Archimedes, Natation of bodies, 1662

Page concordance

< >
Scan Original
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
< >
page |< < of 68 > >|
    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/043.jpg" pagenum="373"/>
              did ſuppoſe that it made an Angle greater than the Angle B, the
                <lb/>
              Poriton did not reſt then neither; It is manifeſt that it ſhall ſtay
                <lb/>
                <arrow.to.target n="marg1253"/>
                <lb/>
              or reſt when it ſhall make an Angle eqnall to B. </s>
              <s>For ſo ſhall I O
                <lb/>
              be equall to
                <foreign lang="grc">Ψ</foreign>
                <emph type="italics"/>
              B
                <emph.end type="italics"/>
              ; and
                <foreign lang="grc">ω</foreign>
              I equall to
                <lb/>
                <figure id="id.073.01.043.1.jpg" xlink:href="073/01/043/1.jpg" number="42"/>
                <lb/>
                <foreign lang="grc">Ψ</foreign>
              R; and P H equall to F: There­
                <lb/>
              fore
                <emph type="italics"/>
              M P
                <emph.end type="italics"/>
              ſhall be ſeſquialter of
                <emph type="italics"/>
              P H,
                <emph.end type="italics"/>
                <lb/>
              and
                <emph type="italics"/>
              P H
                <emph.end type="italics"/>
              double of H M: And there­
                <lb/>
              fore ſince H is the Centre of Gravity
                <lb/>
              of that part of it which is within the
                <lb/>
              Liquid, it ſhall move upwards along
                <lb/>
              the ſame
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              erpendicular according to
                <lb/>
              which the whole
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion moveth;
                <lb/>
              and along the ſame alſo ſhall the part
                <lb/>
              which is above move downwards:
                <lb/>
              The
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion therefore ſhall reſt; for­
                <lb/>
              aſmuch as the parts are not repulſed by each other.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1238"/>
              A</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1239"/>
              B</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1240"/>
              C</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1241"/>
              D</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1242"/>
              E</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1243"/>
              F</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1244"/>
              G</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1245"/>
              (a)
                <emph type="italics"/>
              By 13. of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1246"/>
              H</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1247"/>
              K</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1248"/>
              L</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1249"/>
              M</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1250"/>
              N</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1251"/>
              O</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1252"/>
              P</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1253"/>
              Q</s>
            </p>
            <p type="head">
              <s>COMMANDINE.</s>
            </p>
            <p type="main">
              <s>And let
                <emph type="italics"/>
              C B
                <emph.end type="italics"/>
              be ſeſquialter of
                <emph type="italics"/>
              B R
                <emph.end type="italics"/>
              : C D ſhall alſo be ſeſquialter
                <lb/>
                <arrow.to.target n="marg1254"/>
                <lb/>
              of K R.]
                <emph type="italics"/>
              In the Tranſlation it is read thus:
                <emph.end type="italics"/>
              Sit autem & CB quidem hemeolia
                <lb/>
              ipſius B R: C D autem ipſius K R.
                <emph type="italics"/>
              But we at the reading of this paſſage have thought
                <lb/>
              fit thus to correctit; for it is not ſuppoſed ſo to be, but from the things ſuppoſed is proved to
                <lb/>
              be ſo. </s>
              <s>For if B
                <emph.end type="italics"/>
                <foreign lang="grc">ψ</foreign>
                <emph type="italics"/>
              be double of
                <emph.end type="italics"/>
                <foreign lang="grc">ψ</foreign>
                <emph type="italics"/>
              D, D B ſhall be ſeſquialter of B
                <emph.end type="italics"/>
                <foreign lang="grc">ψ.</foreign>
                <emph type="italics"/>
              And becauſe E B is
                <lb/>
              ſeſquialter of B R, it followeth that the
                <emph.end type="italics"/>
              (a)
                <emph type="italics"/>
              Remainder C D is ſeſquialter of
                <emph.end type="italics"/>
                <foreign lang="grc">ψ</foreign>
                <emph type="italics"/>
              R; that is, of
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1255"/>
                <lb/>
                <emph type="italics"/>
              the Semi-parameter: Wherefore B C ſhall be the Exceſſe by which the Axis is greater than
                <lb/>
              ſeſquialter of the Semi-parameter.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1254"/>
              A</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1255"/>
              (a)
                <emph type="italics"/>
              By 19. of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>And therefore F Q is leſſe than
                <emph type="italics"/>
              B C.] For in regard that the Portion hath
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1256"/>
                <lb/>
                <emph type="italics"/>
              the ſame proportion in Gravity unto the Liquid, as the Square F Q hath to the Square D B;
                <lb/>
              and hath leſſer proportion than the Square made of the Exceſſe by which the Axis
                <lb/>
              is greater than Seſquialter of the Semi parameter, hath to the Square made of the Axis; that
                <lb/>
              is, leßer than the Square C B hath to the Square B D; for the Line B D was ſuppoſed to be
                <lb/>
              equall unto the Axis: Therefore the Square F Q ſhall have to the Square D B leſſer proporti­
                <lb/>
              on than the Sqnare C B to the ſame Square B D: And therefore the Square
                <emph.end type="italics"/>
              (b)
                <emph type="italics"/>
              F Q ſhall be
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1257"/>
                <lb/>
                <emph type="italics"/>
              leße than the Square C B: And, for that reaſon, the Line F Q ſhall be leße than B C.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1256"/>
              B</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1257"/>
              (b)
                <emph type="italics"/>
              By 8 of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>And, for the ſame reaſon, F is leſſe than
                <emph type="italics"/>
              B R.] For C B being ſeſqui-
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1258"/>
                <lb/>
                <emph type="italics"/>
              alter of B R, and F Q ſeſquialter of F
                <emph.end type="italics"/>
              : (c) F
                <emph type="italics"/>
              Q ſhall be likewiſe leſſe than B C; and F
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1259"/>
                <lb/>
                <emph type="italics"/>
              leße than B R.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1258"/>
              C</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1259"/>
              (c)
                <emph type="italics"/>
              By 14 of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Now becauſe it hath been ſuppoſed that the Axis of the
                <emph type="italics"/>
              P
                <emph.end type="italics"/>
              ortion
                <lb/>
                <arrow.to.target n="marg1260"/>
                <lb/>
              doth make an Angle with the Surface of the Liquid greater than
                <lb/>
              the Angle
                <emph type="italics"/>
              B,
                <emph.end type="italics"/>
              the Angle
                <emph type="italics"/>
              P Y I
                <emph.end type="italics"/>
              ſhall be greater than the Angle
                <emph type="italics"/>
              B.]
                <lb/>
              For the Line P Y being parallel to the Surface of the Liquid, that is, to XS
                <emph.end type="italics"/>
              ; (d)
                <emph type="italics"/>
              the Angle
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1261"/>
                <lb/>
                <emph type="italics"/>
              P Y I ſhall be equall to the Angle contained betwixt the Diameter of the Portion N O, and the
                <lb/>
              Line X S: And therefore ſhall be greater than the Angle B.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1260"/>
              D</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1261"/>
              (d)
                <emph type="italics"/>
              By 29 of the
                <lb/>
              firſt.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>Therefore the Square
                <emph type="italics"/>
              P I
                <emph.end type="italics"/>
              hath greater proportion to the Square
                <lb/>
                <arrow.to.target n="marg1262"/>
                <lb/>
              Y I, than the Square E
                <foreign lang="grc">Ψ</foreign>
              hath to the Square
                <foreign lang="grc">Ψ</foreign>
                <emph type="italics"/>
              B] Let the Triangles P I Y
                <lb/>
              and E
                <emph.end type="italics"/>
                <foreign lang="grc">ψ</foreign>
                <emph type="italics"/>
              B, be deſcribed apart: And ſeeing that the Angle P Y I is greater
                <lb/>
              than the Angle E B
                <emph.end type="italics"/>
                <foreign lang="grc">ψ,</foreign>
                <emph type="italics"/>
              unto the Line I Y, and at the Point Y aſſigned in
                <emph.end type="italics"/>
                <lb/>
                <figure id="id.073.01.043.2.jpg" xlink:href="073/01/043/2.jpg" number="43"/>
                <lb/>
                <emph type="italics"/>
              the ſame, make the Angle V Y I equall to the Angle E B
                <emph.end type="italics"/>
                <foreign lang="grc">ψ</foreign>
              ;
                <emph type="italics"/>
              But
                <lb/>
              the Right Angle at I, is equall unto the Right Angle at
                <emph.end type="italics"/>
                <foreign lang="grc">ψ;</foreign>
                <emph type="italics"/>
              therefore the
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
Impressum