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-s844" xml:space="preserve">
              <pb o="(26)" file="0038" n="38"/>
            the points I and H a ſphere be deſcribed which touches the plane AC, I ſay
              <lb/>
            it will alſo touch the ſphere EGF. </s>
            <s xml:id="echoid-s845" xml:space="preserve">From F draw FB to the point of contact
              <lb/>
            of the ſphere and plane, and make the rectangle BFN = the rectangle CFE,
              <lb/>
            and the point N will be in the ſurface of the ſphere EGF, by Lemma IV.
              <lb/>
            </s>
            <s xml:id="echoid-s846" xml:space="preserve">But the rectangle CFE, by conſtruction, = the rectangle IFH; </s>
            <s xml:id="echoid-s847" xml:space="preserve">therefore
              <lb/>
            IFH = BFN, and the point N will be alſo in the ſurſace of the ſphere IHB. </s>
            <s xml:id="echoid-s848" xml:space="preserve">
              <lb/>
            It remains then to be proved that theſe ſpheres touch in N, which is very eaſy
              <lb/>
            to be done. </s>
            <s xml:id="echoid-s849" xml:space="preserve">For from the point F through any point R in the ſpherical ſur-
              <lb/>
            face EGF let the line FR be drawn, which may cut the ſpherical ſurface
              <lb/>
            IBH in L and P, and the plane AC in K. </s>
            <s xml:id="echoid-s850" xml:space="preserve">The rectangle KFR = the rect-
              <lb/>
            angle CFE, by Lemma IV. </s>
            <s xml:id="echoid-s851" xml:space="preserve">= the rectangle IFH, by conſtruction, = the
              <lb/>
            rectangle PFL. </s>
            <s xml:id="echoid-s852" xml:space="preserve">Since then KFR = PFL, and KF is greater than PF, be-
              <lb/>
            cauſe the ſphere IHB touches the plane AC in B, therefore FR is leſs than
              <lb/>
            FL, and the point R is without the ſphere IHB, and the ſame may be ſhewn
              <lb/>
            of every other point in the ſpherical ſurface EGF, except the point N.</s>
            <s xml:id="echoid-s853" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s854" xml:space="preserve">Theſe Lemmas, though they be very eaſy, are very elegant and valuable,
              <lb/>
            eſpecially the IIId and Vth. </s>
            <s xml:id="echoid-s855" xml:space="preserve">In the IIId. </s>
            <s xml:id="echoid-s856" xml:space="preserve">though there be an inſinite num-
              <lb/>
            ber of ſpheres which, paſſing through the points T and S, may touch the
              <lb/>
            ſphere XM, yet they will all alſo touch the ſphere YN, by what is there
              <lb/>
            proved. </s>
            <s xml:id="echoid-s857" xml:space="preserve">In the Vth, though there be an infinite number of ſpheres which,
              <lb/>
            paſſing through the points I and H, may touch the plane AC, yet they will
              <lb/>
            all alſo touch the ſphere EGF, by what is there proved.</s>
            <s xml:id="echoid-s858" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s859" xml:space="preserve">We ſhall now be able to go through the remaining Problems with eaſe.</s>
            <s xml:id="echoid-s860" xml:space="preserve"/>
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        <div xml:id="echoid-div49" type="section" level="1" n="49">
          <head xml:id="echoid-head56" xml:space="preserve">PROBLEM VIII.</head>
          <p>
            <s xml:id="echoid-s861" xml:space="preserve">Let there be given a plane ABC, and two points H and M, and alſo a
              <lb/>
            ſphere DFE; </s>
            <s xml:id="echoid-s862" xml:space="preserve">to find a ſphere which ſhall paſs through the given points, and
              <lb/>
            touch the given plane, and likewiſe the given ſphere.</s>
            <s xml:id="echoid-s863" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s864" xml:space="preserve">Through the center O of the given ſphere let EODB be demitted perpen-
              <lb/>
            dicular to the given plane ABC, and let HE be drawn, and make the rect-
              <lb/>
            angle HEG equal to the rectangle BED, and G will then be given. </s>
            <s xml:id="echoid-s865" xml:space="preserve">Find
              <lb/>
            then a ſphere, by Problem II. </s>
            <s xml:id="echoid-s866" xml:space="preserve">which ſhall paſs through the three points M,
              <lb/>
            H, G, and touch the plane ABC, and it will be the ſphere here required.
              <lb/>
            </s>
            <s xml:id="echoid-s867" xml:space="preserve">For it paſſes through the points M and H, and touches the plane ABC,
              <lb/>
            by conſtruction; </s>
            <s xml:id="echoid-s868" xml:space="preserve">it likewiſe touches the ſphere DFE, by Lemma V. </s>
            <s xml:id="echoid-s869" xml:space="preserve">For
              <lb/>
            ſince the rectangle HEG = the rectangle BED, every ſphere which </s>
          </p>
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