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-div83" type="section" level="1" n="79">
          <p>
            <s xml:id="echoid-s2218" xml:space="preserve">
              <pb o="[27]" file="0097" n="104"/>
            quently, by compound ratio, the rectangle contained by LS and SM is to
              <lb/>
            that contained by SE and SI as the rectangle contained by VO and OY,
              <lb/>
            or its equal, the rectangle contained by AO and OU is to that contained
              <lb/>
            by EO and IO. </s>
            <s xml:id="echoid-s2219" xml:space="preserve">Now the triangles VSL and NSR having the angles at
              <lb/>
            R and L equal, and the angle at S common, are ſimilar; </s>
            <s xml:id="echoid-s2220" xml:space="preserve">and therefore SR
              <lb/>
            is to SN as SL is to SV; </s>
            <s xml:id="echoid-s2221" xml:space="preserve">conſequently, the rectangle contained by SR and
              <lb/>
            SV, or its equal, the rectangle contained by SA and SU is equal to that
              <lb/>
            contained by SN and SL: </s>
            <s xml:id="echoid-s2222" xml:space="preserve">but SN is neceſſarily greater than SM, in con-
              <lb/>
            ſequence whereof the rectangle contained by SN and SL, or its equal, the
              <lb/>
            rectangle contained by AS and SU is greater than that contained by SM
              <lb/>
            and SL; </s>
            <s xml:id="echoid-s2223" xml:space="preserve">wherefore the ratio which the rectangle AS, SU bears to the rect-
              <lb/>
            angle ES, SI is greater than that which the rectangle SM, SL bears to it,
              <lb/>
            and of courſe, greater than the ratio which the rectangle AO, UO bears to
              <lb/>
            the rectangle EO, IO; </s>
            <s xml:id="echoid-s2224" xml:space="preserve">and that, on which ſide ſoever of the point O, S is
              <lb/>
            taken.</s>
            <s xml:id="echoid-s2225" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2226" xml:space="preserve">Again, on YO produced, let fall the perpendiculars EB and IC: </s>
            <s xml:id="echoid-s2227" xml:space="preserve">the
              <lb/>
            triangles EBV and ICY, EBO and ICO are ſimilar, becauſe the angles EVO
              <lb/>
            and IYO are equal, and ſo EO is to IO as EB is to IC, alſo EV is to IY
              <lb/>
            as EB is to IC; </s>
            <s xml:id="echoid-s2228" xml:space="preserve">therefore by equality of ratios EO is to IO as EV is to IY,
              <lb/>
            and (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2229" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s2230" xml:space="preserve">22.) </s>
            <s xml:id="echoid-s2231" xml:space="preserve">the ſquare on EO is to the ſquare on IO as the ſquare
              <lb/>
            on EV is to the ſquare on IY; </s>
            <s xml:id="echoid-s2232" xml:space="preserve">that is (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s2233" xml:space="preserve">III. </s>
            <s xml:id="echoid-s2234" xml:space="preserve">36.) </s>
            <s xml:id="echoid-s2235" xml:space="preserve">as the rectangle con-
              <lb/>
            tained by AE and UE is to that contained by AI and UI.</s>
            <s xml:id="echoid-s2236" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2237" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s2238" xml:space="preserve">E. </s>
            <s xml:id="echoid-s2239" xml:space="preserve">D.</s>
            <s xml:id="echoid-s2240" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div84" type="section" level="1" n="80">
          <head xml:id="echoid-head95" xml:space="preserve">LEMMA III.</head>
          <p>
            <s xml:id="echoid-s2241" xml:space="preserve">If from two points E and I, in the diameter AU, of a circle, AVYU
              <lb/>
            (Fig. </s>
            <s xml:id="echoid-s2242" xml:space="preserve">30.) </s>
            <s xml:id="echoid-s2243" xml:space="preserve">two perpendiculars EV, IY be drawn on the ſame ſide thereof to
              <lb/>
              <handwritten xlink:label="hd-0097-01" xlink:href="hd-0097-01a" number="3"/>
            terminate in the periphery, and if their extremes V and Y be joined by a
              <lb/>
            ſtraight line VY, cutting the ſaid diameter, produced, in O; </s>
            <s xml:id="echoid-s2244" xml:space="preserve">then will the
              <lb/>
            ratio which the rectangle contained by AO and UO bears to the rectangle
              <lb/>
            contained by EO and IO be the greateſt poſſible.</s>
            <s xml:id="echoid-s2245" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2246" xml:space="preserve">*** This, like the ſirſt, is demonſtrated by
              <emph style="sc">Snellius</emph>
            , and needs not be
              <lb/>
            repeated.</s>
            <s xml:id="echoid-s2247" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>
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