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-div53" type="section" level="1" n="53">
          <head xml:id="echoid-head60" xml:space="preserve">PROBLEM XII.</head>
          <p>
            <s xml:id="echoid-s894" xml:space="preserve">
              <emph style="sc">Let</emph>
            there be given a point and three ſpheres, to ſind a ſphere which ſhall
              <lb/>
            paſs through the point, and touch all the three ſpheres.</s>
            <s xml:id="echoid-s895" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s896" xml:space="preserve">
              <emph style="sc">We</emph>
            aſſign no figure to this Problem alſo, becauſe, by help of Lemma III,
              <lb/>
            it may immediately be reduced to Problem IX, where two points and two
              <lb/>
            ſpheres are given.</s>
            <s xml:id="echoid-s897" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div54" type="section" level="1" n="54">
          <head xml:id="echoid-head61" xml:space="preserve">PROBLEM XIII.</head>
          <p>
            <s xml:id="echoid-s898" xml:space="preserve">
              <emph style="sc">Let</emph>
            there be two planes, and alſo two ſpheres given; </s>
            <s xml:id="echoid-s899" xml:space="preserve">to find a ſphere
              <lb/>
            which ſhall touch the planes, as alſo the ſpheres.</s>
            <s xml:id="echoid-s900" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s901" xml:space="preserve">Suppoſe the thing done. </s>
            <s xml:id="echoid-s902" xml:space="preserve">If therefore we imagine another ſpherical ſurface
              <lb/>
            parallel to that which is required, and which we now ſuppoſe found, and
              <lb/>
            whoſe radius is leſs than it's by the radius of the leſſer of the two given
              <lb/>
            ſpheres; </s>
            <s xml:id="echoid-s903" xml:space="preserve">this new ſpherical ſurface will touch two planes parallel to the two
              <lb/>
            given ones, and whoſe diſtance therefrom will be equal to the radius of the
              <lb/>
            leſſer of the given ſpheres; </s>
            <s xml:id="echoid-s904" xml:space="preserve">it will alſo touch a ſphere concentric to the
              <lb/>
            greater given one whoſe radius is leſs than it's by the radius of the leſſer given
              <lb/>
            one; </s>
            <s xml:id="echoid-s905" xml:space="preserve">and it will likewife paſs through the center of the leſſer given one.
              <lb/>
            </s>
            <s xml:id="echoid-s906" xml:space="preserve">The Queſtion is then reduced to Problem X, where a point, two planes and
              <lb/>
            a ſphere are given.</s>
            <s xml:id="echoid-s907" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div55" type="section" level="1" n="55">
          <head xml:id="echoid-head62" xml:space="preserve">PROBLEM XIV.</head>
          <p>
            <s xml:id="echoid-s908" xml:space="preserve">
              <emph style="sc">Having</emph>
            three ſpheres and a plane given; </s>
            <s xml:id="echoid-s909" xml:space="preserve">to find a ſphere which ſhall
              <lb/>
            touch them all.</s>
            <s xml:id="echoid-s910" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s911" xml:space="preserve">
              <emph style="sc">By</emph>
            a like method to what is uſed in the preceeding, and in the VIth Pro-
              <lb/>
            blem, this is reduced to Problem XI, where a point, a plane, and two
              <lb/>
            ſpheres are given.</s>
            <s xml:id="echoid-s912" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div56" type="section" level="1" n="56">
          <head xml:id="echoid-head63" xml:space="preserve">PROBLEM XV.</head>
          <p>
            <s xml:id="echoid-s913" xml:space="preserve">
              <emph style="sc">Having</emph>
            four ſpheres given; </s>
            <s xml:id="echoid-s914" xml:space="preserve">to ſind a ſphere which ſhall touch them all.</s>
            <s xml:id="echoid-s915" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s916" xml:space="preserve">
              <emph style="sc">Suppose</emph>
            the thing done. </s>
            <s xml:id="echoid-s917" xml:space="preserve">As, in the treatiſe of Circular Tangencies, the
              <lb/>
            laſt Problem, where it is required, having three circles given, to find a fourth
              <lb/>
            which ſhall touch them all, is reduced to another, where a point and two
              <lb/>
            circles are given; </s>
            <s xml:id="echoid-s918" xml:space="preserve">ſo alſo this, by a like method, and ſimilar to what has been
              <lb/>
            uſed in the preceding Problems, is reduced to Problem XII, where three
              <lb/>
            ſpheres and a point are given.</s>
            <s xml:id="echoid-s919" xml:space="preserve"/>
          </p>
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