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.
[52.] PROBLEM XI.
[53.] PROBLEM XII.
[54.] PROBLEM XIII.
[55.] PROBLEM XIV.
[56.] PROBLEM XV.
[57.] Synopſis of the PROBLEMS.
[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.
[59.] ADVERTISEMENT.
[60.] EXTRACT from PAPPUS's Preface to his Seventh Book in Dr. HALLEY's Tranſlation. DE SECTIONE DETERMINATA II.
[61.] THE PREFACE.
[62.] PROBLEMS CONCERNING DETERMINATE SECTION. PROBLEM I.
[63.] LEMMA I.
[64.] LEMMA II.
[65.] LEMMA III.
[66.] PROBLEM II.
[67.] LEMMA IV.
[68.] LEMMA V.
[69.] PROBLEM III.
[70.] PROBLEM IV.
[71.] DETERMINATE SECTION. BOOK I. PROBLEM I. (Fig. 1.)
[72.] PROBLEM II. (Fig. 2 and 3.)
[73.] PROBLEM III. (Fig. 4. and 5.)
[74.] PROBLEM IV. (Fig. 6. 7. and 8.)
[75.] PROBLEM V. (Fig. 9. 10. 11. 12. 13. 14. 15. 16.)
[76.] PROBLEM VI. (Fig. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.)
[77.] THE END OF BOOK I.
[78.] DETERMINATE SECTION. BOOK II. LEMMA I.
[79.] LEMMA II.
[80.] LEMMA III.
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            <s xml:id="echoid-s2417" xml:space="preserve">
              <emph style="sc">Case</emph>
            II. </s>
            <s xml:id="echoid-s2418" xml:space="preserve">If the order oſ the given points be retained, but the ratio be of
              <lb/>
            a greater to a leſs, B muſt ſall beyond I with reſpect to A, and C beyond
              <lb/>
            E, as in Fig. </s>
            <s xml:id="echoid-s2419" xml:space="preserve">33, and DH muſt be drawn as before: </s>
            <s xml:id="echoid-s2420" xml:space="preserve">and it may be proved
              <lb/>
            by reaſonings ſimilar to thoſe uſed in the ſirſt Caſe that O will fall between
              <lb/>
            I and U, as was required.</s>
            <s xml:id="echoid-s2421" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2422" xml:space="preserve">
              <emph style="sc">Cases</emph>
            III and IV. </s>
            <s xml:id="echoid-s2423" xml:space="preserve">Theſe caſes are conſtructed exactly in the ſame man-
              <lb/>
            ner as Caſes I and II reſpectively; </s>
            <s xml:id="echoid-s2424" xml:space="preserve">and the reaſonings to prove that O will
              <lb/>
            fall as it ought arethe ſame with thoſe made uſe of in Caſe I, as will appear
              <lb/>
            by inſpecting Fig. </s>
            <s xml:id="echoid-s2425" xml:space="preserve">34. </s>
            <s xml:id="echoid-s2426" xml:space="preserve">and 35.</s>
            <s xml:id="echoid-s2427" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2428" xml:space="preserve">
              <emph style="sc">Case</emph>
            V. </s>
            <s xml:id="echoid-s2429" xml:space="preserve">Let now the order of the given points be A, E, U, I, and the
              <lb/>
            given ratio of a leſs to a greater. </s>
            <s xml:id="echoid-s2430" xml:space="preserve">Then muſt B be made to fall beyond A
              <lb/>
            and C beyond U, as in Fig. </s>
            <s xml:id="echoid-s2431" xml:space="preserve">36, and DH is here to be drawn parallel to
              <lb/>
            to BC: </s>
            <s xml:id="echoid-s2432" xml:space="preserve">and that O will fall as required may be made to appear thus. </s>
            <s xml:id="echoid-s2433" xml:space="preserve">Draw
              <lb/>
            EF and UG perpendicular to BC. </s>
            <s xml:id="echoid-s2434" xml:space="preserve">Now (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2435" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s2436" xml:space="preserve">13. </s>
            <s xml:id="echoid-s2437" xml:space="preserve">17.) </s>
            <s xml:id="echoid-s2438" xml:space="preserve">the ſquare on
              <lb/>
            EF is equal to the rectangle contained by EC and EB, the ſquare on UG
              <lb/>
            to the rectangle contained by UC and UB; </s>
            <s xml:id="echoid-s2439" xml:space="preserve">and, by conſtruction, the
              <lb/>
            ſquare on HO to the rectangle contained by EC and AB, or to the rect-
              <lb/>
            angle contained by UC and IB; </s>
            <s xml:id="echoid-s2440" xml:space="preserve">and ſince, by ſuppoſition, EB is greater
              <lb/>
            than AB, and UB leſs than IB, the rectangle EC, AB, or its equal, the
              <lb/>
            rectangle UC, IB will be leſs than the rectangle EC, EB, and greater than
              <lb/>
            the rectangle UC, UB; </s>
            <s xml:id="echoid-s2441" xml:space="preserve">and conſequently the ſquare on HO will be leſs
              <lb/>
            than the ſquare on EF and greater than the ſquare on UG; </s>
            <s xml:id="echoid-s2442" xml:space="preserve">and therefore
              <lb/>
            HO leſs than EF and greater than UG: </s>
            <s xml:id="echoid-s2443" xml:space="preserve">but this could not be the Caſe
              <lb/>
            unleſs O fell between E and U, as was was required.</s>
            <s xml:id="echoid-s2444" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2445" xml:space="preserve">
              <emph style="sc">Case</emph>
            VI. </s>
            <s xml:id="echoid-s2446" xml:space="preserve">If the order of the points be retained; </s>
            <s xml:id="echoid-s2447" xml:space="preserve">but the given ratio be
              <lb/>
            of a greater to a leſs, B muſt then fall beyond I, and C beyond E; </s>
            <s xml:id="echoid-s2448" xml:space="preserve">and
              <lb/>
            DH is drawn as in the preceding Caſe (See Fig. </s>
            <s xml:id="echoid-s2449" xml:space="preserve">37.) </s>
            <s xml:id="echoid-s2450" xml:space="preserve">moreover, that O
              <lb/>
            will fall between E and U may be made to appear, by reaſonings ſimilar to
              <lb/>
            thoſe there made uſe on the like occaſion.</s>
            <s xml:id="echoid-s2451" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2452" xml:space="preserve">
              <emph style="sc">Case</emph>
            VII. </s>
            <s xml:id="echoid-s2453" xml:space="preserve">Is conſtructed exactly in the ſame manner as Caſe V. </s>
            <s xml:id="echoid-s2454" xml:space="preserve">and,</s>
          </p>
          <p>
            <s xml:id="echoid-s2455" xml:space="preserve">
              <emph style="sc">Case</emph>
            VIII. </s>
            <s xml:id="echoid-s2456" xml:space="preserve">As Caſe VI: </s>
            <s xml:id="echoid-s2457" xml:space="preserve">the truth of which will appear by. </s>
            <s xml:id="echoid-s2458" xml:space="preserve">barely in-
              <lb/>
            ſpecting Fig. </s>
            <s xml:id="echoid-s2459" xml:space="preserve">38 and 39.</s>
            <s xml:id="echoid-s2460" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2461" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            II. </s>
            <s xml:id="echoid-s2462" xml:space="preserve">In this Epitagma there are alſo eight Caſes, viz. </s>
            <s xml:id="echoid-s2463" xml:space="preserve">when the
              <lb/>
            order of the given points is A, U, E, I; </s>
            <s xml:id="echoid-s2464" xml:space="preserve">A, U, I, E; </s>
            <s xml:id="echoid-s2465" xml:space="preserve">U, A, E, I; </s>
            <s xml:id="echoid-s2466" xml:space="preserve">or
              <lb/>
            U, A, I, E; </s>
            <s xml:id="echoid-s2467" xml:space="preserve">and the given ratio of a leſs to a greater: </s>
            <s xml:id="echoid-s2468" xml:space="preserve">and there are
              <lb/>
            four others wherein the order of the points are the ſame as in thoſe, </s>
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