Archimedes, Natation of bodies, 1662

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    <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>
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