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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[81.] LEMMA IV.
[82.] LEMMA V.
[83.] PROBLEM VII. (Fig. 32, 33, 34, &c.)
[84.] PROBLEM I. (Fig. 32 to 45.)
[85.] PROBLEM II. (Fig. 46 to 57.)
[86.] PROBLEM III.
[87.] THE END.
[88.] A SYNOPSIS OF ALL THE DATA FOR THE Conſtruction of Triangles, FROM WHICH GEOMETRICAL SOLUTIONS Have hitherto been in Print.
[89.] By JOHN LAWSON, B. D. Rector of Swanscombe, in KENT. ROCHESTER:
[90.] MDCCLXXIII. [Price One Shilling.]
[91.] ADVERTISEMENT.
[92.] AN EXPLANATION OF THE SYMBOLS made uſe of in this SYNOPSIS.
[93.] INDEX OF THE Authors refered to in the SYNOPSIS.
[94.] Lately was publiſhed by the ſame Author; [Price Six Shillings in Boards.]
[95.] SYNOPSIS.
[96.] Continuation of the Synopsis, Containing ſuch Data as cannot readily be expreſſed by the Symbols before uſed without more words at length.
[97.] SYNOPSIS
[98.] FINIS.
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        <div xml:id="echoid-div70" type="section" level="1" n="66">
          <pb o="[6]" file="0076" n="83"/>
          <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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