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-s1150" xml:space="preserve">
              <pb o="[4]" file="0074" n="81"/>
            two ſolutions; </s>
            <s xml:id="echoid-s1151" xml:space="preserve">if it only touches, then but of one; </s>
            <s xml:id="echoid-s1152" xml:space="preserve">if it neither cuts nor
              <lb/>
            touches, it is then impoſſible.</s>
            <s xml:id="echoid-s1153" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1154" xml:space="preserve">
              <emph style="sc">Demonstration</emph>
            . </s>
            <s xml:id="echoid-s1155" xml:space="preserve">Let the point of interſection then be O or o. </s>
            <s xml:id="echoid-s1156" xml:space="preserve">We ſhall
              <lb/>
            have, by Lemma I. </s>
            <s xml:id="echoid-s1157" xml:space="preserve">AO x ON = MN x NK = MN x AY = AI x AE, by
              <lb/>
            conſtruction. </s>
            <s xml:id="echoid-s1158" xml:space="preserve">Let now from N be ſet off NL = AI in the ſame direction as A
              <lb/>
            is from I; </s>
            <s xml:id="echoid-s1159" xml:space="preserve">then by what has been demonſtrated NL: </s>
            <s xml:id="echoid-s1160" xml:space="preserve">ON:</s>
            <s xml:id="echoid-s1161" xml:space="preserve">: AO: </s>
            <s xml:id="echoid-s1162" xml:space="preserve">AE.</s>
            <s xml:id="echoid-s1163" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1164" xml:space="preserve">And by Diviſion or Compoſition OL: </s>
            <s xml:id="echoid-s1165" xml:space="preserve">ON:</s>
            <s xml:id="echoid-s1166" xml:space="preserve">: OE: </s>
            <s xml:id="echoid-s1167" xml:space="preserve">AE
              <lb/>
            And by Permutation OL: </s>
            <s xml:id="echoid-s1168" xml:space="preserve">OE:</s>
            <s xml:id="echoid-s1169" xml:space="preserve">: ON: </s>
            <s xml:id="echoid-s1170" xml:space="preserve">AE
              <lb/>
            But by what has been proved ON: </s>
            <s xml:id="echoid-s1171" xml:space="preserve">AE:</s>
            <s xml:id="echoid-s1172" xml:space="preserve">: AI: </s>
            <s xml:id="echoid-s1173" xml:space="preserve">AO
              <lb/>
            Therefore by Equality OL: </s>
            <s xml:id="echoid-s1174" xml:space="preserve">OE:</s>
            <s xml:id="echoid-s1175" xml:space="preserve">: AI: </s>
            <s xml:id="echoid-s1176" xml:space="preserve">AO
              <lb/>
            And by Diviſion or Compoſition LE: </s>
            <s xml:id="echoid-s1177" xml:space="preserve">OE:</s>
            <s xml:id="echoid-s1178" xml:space="preserve">: OI: </s>
            <s xml:id="echoid-s1179" xml:space="preserve">AO
              <lb/>
            And LE x AO = OE x OI
              <lb/>
            But LE = NU. </s>
            <s xml:id="echoid-s1180" xml:space="preserve">for NL was put = AI, and IU = AE
              <lb/>
            Hence NU x AO = S x AO = OE x OI
              <lb/>
            And R: </s>
            <s xml:id="echoid-s1181" xml:space="preserve">S:</s>
            <s xml:id="echoid-s1182" xml:space="preserve">: R x AO: </s>
            <s xml:id="echoid-s1183" xml:space="preserve">S x AO or OE x OI
              <lb/>
            Q. </s>
            <s xml:id="echoid-s1184" xml:space="preserve">E. </s>
            <s xml:id="echoid-s1185" xml:space="preserve">D.</s>
            <s xml:id="echoid-s1186" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1187" xml:space="preserve">This Problem may be conſidered as having two
              <emph style="sc">Epitacmas</emph>
            , the firſt,
              <lb/>
            when the ſegment aſſigned for the coefficient of the given external line R is
              <lb/>
            terminated by an extreme point of the three given ones and the point ſought;
              <lb/>
            </s>
            <s xml:id="echoid-s1188" xml:space="preserve">and this again admits of three Caſes. </s>
            <s xml:id="echoid-s1189" xml:space="preserve">The other is when the aforeſaid ſeg-
              <lb/>
            ment is terminated by the middle point of the three given ones and the
              <lb/>
            point ſought.</s>
            <s xml:id="echoid-s1190" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1191" xml:space="preserve">
              <emph style="sc">Epitagma</emph>
            I. </s>
            <s xml:id="echoid-s1192" xml:space="preserve">
              <emph style="sc">Case</emph>
            I. </s>
            <s xml:id="echoid-s1193" xml:space="preserve">Let the aſſigned points be A, E, I. </s>
            <s xml:id="echoid-s1194" xml:space="preserve">A an extreme
              <lb/>
            and E the middle one. </s>
            <s xml:id="echoid-s1195" xml:space="preserve">And let the point O ſought (ſuch that AO x R: </s>
            <s xml:id="echoid-s1196" xml:space="preserve">OE
              <lb/>
            x OI:</s>
            <s xml:id="echoid-s1197" xml:space="preserve">: R: </s>
            <s xml:id="echoid-s1198" xml:space="preserve">S) be required to lie between A and E, or elſe beyond I, which
              <lb/>
            will ariſe from the ſame conſtruction.</s>
            <s xml:id="echoid-s1199" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1200" xml:space="preserve">Here the Homotactical conſtruction is uſed, and IU as likewiſe UN is ſet off
              <lb/>
            in the ſame direction as AI. </s>
            <s xml:id="echoid-s1201" xml:space="preserve">And ſince AO: </s>
            <s xml:id="echoid-s1202" xml:space="preserve">AE:</s>
            <s xml:id="echoid-s1203" xml:space="preserve">: AI: </s>
            <s xml:id="echoid-s1204" xml:space="preserve">ON, and AO + ON
              <lb/>
            is greater than AE + AI or AU, by
              <emph style="sc">Lemma</emph>
            II. </s>
            <s xml:id="echoid-s1205" xml:space="preserve">AO and ON will be the
              <lb/>
            leaſt and greateſt of all; </s>
            <s xml:id="echoid-s1206" xml:space="preserve">and AO will therefore be leſs than AE, as likewiſe
              <lb/>
            Ao (being equal ON by
              <emph style="sc">Lemma</emph>
            I.) </s>
            <s xml:id="echoid-s1207" xml:space="preserve">greater than AI. </s>
            <s xml:id="echoid-s1208" xml:space="preserve">This Caſe is
              <lb/>
            unlimited.</s>
            <s xml:id="echoid-s1209" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1210" xml:space="preserve">
              <emph style="sc">Case</emph>
            II. </s>
            <s xml:id="echoid-s1211" xml:space="preserve">Let the aſſigned points be in the ſame poſition as before, and let
              <lb/>
            the point O ſought be required between E and I.</s>
            <s xml:id="echoid-s1212" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1213" xml:space="preserve">Here the conſtruction is Homotactical, and UN is ſet off the contraty way, viz.
              <lb/>
            </s>
            <s xml:id="echoid-s1214" xml:space="preserve">in the direction IA. </s>
            <s xml:id="echoid-s1215" xml:space="preserve">And ſince AO: </s>
            <s xml:id="echoid-s1216" xml:space="preserve">AE:</s>
            <s xml:id="echoid-s1217" xml:space="preserve">: AI: </s>
            <s xml:id="echoid-s1218" xml:space="preserve">ON, and AO + ON is leſs
              <lb/>
            than AE + AI or AU, by
              <emph style="sc">Lemma</emph>
            II. </s>
            <s xml:id="echoid-s1219" xml:space="preserve">AE and AI will be the leaſt </s>
          </p>
        </div>
      </text>
    </echo>
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