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-div80" type="section" level="1" n="76">
          <p>
            <s xml:id="echoid-s2161" xml:space="preserve">
              <pb o="[25]" file="0095" n="102"/>
            IB, as EI is to IC: </s>
            <s xml:id="echoid-s2162" xml:space="preserve">and hence, by compound ratio, the ſquare on the
              <lb/>
            abovementioned ſum or difference is to the rectangle contained by R and S
              <lb/>
            as the rectangle contained by AI and EI is to the rectangle contained by
              <lb/>
            IB and EC, alſo by permutation, AI is to EI as IB is to IC; </s>
            <s xml:id="echoid-s2163" xml:space="preserve">wherefore,
              <lb/>
            by compoſition or diviſion, AE is to AI as BC is to IB, by permutation,
              <lb/>
            AE is to BC as AI is to IB, therefore by equality, the ſum or difference
              <lb/>
            of R and S is to S as AE is to BC; </s>
            <s xml:id="echoid-s2164" xml:space="preserve">or (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2165" xml:space="preserve">V. </s>
            <s xml:id="echoid-s2166" xml:space="preserve">15.) </s>
            <s xml:id="echoid-s2167" xml:space="preserve">as half AE is to
              <lb/>
            half BC; </s>
            <s xml:id="echoid-s2168" xml:space="preserve">conſequently (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2169" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s2170" xml:space="preserve">22.) </s>
            <s xml:id="echoid-s2171" xml:space="preserve">the ſquare on the above mentioned
              <lb/>
            ſum or diſſerence is to the ſquare on S as the ſquare on half AE is to the
              <lb/>
            ſquare on half BC, or to the rectangle contained by IB and EC. </s>
            <s xml:id="echoid-s2172" xml:space="preserve">Hence
              <lb/>
            exæquo perturbaté, the rectangle contained by R and S is to the ſquare on
              <lb/>
            S as the ſquare on half AE is to the rectangle contained by AI and EI,
              <lb/>
            or (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2173" xml:space="preserve">V. </s>
            <s xml:id="echoid-s2174" xml:space="preserve">15.) </s>
            <s xml:id="echoid-s2175" xml:space="preserve">R is to S as the ſquare on half AE is to the rectangle
              <lb/>
            contained by AI and EI; </s>
            <s xml:id="echoid-s2176" xml:space="preserve">and which is therefore the greateſt ratio which
              <lb/>
            R can have to S in thoſe Caſes.</s>
            <s xml:id="echoid-s2177" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2178" xml:space="preserve">It ought farther to be remarked, that to have Caſe III poſſible, where
              <lb/>
            O is ſought beyond A, and the ratio of a greater to a leſs, it is neceſſary
              <lb/>
            that AI be leſs than IE, and to have Caſe IV. </s>
            <s xml:id="echoid-s2179" xml:space="preserve">poſſible, that it be greater.
              <lb/>
            </s>
            <s xml:id="echoid-s2180" xml:space="preserve">For it is plain from the Conſtruction, that IB muſt in the former caſe be
              <lb/>
            leſs, and in the latter greater than I C; </s>
            <s xml:id="echoid-s2181" xml:space="preserve">but as R is to S ſo is AB to
              <lb/>
            IB, and ſo is EC to IC, wherefore by diviſion, the exceſs of R above S
              <lb/>
            is to S as AI is to IB, and as EI is to IC; </s>
            <s xml:id="echoid-s2182" xml:space="preserve">and ſo by permutation AI is
              <lb/>
            to EI as IB is to IC: </s>
            <s xml:id="echoid-s2183" xml:space="preserve">conſequently when IB is greater than IC, AI will
              <lb/>
            be greater than EI; </s>
            <s xml:id="echoid-s2184" xml:space="preserve">and when leſs, leſs.</s>
            <s xml:id="echoid-s2185" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2186" xml:space="preserve">With reſpect to thoſe caſes wherein the given ratio is that of equality,
              <lb/>
            it may be ſufficient to remark, that none of the Caſes of Epitagma II.
              <lb/>
            </s>
            <s xml:id="echoid-s2187" xml:space="preserve">are poſſible under that ratio: </s>
            <s xml:id="echoid-s2188" xml:space="preserve">that one of Caſes III. </s>
            <s xml:id="echoid-s2189" xml:space="preserve">and IV. </s>
            <s xml:id="echoid-s2190" xml:space="preserve">Epitagma III. </s>
            <s xml:id="echoid-s2191" xml:space="preserve">
              <lb/>
            is always impoſſible when the given ratio of R to S is the ratio of equality; </s>
            <s xml:id="echoid-s2192" xml:space="preserve">
              <lb/>
            and both are ſo if AI be at the ſame time equal to IE. </s>
            <s xml:id="echoid-s2193" xml:space="preserve">Laſtly Caſes V. </s>
            <s xml:id="echoid-s2194" xml:space="preserve">
              <lb/>
            and VI. </s>
            <s xml:id="echoid-s2195" xml:space="preserve">are never poſſible under the ratio of equality, unleſs the ſquare on
              <lb/>
            half AE be equal to, or exceed the rectangle contained by AI and EI; </s>
            <s xml:id="echoid-s2196" xml:space="preserve">
              <lb/>
            all which naturally follows from what has been delivered above.</s>
            <s xml:id="echoid-s2197" xml:space="preserve"/>
          </p>
          <note symbol="*" position="foot" xml:space="preserve">See Prop. A. Book V. of Dr. Simſon’s Euclid.</note>
        </div>
        <div xml:id="echoid-div81" type="section" level="1" n="77">
          <head xml:id="echoid-head92" xml:space="preserve">THE END OF BOOK I.</head>
        </div>
      </text>
    </echo>
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