Archimedes
,
Natation of bodies
,
1662
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(a)
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By 2. of the
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ſixth.
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(b)
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By 30 of the
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firſt.
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<
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>LEMMA. V.</
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>Again, let there be two like Portions, contained betwixt Right
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Lines and the Sections of Right-angled Cones, as in the fore
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going figure, A B C, whoſe Diameter is B D; and E F C,
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whoſe Diameter is F G; and from the Point E, draw the
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Line E H parallel to the Diameters B D and F G; and let it
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cut the Section A B C in K: and from the Point C draw C H
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touching the Section A B C in C, and meeting with the Line
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E H in H; which alſo toucheth the Section E F C in the ſame
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Point C, as ſhall be demonſtrated: I ſay that the Line drawn
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from C
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H
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unto the Section E F C ſo as that it be parallel to
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the Line E H, ſhall be divided in the ſame proportion by the
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Section A B C, in which the
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L
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ine C A is divided by the Section
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E F C; and the part of the
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L
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ine C A which is betwixt the
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two Sections, ſhall anſwer in proportion to the part of the Line
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drawn, which alſo falleth betwixt the ſame Sections: that is,
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as in the foregoing Figure, if D B be produced untill it meet
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with C H in L, that it may interſect the Section E F C in the
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Point M, the
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L
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ine
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L
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B ſhall have to B M the ſame proportion
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that C E hath to E A.</
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For let G F be prolonged untill it meet the ſame Line C H in N, cutting the Section A B C
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in O; and drawing a Line from B to C, which ſhall paſſe by F, as hath been ſhewn, the
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Triangles C G F and C D B ſhall be alike; as
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alſo the Triangles C F N and C B L: Wherefore
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(a)
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as G F is to D B, ſo ſhall C F b to C B:
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And as
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(b)
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C F is to C B, ſo ſhall F N be
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to B L: Therefore G F ſhall be to D B, as F N
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to B L: And,
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Permutando,
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G F ſhall be to
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F N, as D B to B L: But D B is equall to
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B L, by 35 of our Firſt Book of
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Conicks:
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Therefore
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(c)
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G F alſo ſhall be equall to F N:
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And by 33 of the ſame, the Line C H touch
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eth the Section E F C in the ſame Point. </
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<
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>There
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fore, drawing a Line from C to M, prolong it
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untill it meet with the Section A B C in P; and
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from P unto A C draw P Q parallel to B D.
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Becauſe, now, that the Line C H toucheth the
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Section E F C in the Point C; L M ſhall have
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the ſame proportion to M D that C D hath to D E,
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by the Fifth Propoſition of
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Archimedes
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in his
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Book
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De Quadratura Patabolæ:
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And by
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reaſon of the Similitude of the Triangles C M D
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and C P Q, as C M is to C D, ſo ſhall C P
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be to C Q: And,
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Permutando,
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as C M is to
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C P, ſo ſhall C D be to C Q: But as C M is to C P, ſo is C E to C A,; as we have but
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even now demonſtrated: And therefore, as C E is to C A, ſo is C D to C
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that is as the
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whole is to the whole, ſo is the part to the part: The remainder, therefore, D E is to the
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Remainder Q A, as C E is to C A; that is, as C D is to C Q: And,
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Permutando,
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C D
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is to D E, as C Q is to Q A: And L M is alſo to M D, as C D to D E: Therefore L M is
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