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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            to find the center of a circle which will paſs through the two points, and like-
              <lb/>
            wiſe touch the right line, which is the VIIth of the preceeding Problems.</s>
            <s xml:id="echoid-s728" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div39" type="section" level="1" n="39">
          <head xml:id="echoid-head46" xml:space="preserve">PROBLEM III.</head>
          <p>
            <s xml:id="echoid-s729" xml:space="preserve">
              <emph style="sc">Having</emph>
            three points N, O, M, given, as likewiſe a ſphere IG, to de-
              <lb/>
            ſcribe a ſphere which will paſs through the three given points, and likewiſe
              <lb/>
            touch the given ſphere.</s>
            <s xml:id="echoid-s730" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s731" xml:space="preserve">The circle NOM in the ſurface of the ſphere ſought is given, and a per-
              <lb/>
            pendicular to its plane from it’s center FA being drawn, the center of the
              <lb/>
            ſphere required will be in this line. </s>
            <s xml:id="echoid-s732" xml:space="preserve">From I the center of the given ſphere
              <lb/>
            let IB be drawn perpendicular to FA, and through F, ED parallel to IB,
              <lb/>
            which, from what has been before proved, will be in the plane of the circle
              <lb/>
            NOM, and the points E and D will be given.</s>
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          <p>
            <s xml:id="echoid-s734" xml:space="preserve">Suppoſe now the thing done, and that the center of the ſphere required is
              <lb/>
            C. </s>
            <s xml:id="echoid-s735" xml:space="preserve">Then the lines CI, CE, CD, will be in the ſame plane, which is given, as
              <lb/>
            the points I, E, and D are given. </s>
            <s xml:id="echoid-s736" xml:space="preserve">But the point of contact of two ſpheres is
              <lb/>
            in the line joining their centers; </s>
            <s xml:id="echoid-s737" xml:space="preserve">therefore the ſphere ſought will touch the
              <lb/>
            ſphere given in the point G, and the line IC will exceed the lines EC, ED, by
              <lb/>
            IG the radius of the given ſphere: </s>
            <s xml:id="echoid-s738" xml:space="preserve">with center I therefore and this diſtance
              <lb/>
            IG let a circle be deſcribed in the plane of the lines CI, CE, CD, and it
              <lb/>
            will paſs through the point G and be given in magnitude and poſition; </s>
            <s xml:id="echoid-s739" xml:space="preserve">but
              <lb/>
            the points D and E are alſo in the ſame plane; </s>
            <s xml:id="echoid-s740" xml:space="preserve">and therefore the queſtion is
              <lb/>
            reduced to this, Having two points E and D given, as likewiſe a circle
              <lb/>
            IGH, to find the center of a circle which will paſs through the two points
              <lb/>
            and likewiſe touch the circle, which is the XIIth of the preceeding Problems.</s>
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          <head xml:id="echoid-head47" xml:space="preserve">PROBLEM IV.</head>
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            <s xml:id="echoid-s742" xml:space="preserve">
              <emph style="sc">Having</emph>
            four planes AH, AB, BC, HG, given; </s>
            <s xml:id="echoid-s743" xml:space="preserve">it is required to de-
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            ſcribe a ſphere which ſhall touch them all four.</s>
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          </p>
          <p>
            <s xml:id="echoid-s745" xml:space="preserve">
              <emph style="sc">If</emph>
            two planes touch a ſphere, the center of that ſphere will be in a plane
              <lb/>
            beſecting the inclination of the other two. </s>
            <s xml:id="echoid-s746" xml:space="preserve">And if the planes be parallel, it
              <lb/>
            will be in a parallel plane beſecting their interval. </s>
            <s xml:id="echoid-s747" xml:space="preserve">This being allowed,
              <lb/>
            which is too evident to need further proof; </s>
            <s xml:id="echoid-s748" xml:space="preserve">the center of the ſphere ſought
              <lb/>
            will be in a plane biſecting the inclination of two planes CB and BA; </s>
            <s xml:id="echoid-s749" xml:space="preserve">it will
              <lb/>
            likewiſe be in another plane biſecting the inclination of the two planes BA and
              <lb/>
            AH; </s>
            <s xml:id="echoid-s750" xml:space="preserve">and therefore in a right line, which is the common ſection of theſe </s>
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