Archimedes, Natation of bodies, 1662

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s>
                <pb xlink:href="073/01/055.jpg" pagenum="385"/>
                <emph type="italics"/>
              to M D, as C Q to Q A: But L B is to B D, by 5 of
                <emph.end type="italics"/>
              Archimedes,
                <emph type="italics"/>
              before recited, as C D
                <lb/>
              to D A: It is manifeſt therefore, by the precedent Lemma, that C D is to D Q, as L B is to
                <lb/>
              B M: But as C D is to D Q, ſo is C M to M P: Therefore L B is to B M, as C M to M P:
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1343"/>
                <lb/>
                <emph type="italics"/>
              And it haveing been demonſtrated, that C M is to M P, as C E to E A; L B ſhall be to B M,
                <lb/>
              as C E to E A. </s>
              <s>And in like manner it ſhall be demonstrated that ſo is N O to O F; as alſo the
                <lb/>
              Remainders. </s>
              <s>And that alſo H K is to K E, as C E to E A, doth plainly appeare by the ſame
                <emph.end type="italics"/>
                <lb/>
              5.
                <emph type="italics"/>
              of
                <emph.end type="italics"/>
              Archimedes
                <emph type="italics"/>
              : Which is that that we propounded to be demonſtrated.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1340"/>
              (a)
                <emph type="italics"/>
              By 4. of the
                <lb/>
              ſixth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1341"/>
              (b)
                <emph type="italics"/>
              By 11 of the
                <lb/>
              fifth,
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1342"/>
              (c)
                <emph type="italics"/>
              By 14 of the
                <lb/>
              fifth.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1343"/>
                <emph type="italics"/>
              By 2. of the ſixth
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s>LEMMA. VI.</s>
            </p>
            <p type="main">
              <s>And, therefore, let the things ſtand as above; and deſcribe
                <lb/>
              yet another like Portion, contained betwixt a Right Line, and
                <lb/>
              the Section of the Rightangled Cone D R C, whoſe Diameter
                <lb/>
              is R S, that it may cut the Line F G in T; and prolong S R
                <lb/>
              unto the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ine C H in V, which meeteth the Section A B C in
                <lb/>
              X, and E F C in Y. </s>
              <s>I ſay, that B M hath to M D, a propor­
                <lb/>
              tion compounded of the proportion that E A hath to A C;
                <lb/>
              and of that which C D hath to D E.</s>
            </p>
            <p type="main">
              <s>
                <emph type="italics"/>
              For, we ſhall firſt demonſtrate, that the Line C H toucheth the Section D R C in the
                <lb/>
              Point C; and that L M is to M D, as alſo N F to F T, and V Y to Y R, as C D is to E D.
                <lb/>
              And, becauſe now that L B is to B M, as C E is to E A; therefore, Compounding and Conver­
                <lb/>
              ting, B M ſhall be to L M, as E A to A C: And, as L M is to M D, ſo ſhall C D be to
                <lb/>
              D E: The proportion, therefore, of B M to M D, is compounded of the proportion that
                <lb/>
              B M hath to L M, and of the proportion that L M hath to M D: Therefore, the proportion
                <lb/>
              of B M to M D, ſhall alſo be compounded of the proportion that E A hath to A C, and of
                <lb/>
              that which C D hath to D E. </s>
              <s>In the ſame manner it ſhal be demonſtrated, that O F hath to
                <lb/>
              F T, and alſo X Y to Y R, a proportion compounded of thoſe ſame proportions; and ſo in
                <lb/>
              the reſt: Which was to be demonstrated.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="main">
              <s>By which it appeareth that the
                <emph type="italics"/>
              L
                <emph.end type="italics"/>
              ines ſo drawn; which fall betwixt
                <lb/>
              the Sections A B C and D R C, ſhall be divided by the Section E F C
                <lb/>
              in the ſame Proportion.</s>
            </p>
            <p type="main">
              <s>And C B is to B D, as ſix to fifteen.]
                <emph type="italics"/>
              For we have ſuppoſed that B K is
                <emph.end type="italics"/>
                <lb/>
                <arrow.to.target n="marg1344"/>
                <lb/>
                <emph type="italics"/>
              double of K D: Wherefore, by Compoſition B D ſhall be to K D as three to one; that is, as
                <lb/>
              fifteen to five: But B D was to K C as fifteen to four; Therefore B D is to D C as fifteen to nine:
                <lb/>
              And, by Converſion of proportion and Convert­
                <lb/>
              ing, C B is to B D, as ſix to ſifteen.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1344"/>
              N</s>
            </p>
            <figure id="id.073.01.055.1.jpg" xlink:href="073/01/055/1.jpg" number="53"/>
            <p type="main">
              <s>And as C B is to B D, ſo is
                <lb/>
                <arrow.to.target n="marg1345"/>
                <lb/>
              E B to B A; and D Z to D A.]
                <lb/>
                <emph type="italics"/>
              For the Triangles C B E and D B A being
                <lb/>
              alike; As C B is to B E, ſo ſhall D B be to B A:
                <lb/>
              And,
                <emph.end type="italics"/>
              Permutando,
                <emph type="italics"/>
              as C B is to B D, ſo ſhall
                <lb/>
              E B be to B A: Againe, as B C is to C E ſo
                <lb/>
              ſhall B D be to D A, And,
                <emph.end type="italics"/>
              Permutando,
                <emph type="italics"/>
              as
                <lb/>
              C B is to B D, ſo ſhall C E, that is, D Z
                <lb/>
              equall to it, be to D A.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s>
                <margin.target id="marg1345"/>
              O</s>
            </p>
            <p type="main">
              <s>And of D Z and D A, L I and
                <lb/>
                <arrow.to.target n="marg1346"/>
                <lb/>
              L A are double.]
                <emph type="italics"/>
              That the Line L A is
                <lb/>
              double of D A, is manifeſt, for that B D is the Diameter of the Portion. </s>
              <s>And that L I is
                <lb/>
              dovble to D Z ſhall be thus demonſtrated. </s>
              <s>For as much as ZD is to D A, as two to five:
                <lb/>
              therefore, Converting and Dividing, A Z, that is, I Z, ſhall be to Z D, as three to two:
                <emph.end type="italics"/>
              </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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