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

< >
[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.
< >
page |< < ([14]) of 161 > >|
    <echo version="1.0RC">
      <text xml:lang="en" type="free">
        <div xml:id="echoid-div74" type="section" level="1" n="70">
          <pb o="[14]" file="0084" n="91"/>
          <p>
            <s xml:id="echoid-s1662" xml:space="preserve">
              <emph style="sc">Demonstration</emph>
            . </s>
            <s xml:id="echoid-s1663" xml:space="preserve">Through S, any other point taken at pleaſure, draw
              <lb/>
            LSM parallel to VOY, and join VS and produce it to meet the perpendi-
              <lb/>
            cular in N and the circumference in R. </s>
            <s xml:id="echoid-s1664" xml:space="preserve">Produce alfo the perpendicular YI
              <lb/>
            to meet the circumference again in F, and join RF; </s>
            <s xml:id="echoid-s1665" xml:space="preserve">then from ſrmilar tri-
              <lb/>
            angles it appears that the rectangle LSM: </s>
            <s xml:id="echoid-s1666" xml:space="preserve">ESI:</s>
            <s xml:id="echoid-s1667" xml:space="preserve">: VOY i. </s>
            <s xml:id="echoid-s1668" xml:space="preserve">e. </s>
            <s xml:id="echoid-s1669" xml:space="preserve">AOU: </s>
            <s xml:id="echoid-s1670" xml:space="preserve">EOI.
              <lb/>
            </s>
            <s xml:id="echoid-s1671" xml:space="preserve">But the rectangle ASU i. </s>
            <s xml:id="echoid-s1672" xml:space="preserve">e. </s>
            <s xml:id="echoid-s1673" xml:space="preserve">VSR is greater than LSM. </s>
            <s xml:id="echoid-s1674" xml:space="preserve">(for LV i. </s>
            <s xml:id="echoid-s1675" xml:space="preserve">e. </s>
            <s xml:id="echoid-s1676" xml:space="preserve">MY x
              <lb/>
              <emph style="ol">NI + MI</emph>
            together with VS x SN is by the preceeding
              <emph style="sc">Lemma</emph>
            equal to LSM. </s>
            <s xml:id="echoid-s1677" xml:space="preserve">
              <lb/>
            But ASU or VSR is equal to VS x SN, together with VS x NR; </s>
            <s xml:id="echoid-s1678" xml:space="preserve">for which
              <lb/>
            laſt rectangle we may ſubſtitute MY x NF: </s>
            <s xml:id="echoid-s1679" xml:space="preserve">for the triangles VLS and NRF
              <lb/>
            are ſimilar, being each of them ſimilar to VNY; </s>
            <s xml:id="echoid-s1680" xml:space="preserve">therefore VL or MY: </s>
            <s xml:id="echoid-s1681" xml:space="preserve">VS
              <lb/>
            :</s>
            <s xml:id="echoid-s1682" xml:space="preserve">: NR: </s>
            <s xml:id="echoid-s1683" xml:space="preserve">NF and MY x NF = VS x NR. </s>
            <s xml:id="echoid-s1684" xml:space="preserve">Now NF being always greater
              <lb/>
            than
              <emph style="ol">NI + MI</emph>
            , it appears from thence that ASU is greater than LSM.) </s>
            <s xml:id="echoid-s1685" xml:space="preserve">
              <lb/>
            Therefore the ratio of ASU to ESI is greater than that of AOU to EOI. </s>
            <s xml:id="echoid-s1686" xml:space="preserve">
              <lb/>
            And the ſame holds good with regard to any other point S taken between E
              <lb/>
            and I, ſo that the ratio of AOU to EOI is a Minimum, and ſingular, or
              <lb/>
            what the Antients called μοναχ@.</s>
            <s xml:id="echoid-s1687" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1688" xml:space="preserve">Let now VY join the tops of two perpendiculars drawn on the ſame ſide of
              <lb/>
            the diameter, and meet the diameter produced in O; </s>
            <s xml:id="echoid-s1689" xml:space="preserve">I ſay that the ratio of
              <lb/>
            AOU to EOI is a Maximum.</s>
            <s xml:id="echoid-s1690" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1691" xml:space="preserve">For uſing the ſame conſtruction as before, it will appear that the rectangle
              <lb/>
            LSM: </s>
            <s xml:id="echoid-s1692" xml:space="preserve">ESI:</s>
            <s xml:id="echoid-s1693" xml:space="preserve">: VOY or AOU: </s>
            <s xml:id="echoid-s1694" xml:space="preserve">EOI. </s>
            <s xml:id="echoid-s1695" xml:space="preserve">And it may be proved in the ſame
              <lb/>
            manner that VSR or ASU is leſs than LSM. </s>
            <s xml:id="echoid-s1696" xml:space="preserve">(LV i. </s>
            <s xml:id="echoid-s1697" xml:space="preserve">e. </s>
            <s xml:id="echoid-s1698" xml:space="preserve">MY x
              <emph style="ol">NI + MI</emph>
            or
              <lb/>
            (making IK = MI) MY x NK, together with LSM is by the preceding
              <emph style="sc">Lem</emph>
            -
              <lb/>
              <emph style="sc">MA</emph>
            equal to VS x SN. </s>
            <s xml:id="echoid-s1699" xml:space="preserve">But VS x NR, together with VSR, is alſo equal to
              <lb/>
            VS x SN. </s>
            <s xml:id="echoid-s1700" xml:space="preserve">Now VS x NR is equal to MY x NF or LV x NF from the ſimi-
              <lb/>
            larity of the triangles LVS, NRF. </s>
            <s xml:id="echoid-s1701" xml:space="preserve">Therefore now alſo MY x NF together
              <lb/>
            with VSR is proved equal to VS x SN. </s>
            <s xml:id="echoid-s1702" xml:space="preserve">But as NK is leſs than NF, VSR
              <lb/>
            will be a leſs rectangle than LSM) Hence the ratio of LSM to ESI or it's
              <lb/>
            equal AOU to EOI is greater than the ratio of VSR or ASU to ESI. </s>
            <s xml:id="echoid-s1703" xml:space="preserve">And
              <lb/>
            the ſame holds with regard to any other point taken in the diameter pro-
              <lb/>
            duced. </s>
            <s xml:id="echoid-s1704" xml:space="preserve">Therefore the ratio of AOU to EOI is a Maximum.</s>
            <s xml:id="echoid-s1705" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1706" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s1707" xml:space="preserve">E. </s>
            <s xml:id="echoid-s1708" xml:space="preserve">D.</s>
            <s xml:id="echoid-s1709" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>