Archimedes, Natation of bodies, 1662

Page concordance

< >
Scan Original
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
< >
page |< < of 68 > >|
    <archimedes>
      <text>
        <body>
          <chap>
            <pb xlink:href="073/01/037.jpg" pagenum="367"/>
            <p type="margin">
              <s>
                <margin.target id="marg1228"/>
              (a)
                <emph type="italics"/>
              By 4. of the
                <lb/>
              ſixth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1229"/>
              * Or permitting.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1230"/>
              (b)
                <emph type="italics"/>
              By 22. of the
                <lb/>
              ſixth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1231"/>
              (c)
                <emph type="italics"/>
              By
                <emph.end type="italics"/>
              Cor.
                <emph type="italics"/>
              of 20.
                <lb/>
              of the ſixth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>LEMMA IV.</s>
            </p>
            <p type="main">
              <s>The ſame things aſſumed again, and M Q being drawn from the
                <lb/>
              Point M unto the Diameter, let it touch the Section in the
                <lb/>
              Point M. </s>
              <s>I ſay that H Q hath to Q O, the ſame proportion
                <lb/>
              that G H hath to C N.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              For make H R equall to G F; and ſeeing that
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.073.01.037.1.jpg" xlink:href="073/01/037/1.jpg" number="34"/>
                <lb/>
                <emph type="italics"/>
              the Triangles A F C and O P N are alike, and
                <lb/>
              P N equall to F C, we might in like manner de­
                <lb/>
              monſtrate P O and F A to be equall to each other:
                <lb/>
              Wherefore P O ſhall be double to F B: But H O
                <lb/>
              is double to G B: Therefore the Remainder P H
                <lb/>
              is alſo double to the Remainder F G; that is, to
                <lb/>
              R H: And therefore is followeth that P R, R H
                <lb/>
              and F G are equall to one another; as alſo that
                <lb/>
              R G and P F are equall: For P G is common to
                <lb/>
              both R P and G F. </s>
              <s>Since therefore, that H B is
                <lb/>
              to B G, as G B is to B F, by Converſion of Pro­
                <lb/>
              portion, B H ſhall be to H G, as B G is to G F:
                <lb/>
              But Q H is to H B, as H O to B G. </s>
              <s>For by 35
                <lb/>
              of our firſt Book of
                <emph.end type="italics"/>
              Conicks,
                <emph type="italics"/>
              in regard that Q
                <lb/>
              M toucheth the Section in the Point M, H B and
                <lb/>
              B Q ſhall be equall, and Q H double to H B:
                <lb/>
              Therefore,
                <emph.end type="italics"/>
              ex æquali,
                <emph type="italics"/>
              Q H ſhall be to H G, as
                <lb/>
              H O to G F; that is, to H R: and,
                <emph.end type="italics"/>
              Permu­
                <lb/>
              tando,
                <emph type="italics"/>
              Q H ſhall be to H O, as H G to H R: again, by Converſion, H Q ſhall be to Q
                <lb/>
              O, as H G to G R; that is, to P F; and, by the ſame reaſon, to C N: Whichwas to be de­
                <lb/>
              monſtrated.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Theſe things therefore being explained, we come now to that
                <lb/>
              which was propounded. </s>
              <s>I ſay, therefore, firſt that
                <emph type="italics"/>
              N C
                <emph.end type="italics"/>
              hath
                <lb/>
              to C K the ſame proportion that H G hath to G B.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              For ſince that H Q is to Q O, as H G to C N
                <emph.end type="italics"/>
              ;
                <lb/>
                <figure id="id.073.01.037.2.jpg" xlink:href="073/01/037/2.jpg" number="35"/>
                <lb/>
                <emph type="italics"/>
              that is, to A O, equall to the ſaid C N: The Re­
                <lb/>
              mainder G Q ſhall be to the Remainder Q A, as
                <lb/>
              H Q to Q O: and, for the ſame cauſe, the Lines
                <lb/>
              A C and G L prolonged, by the things that wee
                <lb/>
              have above demonstrated, ſhall interſect or meet
                <lb/>
              in the Line Q M. Again, G Q is to Q A,
                <lb/>
              as H Q to Q O: that is, as H G to F P; as
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1232"/>
                <lb/>
              (a)
                <emph type="italics"/>
              was bnt now demonstrated, But unto
                <emph.end type="italics"/>
              (b)
                <emph type="italics"/>
              G
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1233"/>
                <lb/>
                <emph type="italics"/>
              Q two Lines taken together, Q B that is H B, and
                <lb/>
              B G are equall: and to Q A H F is equall; for
                <lb/>
              if from the equall Magnitudes H B and B Q there
                <lb/>
              be taken the equall Magnitudes F B and B A, the
                <lb/>
              Re mainder ſhall be equall; Therefore taking H
                <lb/>
              G from the two Lines H B and B G, there ſhall re­
                <lb/>
              main a Magnitude double to B G; that is, O H:
                <lb/>
              and P F taken from F H, the Remainder is H P:
                <lb/>
              Wherefore
                <emph.end type="italics"/>
              (c)
                <emph type="italics"/>
              O H is to H P, as G Q to Q A:
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1234"/>
                <lb/>
                <emph type="italics"/>
              But as G Q is to Q A, ſo is H Q to Q O;
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
Impressum