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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            <s xml:id="echoid-s869" xml:space="preserve">
              <pb o="(27)" file="0039" n="39"/>
            through the points H and G, and touches the plane ABC, touches likewiſe
              <lb/>
            the ſphere DFE.</s>
            <s xml:id="echoid-s870" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div50" type="section" level="1" n="50">
          <head xml:id="echoid-head57" xml:space="preserve">PROBLEM IX.</head>
          <p>
            <s xml:id="echoid-s871" xml:space="preserve">
              <emph style="sc">Let</emph>
            there be given two ſpheres AB, DE, as alſo two points H and M;
              <lb/>
            </s>
            <s xml:id="echoid-s872" xml:space="preserve">to find a ſphere which ſhall paſs through the two given points, and likewiſe
              <lb/>
            touch the two given ſpheres.</s>
            <s xml:id="echoid-s873" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s874" xml:space="preserve">
              <emph style="sc">Let</emph>
            the right line AF be drawn paſſing through the centers of the
              <lb/>
            ſpheres, and as the radius AB is to the radius DE, ſo make BF to EF, and
              <lb/>
            the point F will be given. </s>
            <s xml:id="echoid-s875" xml:space="preserve">Make the rectangle HFG = the rectangle NFA,
              <lb/>
            and the point G will be given. </s>
            <s xml:id="echoid-s876" xml:space="preserve">Now having given three points M, H, G,
              <lb/>
            as alſo a ſphere DE; </s>
            <s xml:id="echoid-s877" xml:space="preserve">find a ſphere by Problem III, which ſhall paſs through
              <lb/>
            the given points, and touch the given ſphere; </s>
            <s xml:id="echoid-s878" xml:space="preserve">and, by Lemma III, it will be
              <lb/>
            the ſphere here required.</s>
            <s xml:id="echoid-s879" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div51" type="section" level="1" n="51">
          <head xml:id="echoid-head58" xml:space="preserve">PROBLEM X.</head>
          <p>
            <s xml:id="echoid-s880" xml:space="preserve">
              <emph style="sc">Let</emph>
            there be given two planes AB, BD, a point H, and a ſphere
              <lb/>
            EGF; </s>
            <s xml:id="echoid-s881" xml:space="preserve">to find a ſphere which ſhall paſs through the given point, and touch
              <lb/>
            the given ſphere, as alſo the two given planes.</s>
            <s xml:id="echoid-s882" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s883" xml:space="preserve">
              <emph style="sc">Through</emph>
            the center O of the given ſphere let a perpendicular to either of
              <lb/>
            the given planes CEOF be demitted, and make the rectangle HFI = the
              <lb/>
            rectangle CFE. </s>
            <s xml:id="echoid-s884" xml:space="preserve">Then having given the two points H and I, as alſo the
              <lb/>
            two planes AB, BD; </s>
            <s xml:id="echoid-s885" xml:space="preserve">find a ſphere, by Problem VII, which ſhall paſs
              <lb/>
            through the two given points, and likewiſe touch the two given planes; </s>
            <s xml:id="echoid-s886" xml:space="preserve">and,
              <lb/>
            by Lemma V, it will be the ſphere required.</s>
            <s xml:id="echoid-s887" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div52" type="section" level="1" n="52">
          <head xml:id="echoid-head59" xml:space="preserve">PROBLEM XI.</head>
          <p>
            <s xml:id="echoid-s888" xml:space="preserve">
              <emph style="sc">Let</emph>
            there be given a point, a plane, and two ſpheres; </s>
            <s xml:id="echoid-s889" xml:space="preserve">to find a ſphere
              <lb/>
            which ſhall paſs through the point, touch the plane, and alſo the two
              <lb/>
            ſpheres.</s>
            <s xml:id="echoid-s890" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s891" xml:space="preserve">This Problem, by a like method of reaſoning, is immediately reduced to
              <lb/>
            the VIIIth, where two points, a plane, and a ſphere are given, and that by
              <lb/>
            means of the Vth Lemma. </s>
            <s xml:id="echoid-s892" xml:space="preserve">But if you chuſe to uſe the IIId Lemma, it will
              <lb/>
            be reduced to the ſame Problem by a different method, and a different
              <lb/>
            conſtruction.</s>
            <s xml:id="echoid-s893" xml:space="preserve"/>
          </p>
        </div>
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