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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[41.] PROBLEM V.
[42.] PROBLEM VI.
[43.] PROBLEM VII.
[44.] LEMMA I.
[45.] LEMMA II.
[46.] LEMMA III.
[47.] LEMMA IV.
[48.] LEMMA V.
[49.] PROBLEM VIII.
[50.] PROBLEM IX.
[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.
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          <p>
            <s xml:id="echoid-s1268" xml:space="preserve">Therefore AT x AO is greater than o AO</s>
          </p>
          <p>
            <s xml:id="echoid-s1269" xml:space="preserve">Or AT greater than Ao.</s>
            <s xml:id="echoid-s1270" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1271" xml:space="preserve">
              <emph style="sc">Corollary</emph>
            II. </s>
            <s xml:id="echoid-s1272" xml:space="preserve">If the three given points be I, A, E; </s>
            <s xml:id="echoid-s1273" xml:space="preserve">and O falls between
              <lb/>
            A and I, ſo as to make AO x PE: </s>
            <s xml:id="echoid-s1274" xml:space="preserve">IOE:</s>
            <s xml:id="echoid-s1275" xml:space="preserve">: AL: </s>
            <s xml:id="echoid-s1276" xml:space="preserve">LI, I ſay then O will fall
              <lb/>
            beyond L.</s>
            <s xml:id="echoid-s1277" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1278" xml:space="preserve">For let us ſuppoſe that O and L coincide; </s>
            <s xml:id="echoid-s1279" xml:space="preserve">then by hypotbeſis AL: </s>
            <s xml:id="echoid-s1280" xml:space="preserve">LI:</s>
            <s xml:id="echoid-s1281" xml:space="preserve">:
              <lb/>
            AL x PE: </s>
            <s xml:id="echoid-s1282" xml:space="preserve">IL x LE</s>
          </p>
          <p>
            <s xml:id="echoid-s1283" xml:space="preserve">And by the next following
              <emph style="sc">Lemma</emph>
            IV. </s>
            <s xml:id="echoid-s1284" xml:space="preserve">AL x IL: </s>
            <s xml:id="echoid-s1285" xml:space="preserve">IL x PE:</s>
            <s xml:id="echoid-s1286" xml:space="preserve">: AL: </s>
            <s xml:id="echoid-s1287" xml:space="preserve">LE
              <lb/>
            i. </s>
            <s xml:id="echoid-s1288" xml:space="preserve">e. </s>
            <s xml:id="echoid-s1289" xml:space="preserve">AL: </s>
            <s xml:id="echoid-s1290" xml:space="preserve">PE:</s>
            <s xml:id="echoid-s1291" xml:space="preserve">: AL: </s>
            <s xml:id="echoid-s1292" xml:space="preserve">LE</s>
          </p>
          <p>
            <s xml:id="echoid-s1293" xml:space="preserve">Hence PE is equal to LE, a part to the whole, which is manifeſtly abſurd.</s>
            <s xml:id="echoid-s1294" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div71" type="section" level="1" n="67">
          <head xml:id="echoid-head80" xml:space="preserve">LEMMA IV.</head>
          <p>
            <s xml:id="echoid-s1295" xml:space="preserve">If it be as a line to a line ſo a rectangle to a rectangle; </s>
            <s xml:id="echoid-s1296" xml:space="preserve">then I ſay it will be
              <lb/>
            as the flrſt line into the breadth of the ſecond rectangle to the ſecond line into
              <lb/>
            the breadth of the firſt rectangle, ſo the length of the firſt rectangle to the
              <lb/>
            length of the ſecond.</s>
            <s xml:id="echoid-s1297" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1298" xml:space="preserve">Suppoſition. </s>
            <s xml:id="echoid-s1299" xml:space="preserve">AE: </s>
            <s xml:id="echoid-s1300" xml:space="preserve">IO:</s>
            <s xml:id="echoid-s1301" xml:space="preserve">: UYN: </s>
            <s xml:id="echoid-s1302" xml:space="preserve">SRL.</s>
            <s xml:id="echoid-s1303" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1304" xml:space="preserve">Concluſion. </s>
            <s xml:id="echoid-s1305" xml:space="preserve">AE x RL: </s>
            <s xml:id="echoid-s1306" xml:space="preserve">IO x YN:</s>
            <s xml:id="echoid-s1307" xml:space="preserve">: UY: </s>
            <s xml:id="echoid-s1308" xml:space="preserve">SR.</s>
            <s xml:id="echoid-s1309" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1310" xml:space="preserve">
              <emph style="sc">Dem</emph>
            . </s>
            <s xml:id="echoid-s1311" xml:space="preserve">AE: </s>
            <s xml:id="echoid-s1312" xml:space="preserve">IO:</s>
            <s xml:id="echoid-s1313" xml:space="preserve">: AE x YN: </s>
            <s xml:id="echoid-s1314" xml:space="preserve">IO x YN:</s>
            <s xml:id="echoid-s1315" xml:space="preserve">: UYN: </s>
            <s xml:id="echoid-s1316" xml:space="preserve">SRL</s>
          </p>
          <p>
            <s xml:id="echoid-s1317" xml:space="preserve">And by Permutation AE x YN: </s>
            <s xml:id="echoid-s1318" xml:space="preserve">UYN:</s>
            <s xml:id="echoid-s1319" xml:space="preserve">: AE: </s>
            <s xml:id="echoid-s1320" xml:space="preserve">UY:</s>
            <s xml:id="echoid-s1321" xml:space="preserve">: IO x YN: </s>
            <s xml:id="echoid-s1322" xml:space="preserve">SRL</s>
          </p>
          <p>
            <s xml:id="echoid-s1323" xml:space="preserve">But SR: </s>
            <s xml:id="echoid-s1324" xml:space="preserve">AE:</s>
            <s xml:id="echoid-s1325" xml:space="preserve">: SRL:</s>
            <s xml:id="echoid-s1326" xml:space="preserve">: AE x RL</s>
          </p>
          <p>
            <s xml:id="echoid-s1327" xml:space="preserve">Therefore ex æquo perturbatè SR: </s>
            <s xml:id="echoid-s1328" xml:space="preserve">UY:</s>
            <s xml:id="echoid-s1329" xml:space="preserve">: IO x YN: </s>
            <s xml:id="echoid-s1330" xml:space="preserve">AE x RL</s>
          </p>
          <p>
            <s xml:id="echoid-s1331" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s1332" xml:space="preserve">E. </s>
            <s xml:id="echoid-s1333" xml:space="preserve">D.</s>
            <s xml:id="echoid-s1334" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div72" type="section" level="1" n="68">
          <head xml:id="echoid-head81" xml:space="preserve">LEMMA V.</head>
          <p>
            <s xml:id="echoid-s1335" xml:space="preserve">If a right line be cut in two points, I fay the rectangle under the alternate
              <lb/>
            ſegments is equal to that under the whole and the middle ſegment, together
              <lb/>
            with the rectangle under the extremes.</s>
            <s xml:id="echoid-s1336" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1337" xml:space="preserve">
              <emph style="sc">Dem</emph>
            . </s>
            <s xml:id="echoid-s1338" xml:space="preserve">AI x IE + IO x IE = AO x IE.</s>
            <s xml:id="echoid-s1339" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1340" xml:space="preserve">Hence {AI x IE + IO x IE + AE x IO \\ i. </s>
            <s xml:id="echoid-s1341" xml:space="preserve">e. </s>
            <s xml:id="echoid-s1342" xml:space="preserve">AI x IE + AI x IO \\ i. </s>
            <s xml:id="echoid-s1343" xml:space="preserve">e. </s>
            <s xml:id="echoid-s1344" xml:space="preserve">AI x EO} = AO x IE + AE x IO.</s>
            <s xml:id="echoid-s1345" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1346" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s1347" xml:space="preserve">E. </s>
            <s xml:id="echoid-s1348" xml:space="preserve">D.</s>
            <s xml:id="echoid-s1349" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1350" xml:space="preserve">N. </s>
            <s xml:id="echoid-s1351" xml:space="preserve">B. </s>
            <s xml:id="echoid-s1352" xml:space="preserve">Theſe two
              <emph style="sc">Lemmas</emph>
            ſave much Circumlocution and Tautology in
              <lb/>
            the two following Propoſitions, and indeed are highly uſeful in all caſes where
              <lb/>
            compound ratios are concerned.</s>
            <s xml:id="echoid-s1353" xml:space="preserve"/>
          </p>
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