1 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 lefs, lefs a. Again, be5. Def. 5. caufe 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 lefs, lefs: But, if G be greater than L, it has been thewn 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 Fa. Wherefore, ratios that, &c. QE. D. PROP. XII. THEOR. 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 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; ff G be greater than L, H is greater than M, and K greater than N; and if equal, equal; and if lefs, lefs a. Where- a 5. def. 5. fore, ifG be greater than L, then G, H, K together are greater than L, M, N together; and if equal, equal; and if lefs, less. AndG, 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 whole b: For the fame reafon L, and L, M, N are any equi- b 1.5, multiples 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. IF the first has to the second the fame ratio which the set 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 alfo have to the fecond a greater ratio than the fifth has to the fixth. Let A the first, 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: Also the first A fhall 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 fone equimultiples of C and E, and fome of D and F fuch, 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. fuch be taken, and of C, E let G, H be equimultiples, and K, L equimultiples of D, F, fo 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 whatever multiple K is of D, take N the fame multiple of B: Then, becaufe A is to B, as C to D, Book V. D, and of A and C, M and G are equimultiples: And of B and D, N and K are equimuiti,les; if M be greater than N, G is b 5. def. 5. greater than K; and if equal. equal; and if lefs, lef: b; but G is greater than K, therefore M is greater thau N: 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 7. def. 5. B, than E has to Fc. Wherefore, if the firit, &c. Q. E. D. See N. 28.5. COR. And if the first have 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 demonftrated, in like manner, that the firft has a greater ratio to the fe cond, than the fifth has to the fixth. PROP. XIV. THE OR. F the first has to the fecond, the fame ratio which the third has to the fourth; then, if the firft be greater than the third, the fecond fhall be greater than the fourth; and if equal, equal; and if lefs, less. Let the first 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 a: But, as A is to B, fo b 13.5. € 10. 5. d 9.5. A B C D A B C D A B C D is C to D; 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 c: Wherefore D is lefs 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 lefs than C, B fhall be lefs 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 less than D. Therefore, if the firft, &c. Q. E. D. PROP. PROP. XV. THE OR. Book V. M AGNITUDES have the fame ratio to one another Let AB be the fame multiple of C, that DE is ofF: Cis to F, as AB to DE. A D K Because AB is the fame multiple of C, that DE is of F; there 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: G Then the number of the first AG, GH, HB, fhall be equal to the number of the last DK, KL, LE: And becaufe 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 LE a: And as one of the antecedents to its confequent, so are all the antecedents together to all the confequents together b; wherefore, as AG is to DK, fo is AB to DE: But b 12. 5, AG is equal to C, and DK to F: Therefore, as C is to F, fo is AB to DE. Therefore magnitudes, &c. Q. E. D. L BCEF a 7.59 PROP. XVI. THE OR, IF four magnitudes of the fame kind be proportionals they fhail alfo be proportionals when taken alternately. Let the four magnitudes A, B, C, D be proportionals, viz. as A to B, fo 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 beçaufe Book V. because E is the fame multiple of A, that F is of B, and that magnitudes have the fame ratio to one another which their equimultiples have a; therefore A is to B, as E is to F: But as A is to B, fo is C to a 15.5. b II. 5. € 14. 5. D: Wherefore as CE is to D, fob is E to A ·G· C D H D, fo is E to F. Wherefore, as E is to F, fo is G to Hb. But, when four magnitudes are proportionals, if the first be greater than the third, the second shall be greater than the fourth; and if equal, equal; if lefs, lefs c. Wherefore, if E be greater than G, F likewife is greater than H; and if equal, equal; if lefs, lefs: And E, F are any equimultiples whatever of A, B; and G, H any whatever of C, D. Therefore A is to C, as B to d 5.def. 5. D d. If then four magnitudes, &c. Q. E. D. ་ See N. a I. 5. PROP. XVII. THEOR. F magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately; that is, if two magnitudes together have to one of them the fame ratio which two others have to one of thefe, the remaining one of the first two shall have to the other the fame ratio which the remaining one of the last two has to the other of these. Let AB, BE, CD, DF be the magnitudes taken jointly which are proportionals; that is, as AB to BE, fo is CD to DE; they fhall also be proportionals taken feparately, viz. as AE to EB, fo CF to FD. Take of AE, EB, CF, FD any equimultiples whatever GH, HK, LM, MN; and again, of EB, FD, take any equimultiples whatever KX, NP: And becaufe GH is the fame multiple of AE, that HK is of EB, wherefore GH is the fame multiple a of AE, that GK is of AB: But GH is the fame multiple of AE, that LM is of CF; wherefore GK is the fame multiple of AB, that |