Fig. 3. D: And in every one of the cases take H the double of D, K Book V. its triple, and so on, till the multiple of D be that which first w becomes greater than FG: Let L be that multiple of D which is first greater than FG, and K the multiple of D which is next less than L. Then, because L is the multiple of D which is the first that becomes greater than FG, the next preceeding multiple K is not greater than FG; that is, FG is not lets than K: And fince EF is the same multiple of AC, that FG is of CB; FG is the fame multiple of CB, that EG is of AB; wherefore EG and a 1. s. FG are equimultiples of AB and CB: And it was fhewn, that FG was not less than K, Fig. 2. and, by the construc E tion, EF is greater than D; therefore the whole EG is greater than Kand D together : But K together with Dis equal to L; therefore EG is greater than L; but FG Ft A is not greater than L; and EG, FG are equimultiples of AB, BC, GB and L is a multiple of L K HD D; therefore b AB has GB 6 7. def. s. to Da greater ratio than L KD BC has to D. Allo D has to BC a greater ratio than it has to AB: For, having made the same construction, it may be shewn, in like manner, that L is greater than FG, but that it is not greater than EG: And L is a multiple of D; and FG, EG are equimultiples of CB, AB ; therefore D has to CB a greater ratio than it has to AB. Wherefore, of uneqaal magnitudes, &c. Q. E. D. Book V. PRO P. IX. THE O R. Sec N. AGNITUDES which have the same ratio to the same magnitude are equal to one another; and those to which the same magnitude has the fame ratio are equal to one another. M Let A, B have each of them the same 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 fheon in the preceeding proposition, there are some equimultiples of A and B, and 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 that of C. Let such multiples be taken, and let D, E, be the equimultiples of A, B, and F the multiple of C, so that D may be greater than F, and E nor 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 shall also be greater D a s. def. s. than F'; but E is not greater than F, A which is impossible; A therefore and B F B greater ; therefore, as was fhewn in Prop. E 8th, there is some multiple F of C, and fome equimultiples E and D of B and A such, 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 second; F the multiple of the third is greater than D the multiple of the fourth": But Fin not greater than D, which is impossible. Therefore A is equal to B. Wherefore magnitudes which, &c. Q. E. D. c PROP. Book V. PRO P. X. THE O R. TH HAT magnitude which has a greater ratio than an- See N. other has unto the fame magnitude is the greater of the two : And that magnitude to which the same has ? 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 a 7. Def, s. fome multiple of C fuch, 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 D not greater than F: Therefore D is greater than E: And, because D and E are equi multiples of A and B, and D is greater b 4. AX. S. than E; therefore A is greater than B. F 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 fome e B quimultiples E and D of B and A such E that F is greater than E, but is not greater than D: I therefore is less than D; and because I and D are equimultiples of B and A, therefore B is b less than A. That magnitude, therefore, &c. Q. E. D. al cl Rame fotbae are the same to the same ratio, are the 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, lince A is to B, as C to D, and G, H are taken equimultiples of 14 A, Book T. 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“. Again, bea s. def. 5. 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 iş K E L N PRO P. XII. THE O R. 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; ihat is, as A is to B; 1o 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, Ko E F M N 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 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 Km greater than N; and if equal, equal; and if less, less a. Where-a s. def. s.. 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 magnie tudes 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 6: For the same reason L, and L, M, N are any b I. 5. equimultiples 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. IF F the first has to the second the same ratio which the Sec N. third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the fixth; 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 fixth : Also the first A shall have to the second 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 H C- D F Kthe multiple of E is not greater than the multiple of Fa: Let a 7. def. s. 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 fame multiple of A ; and what multiple K is of D, take N the fame multiple of B; Then, because À is to B, as C to D, and of |