14. 15. 16.)
Page: 96 ([19])
[76.] PROBLEM VI. (Fig. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.)
Page: 98 ([21])
[77.] THE END OF BOOK I.
Page: 102 ([25])
[78.] DETERMINATE SECTION. BOOK II. LEMMA I.
Page: 103 ([26])
[79.] LEMMA II.
Page: 103 ([26])
[80.] LEMMA III.
Page: 104 ([27])
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            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>
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        <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>
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