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

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        <div xml:id="echoid-div70" type="section" level="1" n="66">
          <p>
            <s xml:id="echoid-s1219" xml:space="preserve">
              <pb o="[5]" file="0075" n="82"/>
            greateſt of all, and AE will therefore be leſs than AO, and AI greater. </s>
            <s xml:id="echoid-s1220" xml:space="preserve">And
              <lb/>
            the ſame will hold with regard to Ao.</s>
            <s xml:id="echoid-s1221" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1222" xml:space="preserve">Here is a
              <emph style="sc">Limitation</emph>
            , which is this; </s>
            <s xml:id="echoid-s1223" xml:space="preserve">that UN or S the conſequent of the
              <lb/>
            given ratio, ſet off from R, muſt not be given greater than the difference of
              <lb/>
            the ſum of AE and AI and of a line whoſe ſquare is equal to four times their
              <lb/>
            rectangle [i. </s>
            <s xml:id="echoid-s1224" xml:space="preserve">e. </s>
            <s xml:id="echoid-s1225" xml:space="preserve">to expreſs it in the modern manner, UN muſt not exceed AI +
              <lb/>
            AE - √4 AI x AE*.</s>
            <s xml:id="echoid-s1226" xml:space="preserve">] This appears by Fig. </s>
            <s xml:id="echoid-s1227" xml:space="preserve">2. </s>
            <s xml:id="echoid-s1228" xml:space="preserve">to this Caſe, the circle there
              <lb/>
            touching the given indefinite line, and pointing out the Limit.</s>
            <s xml:id="echoid-s1229" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1230" xml:space="preserve">
              <emph style="sc">Case</emph>
            III. </s>
            <s xml:id="echoid-s1231" xml:space="preserve">Let the aſſigned points be ſtill in the ſame poſition, and let the
              <lb/>
            point ſought be now required on the contrary ſide of A.</s>
            <s xml:id="echoid-s1232" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1233" xml:space="preserve">Here the conſtruction is ſtill Homotactical, and UN is ſet off the ſame way as
              <lb/>
            in the laſt Caſe; </s>
            <s xml:id="echoid-s1234" xml:space="preserve">and the
              <emph style="sc">Limitation</emph>
            is, that UN muſt not be given leſs than
              <lb/>
            the ſum of AI, AE, and a line whoſe ſquare is equal to four times their rect-
              <lb/>
            angle [or expreſſing it Algebraically, UN muſt not be leſs than AI + AE +
              <lb/>
            √4 AI x AE*.</s>
            <s xml:id="echoid-s1235" xml:space="preserve">]</s>
          </p>
          <p>
            <s xml:id="echoid-s1236" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            II. </s>
            <s xml:id="echoid-s1237" xml:space="preserve">
              <emph style="sc">Case</emph>
            IV. </s>
            <s xml:id="echoid-s1238" xml:space="preserve">Let now A be the middle point of the given
              <lb/>
            ones, and let O the point ſought be required either between A and one of the
              <lb/>
            extremes, or beyond either of the extremes.</s>
            <s xml:id="echoid-s1239" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1240" xml:space="preserve">Here having ſet off IU = AE toward A, you may ſet off UN either way,
              <lb/>
            and uſing the Antitactical conſtruction, the ſolution will be unlimited. </s>
            <s xml:id="echoid-s1241" xml:space="preserve">The
              <lb/>
            only difference is, that if UN be in the direction UI, two ſolutions will ariſe,
              <lb/>
            whereof in one the point O will fall between A and E, and in the other be-
              <lb/>
            yond I; </s>
            <s xml:id="echoid-s1242" xml:space="preserve">but if UN be in the direction IU, two ſolutions will ariſe, whereof
              <lb/>
            in one the point will fall between A and I, and in the other beyond E. </s>
            <s xml:id="echoid-s1243" xml:space="preserve">In
              <lb/>
            proof of which
              <emph style="sc">Lemma</emph>
            III. </s>
            <s xml:id="echoid-s1244" xml:space="preserve">is to be uſed, as
              <emph style="sc">Lemma</emph>
            II. </s>
            <s xml:id="echoid-s1245" xml:space="preserve">was in Caſe I. </s>
            <s xml:id="echoid-s1246" xml:space="preserve">II.</s>
            <s xml:id="echoid-s1247" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1248" xml:space="preserve">
              <emph style="sc">Corollary</emph>
            I. </s>
            <s xml:id="echoid-s1249" xml:space="preserve">If then the given ratio be that of AT to TI, or of AE to
              <lb/>
            EP ſet off from A the other way, ſo that EP be leſs than AE, I ſay then
              <lb/>
            that O will fall between E and P, as likewiſe ο between T and I, provided o
              <lb/>
            falls beyond I.</s>
            <s xml:id="echoid-s1250" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1251" xml:space="preserve">For by conſtruction IU = AE, and UN = PE. </s>
            <s xml:id="echoid-s1252" xml:space="preserve">therefore IN = AP. </s>
            <s xml:id="echoid-s1253" xml:space="preserve">But by
              <lb/>
              <emph style="sc">Lemma</emph>
            I. </s>
            <s xml:id="echoid-s1254" xml:space="preserve">oN = AO. </s>
            <s xml:id="echoid-s1255" xml:space="preserve">therefore (o falling beyond I by hypotbeſis) O will fall
              <lb/>
            beyond P; </s>
            <s xml:id="echoid-s1256" xml:space="preserve">but by hypotbeſis it falls ſhort of E; </s>
            <s xml:id="echoid-s1257" xml:space="preserve">therefore O falls between
              <lb/>
            P and E.</s>
            <s xml:id="echoid-s1258" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1259" xml:space="preserve">Next to ſhew that ο will fall between T and I, we have AT: </s>
            <s xml:id="echoid-s1260" xml:space="preserve">TI:</s>
            <s xml:id="echoid-s1261" xml:space="preserve">: AE: </s>
            <s xml:id="echoid-s1262" xml:space="preserve">EP</s>
          </p>
          <p>
            <s xml:id="echoid-s1263" xml:space="preserve">And by Diviſion AT: </s>
            <s xml:id="echoid-s1264" xml:space="preserve">AI:</s>
            <s xml:id="echoid-s1265" xml:space="preserve">: AE: </s>
            <s xml:id="echoid-s1266" xml:space="preserve">AP</s>
          </p>
          <p>
            <s xml:id="echoid-s1267" xml:space="preserve">Hence AT x AP = IAE or o AO</s>
          </p>
        </div>
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    </echo>
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