Archimedes
,
Natation of bodies
,
1662
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[Figure 61]
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(a)
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By 4. of the
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ſixth.
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* Or permitting.</
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(b)
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By 22. of the
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(c)
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By
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Cor.
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of 20.
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of the ſixth.
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>LEMMA IV.</
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>The ſame things aſſumed again, and M Q being drawn from the
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Point M unto the Diameter, let it touch the Section in the
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Point M. </
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>I ſay that H Q hath to Q O, the ſame proportion
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that G H hath to C N.</
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For make H R equall to G F; and ſeeing that
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the Triangles A F C and O P N are alike, and
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P N equall to F C, we might in like manner de
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monſtrate P O and F A to be equall to each other:
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Wherefore P O ſhall be double to F B: But H O
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is double to G B: Therefore the Remainder P H
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is alſo double to the Remainder F G; that is, to
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R H: And therefore is followeth that P R, R H
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and F G are equall to one another; as alſo that
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R G and P F are equall: For P G is common to
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both R P and G F. </
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>Since therefore, that H B is
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to B G, as G B is to B F, by Converſion of Pro
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portion, B H ſhall be to H G, as B G is to G F:
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But Q H is to H B, as H O to B G. </
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>For by 35
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of our firſt Book of
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Conicks,
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in regard that Q
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M toucheth the Section in the Point M, H B and
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B Q ſhall be equall, and Q H double to H B:
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Therefore,
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ex æquali,
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Q H ſhall be to H G, as
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H O to G F; that is, to H R: and,
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Permu
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tando,
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Q H ſhall be to H O, as H G to H R: again, by Converſion, H Q ſhall be to Q
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O, as H G to G R; that is, to P F; and, by the ſame reaſon, to C N: Whichwas to be de
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monſtrated.
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>Theſe things therefore being explained, we come now to that
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which was propounded. </
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>I ſay, therefore, firſt that
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N C
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hath
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to C K the ſame proportion that H G hath to G B.</
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For ſince that H Q is to Q O, as H G to C N
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;
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that is, to A O, equall to the ſaid C N: The Re
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mainder G Q ſhall be to the Remainder Q A, as
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H Q to Q O: and, for the ſame cauſe, the Lines
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A C and G L prolonged, by the things that wee
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have above demonstrated, ſhall interſect or meet
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in the Line Q M. Again, G Q is to Q A,
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as H Q to Q O: that is, as H G to F P; as
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(a)
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was bnt now demonstrated, But unto
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(b)
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G
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Q two Lines taken together, Q B that is H B, and
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B G are equall: and to Q A H F is equall; for
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if from the equall Magnitudes H B and B Q there
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be taken the equall Magnitudes F B and B A, the
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Re mainder ſhall be equall; Therefore taking H
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G from the two Lines H B and B G, there ſhall re
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main a Magnitude double to B G; that is, O H:
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and P F taken from F H, the Remainder is H P:
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Wherefore
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(c)
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O H is to H P, as G Q to Q A:
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But as G Q is to Q A, ſo is H Q to Q O;
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