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-s750" xml:space="preserve">
              <pb o="(22)" file="0034" n="34"/>
            biſecting planes; </s>
            <s xml:id="echoid-s751" xml:space="preserve">let this right line be EF. </s>
            <s xml:id="echoid-s752" xml:space="preserve">Moreover, the center of the
              <lb/>
            ſphere ſought will alſo be in a plane biſecting the inclination of the two planes
              <lb/>
            AH and GH, and the interſection of this laſt biſecting plane with the right
              <lb/>
            line EF will give a point D, which will be the center of the ſphere required.</s>
            <s xml:id="echoid-s753" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div41" type="section" level="1" n="41">
          <head xml:id="echoid-head48" xml:space="preserve">PROBLEM V.</head>
          <p>
            <s xml:id="echoid-s754" xml:space="preserve">
              <emph style="sc">Having</emph>
            three planes AB, BC, CD, given, and alſo a point H; </s>
            <s xml:id="echoid-s755" xml:space="preserve">to find
              <lb/>
            a ſphere which ſhall paſs through the given point, and likewiſe touch the
              <lb/>
            three given planes.</s>
            <s xml:id="echoid-s756" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s757" xml:space="preserve">
              <emph style="sc">Suppose</emph>
            it done. </s>
            <s xml:id="echoid-s758" xml:space="preserve">The three planes, by what was ſaid under the laſt pro-
              <lb/>
            poſition, will give a right line in poſition, in which will be the center of the
              <lb/>
            ſphere required. </s>
            <s xml:id="echoid-s759" xml:space="preserve">Let this right line be GE, perpendicular to which from H
              <lb/>
            the given point let HI be drawn, which therefore will be given in magnitude
              <lb/>
            and poſition. </s>
            <s xml:id="echoid-s760" xml:space="preserve">Let HI be produced, and FI taken equal to HI; </s>
            <s xml:id="echoid-s761" xml:space="preserve">the point
              <lb/>
            F will then be given. </s>
            <s xml:id="echoid-s762" xml:space="preserve">Now ſince the center of the ſphere required is in the
              <lb/>
            line GE, and FH is perpendicular thereto and biſected thereby, and one
              <lb/>
            extreme H is by hypotheſis in the ſurface of the ſaid ſphere, the other extreme
              <lb/>
            F will be ſo too. </s>
            <s xml:id="echoid-s763" xml:space="preserve">Nay even a circle deſcribed with I center and IH radius
              <lb/>
            in a plane perpendicular to GE will be in the ſaid ſpherical ſurface. </s>
            <s xml:id="echoid-s764" xml:space="preserve">Here
              <lb/>
            then we have a circle given in magnitude and poſition, and taking any one
              <lb/>
            of the given planes AB, by an evident corollary from Problem II. </s>
            <s xml:id="echoid-s765" xml:space="preserve">of this
              <lb/>
            Supplement, a ſphere may be deſcribed which will touch the given plane,
              <lb/>
            and likewiſe have the given circle in it’s ſurface; </s>
            <s xml:id="echoid-s766" xml:space="preserve">and ſuch a ſphere will
              <lb/>
            anſwer every thing here required.</s>
            <s xml:id="echoid-s767" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div42" type="section" level="1" n="42">
          <head xml:id="echoid-head49" xml:space="preserve">PROBLEM VI.</head>
          <p>
            <s xml:id="echoid-s768" xml:space="preserve">
              <emph style="sc">Having</emph>
            three planes ED, DB, BC, given, and alſo a ſphere RM, to
              <lb/>
            conſtruct a ſphere which ſhall touch the given one, and likewiſe the three
              <lb/>
            given planes.</s>
            <s xml:id="echoid-s769" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s770" xml:space="preserve">
              <emph style="sc">Suppose</emph>
            it done, and that the ſphere ERCA is the required one, viz.
              <lb/>
            </s>
            <s xml:id="echoid-s771" xml:space="preserve">touches the ſphere in R, and the planes in E, A, C. </s>
            <s xml:id="echoid-s772" xml:space="preserve">Let the center of this
              <lb/>
            ſphere be O; </s>
            <s xml:id="echoid-s773" xml:space="preserve">then drawing RO, EO, AO, CO, they will all be equal, and
              <lb/>
            RO will paſs through M the center of the given ſphere; </s>
            <s xml:id="echoid-s774" xml:space="preserve">and EO, AO, CO,
              <lb/>
            will be perpendicular to the planes ED, DB, BC. </s>
            <s xml:id="echoid-s775" xml:space="preserve">Let OU, OG, OI, be
              <lb/>
            made each equal to OM; </s>
            <s xml:id="echoid-s776" xml:space="preserve">and through the points U, G and I, let the planes
              <lb/>
            UP, GH, IN, be ſuppoſed drawn parallel to the given ones ED, DB, BC,
              <lb/>
            reſpectively. </s>
            <s xml:id="echoid-s777" xml:space="preserve">Since OR is equal to OE, and OM equal to OU, RM will </s>
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