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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          <head xml:id="echoid-head82" xml:space="preserve">PROBLEM III.</head>
          <p>
            <s xml:id="echoid-s1354" xml:space="preserve">To cut a given indefinite right line in one point, ſo that of the three ſeg-
              <lb/>
            ments intercepted between the ſame, and three points given, the rectangle
              <lb/>
            under two of them may be to the ſquare of the remaining one in a given ratio.</s>
            <s xml:id="echoid-s1355" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1356" xml:space="preserve">In the indefinite line let the three points be A, E, I. </s>
            <s xml:id="echoid-s1357" xml:space="preserve">it is then required to be
              <lb/>
            cut again in O, ſo that OA x OE may be to
              <emph style="ol">OI</emph>
              <emph style="sub">2</emph>
            (let the ſituation of I be
              <lb/>
            what it may) in a given ratio, which ratio let be expreſſed by EL to LI.
              <lb/>
            </s>
            <s xml:id="echoid-s1358" xml:space="preserve">[And here I cannot but obſerve with
              <emph style="sc">Hugo</emph>
            D'
              <emph style="sc">Omerique</emph>
            , page 113. </s>
            <s xml:id="echoid-s1359" xml:space="preserve">that
              <lb/>
            this Problem, viz. </s>
            <s xml:id="echoid-s1360" xml:space="preserve">‘To exhibit two lines in a given ratio whoſe ſum, or whoſe
              <lb/>
            difference is given,’ ought to have had a place in the Elements as a Propoſition; </s>
            <s xml:id="echoid-s1361" xml:space="preserve">
              <lb/>
            or at leaſt to have been annext as a Scholium to the 9th or 10th of the VIth
              <lb/>
            Book.</s>
            <s xml:id="echoid-s1362" xml:space="preserve">] And be the ſituation of L alſo what it may, either between A and E,
              <lb/>
            or between A and I, or between E and I, or beyond either extreme. </s>
            <s xml:id="echoid-s1363" xml:space="preserve">To the three
              <lb/>
            points E, L, I, and the right line AI, let be found, by
              <emph style="sc">Problem</emph>
            II, a fourth
              <lb/>
            point O ſuch, that AI x OE: </s>
            <s xml:id="echoid-s1364" xml:space="preserve">OI x OL:</s>
            <s xml:id="echoid-s1365" xml:space="preserve">: EI: </s>
            <s xml:id="echoid-s1366" xml:space="preserve">IL. </s>
            <s xml:id="echoid-s1367" xml:space="preserve">And let ſuch a Caſe be
              <lb/>
            choſen of
              <emph style="sc">Problem</emph>
            II, that, according as AO is greater or leſs than AI, ſo of
              <lb/>
            the three rectangles, deſcribed in
              <emph style="sc">Lemma</emph>
            V, made by the four points E, O,
              <lb/>
            I, L, that of IO x EL may accordingly be greater or leſs than that of EI x OL.</s>
            <s xml:id="echoid-s1368" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1369" xml:space="preserve">D
              <emph style="sc">EMONSTRATION</emph>
            .</s>
            <s xml:id="echoid-s1370" xml:space="preserve">On ſuppoſition then that ſuch a Caſe of
              <emph style="sc">Problem</emph>
            II. </s>
            <s xml:id="echoid-s1371" xml:space="preserve">is
              <lb/>
            made uſe of, we have
              <lb/>
            AI x OE: </s>
            <s xml:id="echoid-s1372" xml:space="preserve">OI x OL:</s>
            <s xml:id="echoid-s1373" xml:space="preserve">: EI: </s>
            <s xml:id="echoid-s1374" xml:space="preserve">IL</s>
          </p>
          <p>
            <s xml:id="echoid-s1375" xml:space="preserve">And by
              <emph style="sc">Lemma</emph>
            IV, OL x EI: </s>
            <s xml:id="echoid-s1376" xml:space="preserve">OE x IL:</s>
            <s xml:id="echoid-s1377" xml:space="preserve">: AI: </s>
            <s xml:id="echoid-s1378" xml:space="preserve">OI</s>
          </p>
          <p>
            <s xml:id="echoid-s1379" xml:space="preserve">And by Diviſion or Compoſition EL x OI: </s>
            <s xml:id="echoid-s1380" xml:space="preserve">OE x IL:</s>
            <s xml:id="echoid-s1381" xml:space="preserve">: AO: </s>
            <s xml:id="echoid-s1382" xml:space="preserve">OI</s>
          </p>
          <p>
            <s xml:id="echoid-s1383" xml:space="preserve">This appears from
              <emph style="sc">Lemma</emph>
            V.</s>
            <s xml:id="echoid-s1384" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1385" xml:space="preserve">Then again by
              <emph style="sc">Lemma</emph>
            IV, AO x OE: </s>
            <s xml:id="echoid-s1386" xml:space="preserve">
              <emph style="ol">OI</emph>
              <emph style="sub">2</emph>
            :</s>
            <s xml:id="echoid-s1387" xml:space="preserve">: EL: </s>
            <s xml:id="echoid-s1388" xml:space="preserve">IL.</s>
            <s xml:id="echoid-s1389" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1390" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s1391" xml:space="preserve">E. </s>
            <s xml:id="echoid-s1392" xml:space="preserve">D.</s>
            <s xml:id="echoid-s1393" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1394" xml:space="preserve">This Problem has two
              <emph style="sc">Epitagmas</emph>
            . </s>
            <s xml:id="echoid-s1395" xml:space="preserve">The firſt wherein OI, whoſe ſquare is
              <lb/>
            ſought, is bounded by I an extreme point of the three given ones. </s>
            <s xml:id="echoid-s1396" xml:space="preserve">And this
              <lb/>
            again admits of three Caſes. </s>
            <s xml:id="echoid-s1397" xml:space="preserve">The ſecond is when the point I is the middle
              <lb/>
            point. </s>
            <s xml:id="echoid-s1398" xml:space="preserve">And this again has three caſes. </s>
            <s xml:id="echoid-s1399" xml:space="preserve">And there remain two Anomalous
              <lb/>
            Caſes, wherein Problem II. </s>
            <s xml:id="echoid-s1400" xml:space="preserve">is of no uſe, which muſt therefore be conſtructed
              <lb/>
            by themſelves.</s>
            <s xml:id="echoid-s1401" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1402" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            I. </s>
            <s xml:id="echoid-s1403" xml:space="preserve">
              <emph style="sc">Case</emph>
            I. </s>
            <s xml:id="echoid-s1404" xml:space="preserve">Let the ratio given, EL to LI, be inequalitatis
              <lb/>
            majoris, i. </s>
            <s xml:id="echoid-s1405" xml:space="preserve">e. </s>
            <s xml:id="echoid-s1406" xml:space="preserve">of a greater to a leſs; </s>
            <s xml:id="echoid-s1407" xml:space="preserve">and the point O ſought be required to lie
              <lb/>
            between I and the next point to it E, or elſe to lie beyond I the other way;
              <lb/>
            </s>
            <s xml:id="echoid-s1408" xml:space="preserve">for the ſame conſtruction ſerves for both. </s>
            <s xml:id="echoid-s1409" xml:space="preserve">Here
              <emph style="sc">Case</emph>
            I. </s>
            <s xml:id="echoid-s1410" xml:space="preserve">of
              <emph style="sc">Problem</emph>
            II. </s>
            <s xml:id="echoid-s1411" xml:space="preserve">is </s>
          </p>
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