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

Page concordance

< >
Scan Original
21 (9)
22 (10)
23 (11)
24 (12)
25 (13)
26 (14)
27 (15)
28 (16)
29 (17)
30
31 (19)
32 (20)
33 (21)
34 (22)
35 (23)
36 (24)
37 (25)
38 (26)
39 (27)
40 (28)
41 (29)
42
43
44
45
46
47
48
49
50
< >
page |< < ([15]) 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="[15]" file="0085" n="92"/>
        </div>
        <div xml:id="echoid-div75" type="section" level="1" n="71">
          <head xml:id="echoid-head84" xml:space="preserve">DETERMINATE SECTION.</head>
          <head xml:id="echoid-head85" xml:space="preserve">BOOK I.</head>
          <head xml:id="echoid-head86" xml:space="preserve">PROBLEM I. (Fig. 1.)</head>
          <p>
            <s xml:id="echoid-s1710" xml:space="preserve">In any indefinite ſtraight line, let the Point A be aſſigned; </s>
            <s xml:id="echoid-s1711" xml:space="preserve">it is required
              <lb/>
            to cut it in ſome other point O, ſo that the ſquare on the ſegment AO
              <lb/>
            may be to the ſquare on a given line, P, in the ratio of two given ſtraight
              <lb/>
            lines R and S.</s>
            <s xml:id="echoid-s1712" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1713" xml:space="preserve">
              <emph style="sc">Analysis</emph>
            . </s>
            <s xml:id="echoid-s1714" xml:space="preserve">Since, by Hypotheſis, the ſquare on AO muſt be to the
              <lb/>
            ſquare on P as R is to S, the ſquare on AO will be to the Square on P as
              <lb/>
            the ſquare on R is to the rectangle contained by R and S (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1715" xml:space="preserve">V. </s>
            <s xml:id="echoid-s1716" xml:space="preserve">15.)
              <lb/>
            </s>
            <s xml:id="echoid-s1717" xml:space="preserve">Let there be taken AD, a mean proportional between AB (R) and AC
              <lb/>
            (S); </s>
            <s xml:id="echoid-s1718" xml:space="preserve">then the Square on AO is to the ſquare on P as the ſquare on R is
              <lb/>
            to the ſquare on AD, or (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1719" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1720" xml:space="preserve">22) AO is to P as R to AD; </s>
            <s xml:id="echoid-s1721" xml:space="preserve">conſe-
              <lb/>
            quently, AO is given by
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1722" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1723" xml:space="preserve">12.</s>
            <s xml:id="echoid-s1724" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1725" xml:space="preserve">
              <emph style="sc">Synthesis</emph>
            . </s>
            <s xml:id="echoid-s1726" xml:space="preserve">Make AB equal to R, AC equal to S, and deſcribe on
              <lb/>
            BC a ſemi-circle; </s>
            <s xml:id="echoid-s1727" xml:space="preserve">erect at A the indefinite perpendicular AF, meeting the
              <lb/>
            circle in D, and take AF equal to P; </s>
            <s xml:id="echoid-s1728" xml:space="preserve">draw DB, and parallel thereto FO,
              <lb/>
            meeting the indefinite line in O, the point required.</s>
            <s xml:id="echoid-s1729" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1730" xml:space="preserve">For, by reaſon of the ſimilar triangles ADB, AFO, AO is to AF (P) as
              <lb/>
            AB (R) is to AD; </s>
            <s xml:id="echoid-s1731" xml:space="preserve">therefore (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1732" xml:space="preserve">VI. </s>
            <s xml:id="echoid-s1733" xml:space="preserve">22.) </s>
            <s xml:id="echoid-s1734" xml:space="preserve">the ſquare on AO is to the
              <lb/>
            ſquare on P as the ſquare on R is to the ſquare on AD; </s>
            <s xml:id="echoid-s1735" xml:space="preserve">but the ſquare on
              <lb/>
            AD is equal to the rectangle contained by AB (R) and AC (S) by
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1736" xml:space="preserve">VI.
              <lb/>
            </s>
            <s xml:id="echoid-s1737" xml:space="preserve">13. </s>
            <s xml:id="echoid-s1738" xml:space="preserve">17; </s>
            <s xml:id="echoid-s1739" xml:space="preserve">and ſo the ſquare on AO is to the ſquare on P as the ſquare on R
              <lb/>
            is to the rectangle contained by R and S; </s>
            <s xml:id="echoid-s1740" xml:space="preserve">that is (
              <emph style="sc">Eu</emph>
            . </s>
            <s xml:id="echoid-s1741" xml:space="preserve">V. </s>
            <s xml:id="echoid-s1742" xml:space="preserve">15.) </s>
            <s xml:id="echoid-s1743" xml:space="preserve">as R is to S.</s>
            <s xml:id="echoid-s1744" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1745" xml:space="preserve">Q. </s>
            <s xml:id="echoid-s1746" xml:space="preserve">E. </s>
            <s xml:id="echoid-s1747" xml:space="preserve">D.</s>
            <s xml:id="echoid-s1748" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1749" xml:space="preserve">
              <emph style="sc">Scholium</emph>
            . </s>
            <s xml:id="echoid-s1750" xml:space="preserve">Here are no limitations, nor any precautions whatever to be
              <lb/>
            obſerved, except that AB (R) muſt be ſet off from A that way which O
              <lb/>
            is required to fall.</s>
            <s xml:id="echoid-s1751" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>
Impressum