Book V. PROP. X. THEOR. THAT magnitude which has a greater ratio than See N. 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. 5. Let A have to C a greater ratio than B has to C: A is greater than B: for, because A has a greater ratio to C than B has to C, there are some equimultiples of A and B, and a 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 equimultiples of A, B, and F a multiple of C such that D is greater than F, but E is not greater than F: therefore D is greater than E: and, because D and E are equi- A Next, Let C have a greater ratio to B than it has to A; B is less than A: for a 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 is not greater than D: E therefore is less than D; and because E and D are equimultiples of B and A, therefore B is less than A. That magnitude, therefore, &c. Q. E. D. B 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, as C to D, so let E be to F; A is to B as E to 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 is to B as C to D, and G, H are taken equimultiples of Book V. A, C, and L, M of B, D; if G be greater than L, H is greater than M; and if equal, equal; and if less, less 2. Again, bea 5. def. cause C is to D as E is to F, 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 5. 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, as A is to B so is E to Fa. Wherefore, ratios that, &c. Q. E. D. : 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 Book V. B, D, F; if G be greater than L, H is greater than M, and Ked greater than N; and if equal, equal; and if less, less. Where-a5. def. 5. fore, if G be greater than L, then G, H, K together are greater than L, M, N together; and if equal, equal; and if less, less. And 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 equi- b 1. 5. multiples of B, and B, D, F: as therefore A is to B, so are A, C, E together to B, D, F together. Wherefore, if any number, &c. Q. E. D. PROP. XIII. THEOR. IF the first has to the second the same ratio which see N. 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 to F the sixth: also, the first A ́ shall have to the second B, a greater ratio than the fifth E to the sixth F. Because C has a greater ratio to D, than E 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 is not greater than the multiple of Fa: let a 7.def. 5. such be taken, and of C, E let G, H be equimultiples, and K, L equimultiples of D, F, so that G 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 A is to B, as C to D, and S Book V. of A and C, M and G are equimultiples; and of B and D, N and K are equimultiples; if M be greater than N, G is greater b 5.def. 5. than K; and if equal, equal; and if less, lessb; but G is greater than K, therefore M is greater than N: but H is not greater than L; and M, H are equimultiples of A, E; and N, L equimultiples c7.def. 5. of B, F: therefore A has a greater ratio to B than E has to Fe. Wherefore, if the first, &c. Q. E. D. COR. And if the first has a greater ratio to the second, than the third has to the fourth, but the third the same ratio to the fourth, which the fifth has to the sixth; it may be demonstrated, in like manner, that the first has a greater ratio to the second, than the fifth has to the sixth. See N. a 8. 5. b 13. 5. c 10. 5. d 9. 5. PROP. XIV. THEOR. IF the first has 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 the first A have to the second B, the same ratio which the third C has to the fourth D; if A be greater than C, B is greater than D. Because A is greater than C, and B is any other magnitude, A has to B a greater ratio than C to Ba: but, as A is to B, so is C to D; therefore also C has to D a greater ratio than C has to Bb: but of two magnitudes, that to which the same has the greater ratio is the lesser : wherefore D is less than B; that is, B is greater than D. Secondly, If A be equal to C, B is equal to D: for A is to B, as C, that is, A, to D; B therefore is equal to Dd. Thirdly, If A be less than C, B shall be less than D: for C is greater than A, and because C is to D, as A is to B, D is greater than B, by the first case; wherefore B is less than D. ThereQ. E. D. fore, if the first, &c. PROP. XV. THEOR. Book V. MAGNITUDES have the same ratio to one another which their equimultiples have. Let AB be the same multiple of C that DE is of F: C is to F as AB to DE. A K D there Because AB is the same multiple of C that DE is of F, are as many magnitudes in AB equal to C as there are in DE equal to F: let AB be divided into magnitudes, each equal to C, viz. AG, GH, HB; and DE into magnitudes, each equal to F, viz. DK, KL, LE; then the number of the first AG, GH, HB shall be equal to the number of the last DK, KL, LE: and because AG, GH, HB are H. all equal, and that DK, KL, LE are also equal to one another: therefore AG is to DK as GH to KL, and as HB to LEa: and as one of the antecedents to its consequent, so are all the antecedents together to all the consequents together; wherefore, as AG is to DK so is AB to DE: but b 12. 5, AG is equal to C, and DK to F: therefore, as C is to F so is AB to DĒ. Therefore, magnitudes, &c. Q. E. D, B a 7.5. CE F PROP. XVI. THEOR, IF four magnitudes of the same kind be propor tionals, they shall also be proportionals when taken alternately. Let the four magnitudes A, B, C, D be proportionals, viz. as A to B so C to D; they shall also be proportionals when taken alternately, that is, A is to C as B to D. Take of A and B any equimultiples whatever E and F; and of C and D take any equimultiples whatever G and H and : |