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-div42" type="section" level="1" n="42">
          <p>
            <s xml:id="echoid-s777" xml:space="preserve">
              <pb o="(23)" file="0035" n="35"/>
            equal to UE: </s>
            <s xml:id="echoid-s778" xml:space="preserve">but RM is given in magnitude, being the radius of the given
              <lb/>
            ſphere, therefore UE is alſo given in magnitude. </s>
            <s xml:id="echoid-s779" xml:space="preserve">And ſince OE is perpen-
              <lb/>
            dicular to the plane DE, it will be alſo to the plane PU which is parallel
              <lb/>
            thereto. </s>
            <s xml:id="echoid-s780" xml:space="preserve">UE then being given in magnitude, and being the interval be-
              <lb/>
            tween two parallel planes DE, PU, whereof DE is given in poſition by hypo-
              <lb/>
            theſis, the other PU will alſo be given in poſition. </s>
            <s xml:id="echoid-s781" xml:space="preserve">In the ſame manner it
              <lb/>
            may be proved that the planes GH, IN, are given in poſition, and that the
              <lb/>
            lines OG, OI, are perpendicular thereto reſpectively, and each alſo equal to
              <lb/>
            OM. </s>
            <s xml:id="echoid-s782" xml:space="preserve">A ſphere therefore deſcribed with center O and OM radius will touch
              <lb/>
            the three planes PU, GH, IN, given in poſition: </s>
            <s xml:id="echoid-s783" xml:space="preserve">but the point M is given,
              <lb/>
            being the center of the given ſphere. </s>
            <s xml:id="echoid-s784" xml:space="preserve">The queſtion is then reduced to this,
              <lb/>
            Having three planes given PU, GH, IN, and a point M, to find the radius
              <lb/>
            of a ſphere which ſhall touch the given planes, and paſs through the given
              <lb/>
            point; </s>
            <s xml:id="echoid-s785" xml:space="preserve">which is the ſame as the preceeding Problem. </s>
            <s xml:id="echoid-s786" xml:space="preserve">[And this radius be-
              <lb/>
            ing increaſed or diminiſhed by MR, according as R is taken in the further or
              <lb/>
            nearer ſurface of the given ſphere, will give the radius of a ſphere which will
              <lb/>
            touch the three given planes DE, DB, BC, and likewiſe the given
              <lb/>
            ſphere.</s>
            <s xml:id="echoid-s787" xml:space="preserve">]</s>
          </p>
          <p>
            <s xml:id="echoid-s788" xml:space="preserve">
              <emph style="sc">By</emph>
            a like method, when among the Data there are no points, but only
              <lb/>
            planes and ſpheres, we ſhall always be able to ſubſtitute a given point in the
              <lb/>
            place of a given ſphere.</s>
            <s xml:id="echoid-s789" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div43" type="section" level="1" n="43">
          <head xml:id="echoid-head50" xml:space="preserve">PROBLEM VII.</head>
          <p>
            <s xml:id="echoid-s790" xml:space="preserve">
              <emph style="sc">Having</emph>
            two points H, M, as alſo two planes AB, BC, given, to find a
              <lb/>
            ſphere which ſhall paſs through the given points, and touch the given
              <lb/>
            planes.</s>
            <s xml:id="echoid-s791" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s792" xml:space="preserve">
              <emph style="sc">Draw</emph>
            HM and biſect it in I, the point I will be given, through the
              <lb/>
            point I let a plane be erected perpendicular to the right line HM, this plane
              <lb/>
            will be given in poſition, and the center of the ſphere required will be in this
              <lb/>
            plane. </s>
            <s xml:id="echoid-s793" xml:space="preserve">But becauſe it is alſo to touch the planes AB, BC, its center will be
              <lb/>
            alſo in another plane given in poſition (by what has been proved, Prob. </s>
            <s xml:id="echoid-s794" xml:space="preserve">IV.)
              <lb/>
            </s>
            <s xml:id="echoid-s795" xml:space="preserve">and therefore in a right line which is their interſection, given in poſition,
              <lb/>
            which let be GE; </s>
            <s xml:id="echoid-s796" xml:space="preserve">to which line GE from one of the given points M demit-
              <lb/>
            ting a perpendicular MF, it will be given in magnitude and poſition, and
              <lb/>
            being continued to D ſo that FD equals MF, the point D will be given; </s>
            <s xml:id="echoid-s797" xml:space="preserve">
              <lb/>
            and, from what has been proved before, will be in the ſpherical ſurface.</s>
            <s xml:id="echoid-s798" xml:space="preserve"/>
          </p>
        </div>
      </text>
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