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

< >
[91.] ADVERTISEMENT.
Page: 135
[92.] AN EXPLANATION OF THE SYMBOLS made uſe of in this SYNOPSIS.
Page: 137
[93.] INDEX OF THE Authors refered to in the SYNOPSIS.
Page: 139 ([i])
[94.] Lately was publiſhed by the ſame Author; [Price Six Shillings in Boards.]
Page: 140 ([ii])
[95.] SYNOPSIS.
Page: 141 ([1])
[96.] Continuation of the Synopsis, Containing ſuch Data as cannot readily be expreſſed by the Symbols before uſed without more words at length.
Page: 149 ([9])
[97.] SYNOPSIS
Page: 152 ([12])
[98.] FINIS.
Page: 156 ([16])
< >
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>
Impressum