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

Table of contents

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[31.] PROBLEM II.
[32.] PROBLEM III.
[33.] PROBLEM IV.
[34.] PROBLEM V.
[35.] PROBLEM VI.
[36.] General Solution.
[37.] A SECOND SUPPLEMENT, BEING Monſ. DE FERMAT’S Treatiſe on Spherical Tangencies. PROBLEM I.
[38.] PROBLEM II.
[39.] PROBLEM III.
[40.] PROBLEM IV.
[41.] PROBLEM V.
[42.] PROBLEM VI.
[43.] PROBLEM VII.
[44.] LEMMA I.
[45.] LEMMA II.
[46.] LEMMA III.
[47.] LEMMA IV.
[48.] LEMMA V.
[49.] PROBLEM VIII.
[50.] PROBLEM IX.
[51.] PROBLEM X.
[52.] PROBLEM XI.
[53.] PROBLEM XII.
[54.] PROBLEM XIII.
[55.] PROBLEM XIV.
[56.] PROBLEM XV.
[57.] Synopſis of the PROBLEMS.
[58.] THE TWO BOOKS OF APOLLONIUS PERGÆUS, CONCERNING DETERMINATE SECTION, As they have been Reſtored by WILLEBRORDUS SNELLIUS. By JOHN LAWSON, B. D. Rector of Swanſcombe, Kent. TO WHICH ARE ADDED, THE SAME TWO BOOKS, BY WILLIAM WALES, BEING AN ENTIRE NEW WORK. LONDON: Printed by G. BIGG, Succeſſor to D. LEACH. And ſold by B. White, in Fleet-Street; L. Davis, in Holborne; J. Nourse, in the Strand; and T. Payne, near the Mews-Gate. MDCC LXXII.
[59.] ADVERTISEMENT.
[60.] EXTRACT from PAPPUS's Preface to his Seventh Book in Dr. HALLEY's Tranſlation. DE SECTIONE DETERMINATA II.
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            through the points H and G, and touches the plane ABC, touches likewiſe
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            the ſphere DFE.</s>
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          <head xml:id="echoid-head57" xml:space="preserve">PROBLEM IX.</head>
          <p>
            <s xml:id="echoid-s871" xml:space="preserve">
              <emph style="sc">Let</emph>
            there be given two ſpheres AB, DE, as alſo two points H and M;
              <lb/>
            </s>
            <s xml:id="echoid-s872" xml:space="preserve">to find a ſphere which ſhall paſs through the two given points, and likewiſe
              <lb/>
            touch the two given ſpheres.</s>
            <s xml:id="echoid-s873" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s874" xml:space="preserve">
              <emph style="sc">Let</emph>
            the right line AF be drawn paſſing through the centers of the
              <lb/>
            ſpheres, and as the radius AB is to the radius DE, ſo make BF to EF, and
              <lb/>
            the point F will be given. </s>
            <s xml:id="echoid-s875" xml:space="preserve">Make the rectangle HFG = the rectangle NFA,
              <lb/>
            and the point G will be given. </s>
            <s xml:id="echoid-s876" xml:space="preserve">Now having given three points M, H, G,
              <lb/>
            as alſo a ſphere DE; </s>
            <s xml:id="echoid-s877" xml:space="preserve">find a ſphere by Problem III, which ſhall paſs through
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            the given points, and touch the given ſphere; </s>
            <s xml:id="echoid-s878" xml:space="preserve">and, by Lemma III, it will be
              <lb/>
            the ſphere here required.</s>
            <s xml:id="echoid-s879" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div51" type="section" level="1" n="51">
          <head xml:id="echoid-head58" xml:space="preserve">PROBLEM X.</head>
          <p>
            <s xml:id="echoid-s880" xml:space="preserve">
              <emph style="sc">Let</emph>
            there be given two planes AB, BD, a point H, and a ſphere
              <lb/>
            EGF; </s>
            <s xml:id="echoid-s881" xml:space="preserve">to find a ſphere which ſhall paſs through the given point, and touch
              <lb/>
            the given ſphere, as alſo the two given planes.</s>
            <s xml:id="echoid-s882" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s883" xml:space="preserve">
              <emph style="sc">Through</emph>
            the center O of the given ſphere let a perpendicular to either of
              <lb/>
            the given planes CEOF be demitted, and make the rectangle HFI = the
              <lb/>
            rectangle CFE. </s>
            <s xml:id="echoid-s884" xml:space="preserve">Then having given the two points H and I, as alſo the
              <lb/>
            two planes AB, BD; </s>
            <s xml:id="echoid-s885" xml:space="preserve">find a ſphere, by Problem VII, which ſhall paſs
              <lb/>
            through the two given points, and likewiſe touch the two given planes; </s>
            <s xml:id="echoid-s886" xml:space="preserve">and,
              <lb/>
            by Lemma V, it will be the ſphere required.</s>
            <s xml:id="echoid-s887" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div52" type="section" level="1" n="52">
          <head xml:id="echoid-head59" xml:space="preserve">PROBLEM XI.</head>
          <p>
            <s xml:id="echoid-s888" xml:space="preserve">
              <emph style="sc">Let</emph>
            there be given a point, a plane, and two ſpheres; </s>
            <s xml:id="echoid-s889" xml:space="preserve">to find a ſphere
              <lb/>
            which ſhall paſs through the point, touch the plane, and alſo the two
              <lb/>
            ſpheres.</s>
            <s xml:id="echoid-s890" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s891" xml:space="preserve">This Problem, by a like method of reaſoning, is immediately reduced to
              <lb/>
            the VIIIth, where two points, a plane, and a ſphere are given, and that by
              <lb/>
            means of the Vth Lemma. </s>
            <s xml:id="echoid-s892" xml:space="preserve">But if you chuſe to uſe the IIId Lemma, it will
              <lb/>
            be reduced to the ſame Problem by a different method, and a different
              <lb/>
            conſtruction.</s>
            <s xml:id="echoid-s893" xml:space="preserve"/>
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