5. of C such, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than that of C. Let these multiples be taken, and let D, E, be the equimultiples of A, B, and F the multiple of C, such that D may be greater than F, but E not Hyp. greater than F: then, because* A is to C, as B is to C, and of A, B, are taken equimultiples D, E, and of C is taken a multiple F; a 5 Def. E is also greater than F: but this is B and B are not unequal; that is, they are equal. For, if they are not equal, one of them must be greater than the other; let A be the greater; then, as was proved in Prop. 8th, there is some multiple 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; and *Hyp. because* C is to B, as C is to A, and that F the multiple of the first, is greater than E the multiple of the second; therefore F the multiple of the third, is greater *5 Def than D the multiple of the fourth:* but this is impossible, because F is not greater than D. Therefore, A is equal to B. Wherefore, magnitudes which, &c. Q. E. D. 5. PROP. X. THEOR. That magnitude which has a greater ratio than another has unto the same magnitude, is the greater of the two: and that magnitude to which the same has a greater ratio than it has unto another magnitude, is the lesser of the two. Let A have to C a greater ratio than B has to C; A shall be greater than B. 5. For, because A has to C a greater ratio than B has to C, there are some equimultiples of A and B, and *7 Def. some multiple of C such, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than it: let them be taken, and let D, E be the equimultiples of A, B, and F the multiple of C such, that D is greater 4 than F, but E is not greater than F: therefore D is greater than E: and be cause D and E are equimultiples of A B and B, and that D is greater than E; therefore A is greater than B. 디 E Next, let C have to B a greater ratio than C has to A; B shall be less than A. * 4 Ax. 5. 5. For there is some multiple F of C, and some equi- * 7 Def. multiples 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, therefore B is less than A. Therefore, that * 4 Ax. magnitude, &c. Q. E. D. PROP. XI. THEOR. Ratios that are the same to the same ratio, are the same to one another. Let A be to B as C is to D; and let C be to D as E is to F; A shall be to B, as E to F. 5. Take of A, C, E, any equimultiples whatever G, H, K; and of B, D, F, any equimultiples whatever L, M, N. And since A is to B, as C to D, and that G, H are taken equimultiples of A, C, and L, M of B, D; therefore if G be greater than L, H is greater than M; and if equal, equal; and if less, less.* Again, because * 5 Def. C is to D, as E is to F, and that H, K are taken equimultiples of C, E; and M, N, of D, F; therefore if H be greater than M, K is greater than N; and if equal, 5. equal; and if less, less; but it has been proved that if G be greater than L, 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: but G, K are any equimultiples whatever of A, E; and L, N any whatever of *5 Def. B, F: therefore, as A is to B, so is E to F.* Wherefore, ratios that, &c. Q. E. D. 5. PROP. XII. THEOR. If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so shall all the antecedents taken together be to all the consequents. Let any number of magnitudes A, B, C, D, E, F, be proportionals; that is, as A is to B, so C to D, and E to F: as A is to B, so shall A, C, E together, be to B, D, F together. 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 is to B, as C is to D, and as E to F; and that G, H, K are equimultiples of A, C, E, and L, M, N equimultiples of B, D, F; therefore if G be greater than L, H is greater than M, and K greater there * 5 Def. than N; and if equal, equal; and if less, less: fore if G be greater than L, then G, H, K together, 5. are greater than L, M, N together; and if equal, equal; and if less, less. But G, and G, H, K together, are any equimultiples of A, and A, C, E together; because, if there be any number of magnitudes equimultiples of as many, each of each, whatever multiple one of them is of its part, the same multiple is the whole of the whole:* for the same reason. L. and L, M, N are any * 1. 5. equimultiples of B, and B, D, F: therefore as A is to B, so are A, C, E together to B, D, F together. * 5 Def. Wherefore, if any number, &c. Q. E. D. 5. If the first has to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth; the first shall also have to the second a greater ratio than the fifth has to the sixth. Let A the first have the same ratio to B the second, which C the third has to D the fourth, but C the third to D the fourth, a greater ratio than E the fifth has to F the sixth: A the first shall have to B the second, a greater ratio than E the fifth has to F the sixth. Because C has to D a greater ratio than E has to F, there are some equimultiples of C and E, and some of D and F such, that the multiple of C is greater than the multiple of D, but the multiple of E not greater 5. than the multiple of F: let such be taken, and let * 7 Def. G, H be equimultiples of C, E and K, L equimultiples of D, F, such that G may be greater than K, but H not greater than L; and whatever multiple G is of C, take M the same multiple of A; and whatever multiple K is of D, take N the same multiple of B: then, because Hyp. A is to B,* as C to D, and of A and C, M and G are equimultiples: and of B and D, N and K are equimultiples; therefore if M be greater than N, G is *5 Def. greater than K; and if equal, equal; and if less, less;* *Const. but G is greater than K, therefore M is greater than *Const. N: but H is not greater* than L; and M H are equimultiples of A, E; and N, L equimultiples of B, F: *7 Def. therefore A has to B a greater ratio than E has to F.* Wherefore, if the first, &c. Q. E. D. 5. 5. * COR. And if the first have to the second a greater ratio than the third has to the fourth, but the third to the fourth the same ratio which the fifth has to the sixth; it may be demonstrated, in like manner, that the first has to the second a greater ratio than the fifth has to the sixth. PROP. XIV. THEOR. If the first have to the second the same ratio which the third has to the fourth; then, if the first be greater than the third, the second shall be greater than the fourth; and if equal, equal; and if less, less. Let A the first have to B the second, the same ratio which C the third has to D the fourth; if A be greater than C, B shall be greater than D. Because A is greater than C, and B is any other magnitude, A has to B a greater ratio than C has to * 8. 5. B: but, as A is to B, so is C to D; therefore also C 13. 5. has to D a greater ratio than C has to B: but of two |