3. Book V. See N. PROP. IX. THEOR. MAGNITUDES which have the fame ratio to the fame magnitude are equal to one another; and thofe to which the fame magnitude has the fame ratio are equal to one another. Let A, B have each of them the fame ratio to C; A is equal to B. for if they are not equal, one of them is greater than the other; let A be the greater; then, by what was fhewn in the preceeding Propofition, there are fome equimultiples of A and B, and fome 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 that of C. Let fuch multiples be taken, and let D, E, be the equimultiples of A, B, and F the multiple of C fo that D may be greater than F, and E not greater than F. but 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; and that D is greater than F; E fhall also be §. Def. 5. greater than Fa; but È is not greater than F, which is impoffible. A therefore and B are not unequal; that is, they are equal. A B D F E Next, Let C have the fame ratio to each of the magnitudes A and B; A is equal to B. for if they are not, one of them is greater than the other; let A be the greater, therefore, as was fhewn in Prop. 8th, there is fome multiple F of C, and fome equimultiples E and D of B and A fuch, that F is greater than E, and not greater than D. but 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 fecond; F the multiple of the third is greater than D the multiple of the fourth. but F is not greater than D, which is impoffible. Therefore A is equal to B. Wherefore magnitudes which, &c. Q. E. D. PROP. PROP. X. THEOR. Book V. THAT magnitude which has a greater ratio than ano- See N. ther has unto the fame magnitude is the greater of the two. and that magnitude to which the fame has a greater ratio than it has unto another magnitude is the leffer of the two. 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 a fome equimultiples of A and B, and fome multiple of Ca. 7. Def. 5. fuch, that the multiple of A is greater than the multiple of C, but the multiple of B is not greater than it. let than B. greater B D b. 4. Ax. 5 E Next, Let C have a greater ratio to B than it has to A; B is lefs than A. for there is fome multiple F of C, and fome equimultiples E and D of B and A fuch, 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 Band A, therefore B is blefs than A. That magnitude therefore, &c. Q. E. D. PROP. XI. THEOR. RATIOS that are the fame to the same ratio, are the fame to one another. Let A be to B, as C is to D; and as C to D, fo 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 fince A is to B, as C to D, and of A, C are taken equimultiples G, H; and Book V. H; and L, M of B, D; if G be greater than L, H is greater than WM; and if equal, equal; and if lefs, lefs. Again, because C is to a. 5. Def. 5. 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 lefs, lefs. but if G be greater than L, it has been fhewn that H is greater than M; and if equal, equal; and if lefs, lefs; therefore if G be greater than L, K is greater than N; and if equal, equal; and if lefs, lefs. and G, K, are any equimultiples whatever of A, E; and L, N any whatever of B, F. Therefore as A is to B, fo is E to F2. Wherefore ratios that, &c. Q. E. D.' PRO P. XII. THE OR. IF any number of magnitudes be proportionals, as one of the antecedents is to its confequent, fo fhall all the antecedents taken together be to all the confequents. Let any number of magnitudes A, B, C, D, E, F, be proportionals; that is, as A is to B, fo C to D, and E to F. as A is to B, fo fhall 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 of bf A, Ĉ, E, and L, M, N equimultiples of B, D, F; if G be greater Book V. than L, H is greater than M, and K greater than N; and if equal, equal; and if lefs lefs. Wherefore if G be greater than L, then G, a. s. Def. §i H, K together are greater than L, M, N together; and if equal, equal; and if less, lefs. 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 fame multiple is the whole of the wholeb. for the fame reason L, and L, M, N are any equimultiples b. 1. §. of B, and B, D, F. as therefore A is to B, fo are A, C, E together to B, D, F together. Wherefore if any number, &c. Q. E. D. PROP. XIII. THEOR. [F the firft has to the fecond the fame ratio which the Set N. If fint has to fecond the fame ratio which greater ratio than the fifth has to the fixth; the first shall also have to the fecond a greater ratio than the fifth has to the fixth. Let A the firft have the fame ratio to B the fecond 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 fixth. alfo the first A shall have to the fecond B a greater ratio than the fifth E to the fixth F. Because C has a greater ratio to D, than E to F, there are some equimultiples of C and E, and fome of D and F fuch, that the mul tiple of C is greater than the multiple of D, but the multiple of E is not greater than the multiple of Fa. let fuch be taken, and of C, E a. 7. Def. §. let G, H be equimultiples, and K, L equimultiples of D, F fo that G be greater than K, bat H not greater than L; and whatever mul tiple G is of C take M the fame multiple of A; and what multiple K is of D, take N the fame multiple of B. then becaufe A is to B, as C to D, and of A and C, M and G are equimultiples, and of Book V. B and D, N and K are equimultiples; if M be greater than N, G is greater than K; and if equal, equal; and if lefs, lefs b; but G is but H is not greater than L; and M, H are equimultiples of A, E; and N, L equimultiples of B, F. Therefore A has a greater ratio to B, than E has to . 7. Def. 5. F. Wherefore if the firft, &c. Q. E. D. b. 5. Def. 5. greater than K, therefore M is greater than N. Ecc N. 1. 8. S. COR. And if the firft has a greater ratio to the second, than the third has to the fourth; but the third the fame ratio to the fourth, which the fifth has to the fixth; it may be demonstrated in like manner that the first has a greater ratio to the second than the fifth has to the fixth. PROP. XIV. THEOR. F the firft has to the fecond the fame ratio, which the If first has to theft, then, if the first be greater than the third, the fecond fhall be greater than the fourth; and if equal, equal; and if lefs, lefs. Let the firft A have to the fecond B the fame 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 B. but as A is to B, fo is C to D; 6. 13. 5. C. 10. 5. A B C D A B C D AB CD therefore alfo C has to D a greater ratio than C has to Bb. but of two magnitudes, that to which the fame has the greater ratio is the leffer. wherefore D is lefs than B; that is, B is greater than D. Secondly, If A be equal to C, B is equal D. for A is to B, as 9. 5. C, that is A, to D; B therefore is equal to Da. Thirdly, If A be lefs 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 cafe; wherefore B is lefs than D. Therefore if the firft, &c. Q. E. D. PROP |