Archimedes, Natation of bodies, 1662

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/052.jpg" pagenum="382"/>
                <figure id="id.073.01.052.1.jpg" xlink:href="073/01/052/1.jpg" number="49"/>
                <lb/>
                <emph type="italics"/>
              from K to C, cutting the Diameter F G in L:
                <lb/>
              and, thorow L, unto the Section E F. G, on the
                <lb/>
              part E, draw the Line L M, parallel unto the
                <lb/>
              ſame Baſe A C. And, of the Section A B C,
                <lb/>
              let the Line B N be the Parameter; and, of the
                <lb/>
              Section E F C, let F O be the Parameter. </s>
              <s>And,
                <lb/>
              becauſe the Triangles C B D and C F G are alike
                <emph.end type="italics"/>
              ;
                <lb/>
              (b)
                <emph type="italics"/>
              therefore, as B C is to C F, ſo ſhall D C be
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1331"/>
                <lb/>
                <emph type="italics"/>
              to C G, and B D to F G. Again, becauſe the
                <lb/>
              Triangles C K B and C L F, are alſo alike to
                <lb/>
              one another; therefore, as B C is to C F, that is,
                <lb/>
              as B D is to F G, ſo ſhall K C be to C L, and B K to F L: Wherefore, K C to C L, and,
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1332"/>
                <lb/>
                <emph type="italics"/>
              B K to F L, are as D C to C G; that is,
                <emph.end type="italics"/>
              (c)
                <emph type="italics"/>
              as their duplicates A C and C E: But as
                <lb/>
              B D is to F G, ſo is D C to C G; that is, A D to E G: And,
                <emph.end type="italics"/>
              Permutando,
                <emph type="italics"/>
              as B D is to
                <lb/>
              A D, ſo is F G to E G: But the Square A D, is equall to the Rectangle D B N, by the 11
                <lb/>
              of our firſt of
                <emph.end type="italics"/>
              Conicks:
                <emph type="italics"/>
              Therefore, the
                <emph.end type="italics"/>
              (d)
                <emph type="italics"/>
              three Lines B D, A D and B N are
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1333"/>
                <lb/>
                <emph type="italics"/>
              Proportionalls. </s>
              <s>By the ſame reaſon, likewiſe, the Square E G being equall to the Rectangle
                <lb/>
              G F O, the three other Lines F G, E G and F O, ſhall be alſo Proportionals: And, as B D is
                <lb/>
              to A D, ſo is F G to E G: And, therefore, as A D is to B N, ſo is E G to F O:
                <emph.end type="italics"/>
              Ex equali,
                <lb/>
                <emph type="italics"/>
              therefore, as D B is to B N, ſo is G F to F O: And,
                <emph.end type="italics"/>
              Permutando,
                <emph type="italics"/>
              as D B is to G F, ſo is
                <lb/>
              B N to F O: But as D B is to G F, ſo is B K to F L: Therefore, B K is to F L, as
                <lb/>
              B N is to F O: And,
                <emph.end type="italics"/>
              Permutando,
                <emph type="italics"/>
              as B K is to B N, ſo is F L to F O. Again,
                <lb/>
              becauſe the
                <emph.end type="italics"/>
              (e)
                <emph type="italics"/>
              Square H K is equall to the Rectangle B N; and the Square M L, equall
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1334"/>
                <lb/>
                <emph type="italics"/>
              to the Rectangle L F O, therefore, the three Lines B K, K H and B N ſhall be Proportionals:
                <lb/>
              and F L, L M, and F O ſhall alſo be Proportionals: And, therefore,
                <emph.end type="italics"/>
              (f)
                <emph type="italics"/>
              as the Line
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1335"/>
                <lb/>
                <emph type="italics"/>
              B K is to the Line B N, ſo ſhall the Square B K, be to the Square H K: And, as the
                <lb/>
              Line F L is to the Line F O, ſo ſhall the Square F L be to the Square L M:
                <lb/>
              Therefore, becauſe that as B K is to B N, ſo is F L to F O; as the Square
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1336"/>
                <lb/>
                <emph type="italics"/>
              B K is to the Square K H, ſo ſhall the Square F L be to the Square L M: Therefore,
                <emph.end type="italics"/>
                <lb/>
              (g)
                <emph type="italics"/>
              as the Line B K is to the Line K H, ſo is the Line F L to L M: And,
                <emph.end type="italics"/>
              Permutando,
                <lb/>
                <emph type="italics"/>
              as B K is to F L, ſo is K H to L M: But B K was to F L, as K C to C L: Therefore,
                <lb/>
              K H is to L M, as K C to C L: And, therefore, by the preceding Lemma, it is manifeſt that
                <lb/>
              the Line H C alſo ſhall paſs thorow the Point M: As K C, therefore, is to C L, that is,
                <lb/>
              as A C to C E, ſo is H C to C M; that is, to the ſame part of it ſelf, that lyeth betwixt C and
                <lb/>
              the Section E F C. And, in like manner might we demonſtrate, that the ſame happeneth
                <lb/>
              in other Lines, that are produced from the Point C, and the Sections E B C. And, that
                <lb/>
              B C hath the ſame proportion to C F, plainly appeareth; for B C is to C F, as D C to C G;
                <lb/>
              that is, as their Duplicates A C to C E.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1330"/>
              (a)
                <emph type="italics"/>
              By 15. of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1331"/>
              (b)
                <emph type="italics"/>
              By 4. of the
                <lb/>
              ſixth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1332"/>
              (c)
                <emph type="italics"/>
              By 15. of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1333"/>
              (d)
                <emph type="italics"/>
              By 17. of the
                <lb/>
              ſixth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1334"/>
              (e)
                <emph type="italics"/>
              By 11 of our
                <lb/>
              firſt of
                <emph.end type="italics"/>
              Conicks.</s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1335"/>
              (f)
                <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="margin">
              <s>
                <margin.target id="marg1336"/>
              (g)
                <emph type="italics"/>
              By 23. of the
                <lb/>
              ſixth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>From whence it is manifeſt, that all Lines ſo drawn, ſhall be cut by the
                <lb/>
              ſaid Section in the ſame proportion. </s>
              <s>For, by Diviſion and Converſion,
                <lb/>
              C M is to M H, and C F to F B, as C E to E A.</s>
            </p>
            <p type="head">
              <s>LEMMA. III.</s>
            </p>
            <p type="main">
              <s>And, hence it may alſo be proved, that the Lines which are
                <lb/>
              drawn in like Portions, ſo, as that with the Baſes, they con­
                <lb/>
              tain equall Angles, ſhall alſo cut off like Portions; that is,
                <lb/>
              as in the foregoing Figure, the Portions H B C and M F C,
                <lb/>
              which the Lines C H and C M do cut off, are alſo alike to
                <lb/>
              each other.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              For let C H and C M be divided in the midst in the Points P and
                <expan abbr="q;">que</expan>
              and thorow thoſe
                <lb/>
              Points draw the Lines R P S and T Q V parallel to the Diameters. </s>
              <s>Of the Portion
                <lb/>
              H S C the Diameter ſhall be P S, and of the Portion M V C the Diameter ſhall be
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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