Archimedes
,
Natation of bodies
,
1662
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<
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from K to C, cutting the Diameter F G in L:
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and, thorow L, unto the Section E F. G, on the
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part E, draw the Line L M, parallel unto the
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ſame Baſe A C. And, of the Section A B C,
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let the Line B N be the Parameter; and, of the
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Section E F C, let F O be the Parameter. </
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>And,
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becauſe the Triangles C B D and C F G are alike
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;
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(b)
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therefore, as B C is to C F, ſo ſhall D C be
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to C G, and B D to F G. Again, becauſe the
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Triangles C K B and C L F, are alſo alike to
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one another; therefore, as B C is to C F, that is,
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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,
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B K to F L, are as D C to C G; that is,
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(c)
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as their duplicates A C and C E: But as
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B D is to F G, ſo is D C to C G; that is, A D to E G: And,
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Permutando,
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as B D is to
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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
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of our firſt of
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Conicks:
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Therefore, the
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(d)
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three Lines B D, A D and B N are
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Proportionalls. </
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>By the ſame reaſon, likewiſe, the Square E G being equall to the Rectangle
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G F O, the three other Lines F G, E G and F O, ſhall be alſo Proportionals: And, as B D is
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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:
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Ex equali,
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therefore, as D B is to B N, ſo is G F to F O: And,
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Permutando,
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as D B is to G F, ſo is
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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
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B N is to F O: And,
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Permutando,
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as B K is to B N, ſo is F L to F O. Again,
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becauſe the
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(e)
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Square H K is equall to the Rectangle B N; and the Square M L, equall
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to the Rectangle L F O, therefore, the three Lines B K, K H and B N ſhall be Proportionals:
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and F L, L M, and F O ſhall alſo be Proportionals: And, therefore,
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(f)
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as the Line
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B K is to the Line B N, ſo ſhall the Square B K, be to the Square H K: And, as the
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Line F L is to the Line F O, ſo ſhall the Square F L be to the Square L M:
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Therefore, becauſe that as B K is to B N, ſo is F L to F O; as the Square
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B K is to the Square K H, ſo ſhall the Square F L be to the Square L M: Therefore,
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(g)
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as the Line B K is to the Line K H, ſo is the Line F L to L M: And,
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Permutando,
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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,
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K H is to L M, as K C to C L: And, therefore, by the preceding Lemma, it is manifeſt that
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the Line H C alſo ſhall paſs thorow the Point M: As K C, therefore, is to C L, that is,
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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
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the Section E F C. And, in like manner might we demonſtrate, that the ſame happeneth
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in other Lines, that are produced from the Point C, and the Sections E B C. And, that
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B C hath the ſame proportion to C F, plainly appeareth; for B C is to C F, as D C to C G;
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that is, as their Duplicates A C to C E.
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(a)
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By 15. of the
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fifth.
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(b)
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By 4. of the
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ſixth.
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(c)
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By 15. of the
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fifth.
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(d)
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By 17. of the
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ſixth.
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(e)
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By 11 of our
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firſt of
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Conicks.</
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(f)
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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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(g)
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By 23. of the
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ſixth.
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<
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>From whence it is manifeſt, that all Lines ſo drawn, ſhall be cut by the
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ſaid Section in the ſame proportion. </
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<
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>For, by Diviſion and Converſion,
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C M is to M H, and C F to F B, as C E to E A.</
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<
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>LEMMA. III.</
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>And, hence it may alſo be proved, that the Lines which are
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drawn in like Portions, ſo, as that with the Baſes, they con
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tain equall Angles, ſhall alſo cut off like Portions; that is,
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as in the foregoing Figure, the Portions H B C and M F C,
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which the Lines C H and C M do cut off, are alſo alike to
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each other.</
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For let C H and C M be divided in the midst in the Points P and
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and thorow thoſe
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Points draw the Lines R P S and T Q V parallel to the Diameters. </
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<
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>Of the Portion
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H S C the Diameter ſhall be P S, and of the Portion M V C the Diameter ſhall be
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