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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1843 Excerpt: ...F of C, and some equimultiples E and D of B and A, such that F is greater than E, but not greater than D: therefore E is less than D: and because E and D are equimultiples of B and A, and that E is less than D, therefore B is less than A. Proposition XL Theorem. (488) Ratios that are equal to the same ratio are equal to one another. Let A: B = C: D; and E: F = C: D: then A: B = E: F. Take of A, C, E any equimultiples whatever G, H, K; and of B, D, F any equimultiples whatever L, M, N. Therefore, since A: B = C: D, and G, H are taken equimultiples of A, C, ani L, M, of B, D; if G be greater than L, H is greater than M; and if equal, equal; and it' less, less. (Def. V.) Again, because E: F = C: D, and H, K are taken equimultiples of C, E; and M, N, of D, F; if H be greater than M, K is greater than N; and if equal, equal: and if less, less: but if G be greater than L, it has been shown that H is greater than M; and if equal, equal; and if less, less: therefore if G be greater than L, K is greater than N; and if equal, equal; and if less, less: and G, K are any equimultiples whatever of A, E; and L, N any whatever of B, F: therefore (Def. V.) A: B = E: F. This proposition is to ratios what Axiom I. Book I. is to mag-' nitudes. Proposition XII. Theorem. (489) If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so are all the antecedents taken together to all the consequents. Let any number of magnitudes A, B, C, D, E, F, be proportionals; that is, A: B = C: D = E: F. Then A: B = A + C + E: B + D + F. Take of A, C, E any equimultiples whatever G, H, K; and of B, D, F any equimultiples whatever L, M, N: then, because A: B = C: D = E: F; and that G, H, K are equimultiples of A, C, E, and L, M, N equimultiples of B, D, F; i...
