BOOK V. than L, H is greater than M; and if equal, equal; and if less, lessa. Again, because C is to D, as E is to F, and H, * 5 Def. 5. 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 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 E, 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, Nany whatever of B, F: Therefore, as A is to B, so is E to Fa. Wherefore, ratios that, &c. Q.E.D. G IF PROP. XII. THEOR. Ir 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; A B L 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, equimul- Book V. tiples of B, D, F; if G be greater than L, H is greater than M, and K greater than N; and if equal, equal; and if less, less. Wherefore, if G be greater than L, then G, H, K 5 Def. 5. 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 b: For 1.5. the same reason L, and L, M, N are any 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 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 such be taken, and of C, E, let G, H be equi- a 7 Def. 5. multiples, 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 K Book V. B: Then, because 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; if M be greater than N, G is greater than 5 Def. 5. K; and if equal, equal; and if less, less; 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 of B, F: Therefore A has a greater 7 Def. 5. ratio to B, than E has to Fc. Wherefore, if the first, &c. Q.E.D. COR. And if the first have 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. PROP. XIV. THEOR. See N. 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 mag- a. 5. nitude, 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 A B € D A B C D A B C D 13. 5. ratio than C has to Bb. But of two magnitudes, that to 10. 5. 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 9. 5. to B, as C, that is, A to D: B therefore is eqnal 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 Therefore, if the first, &c. Q.E.D. BOOK V. PROP. XV. THEOR. 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. D Because AB is the same multiple of C, that DE is of F; there are as many magnitudes in AB equal A 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 H number of the last DK, KL, LE: And because AG, GH, HB are all equal, and that DK, KL, LE, are also equal to one KH another; therefore AG is to DK as GH to B C E F KL, and as HB to LEa: And as one of the antecedents to ⚫ 7. 5. its consequent, so are all the antecedents together to all the consequents together; wherefore, as AG is to DK, so is ↳ 12. 5. AB to DE: But AG is equal to C, and DK to F: Therefore, as C is to F, so is AB to DE. Therefore magnitudes, &c. Q.E.D. PROP. XVI. THEOR. IF four magnitudes of the same kind be proportionals, 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: E Book V. and because E is the same multiple of A, that F is of B, and that magnitudes have the same ratio to one another 15.5. which their equimultiples have; therefore A is to B, as E is to F; But as A is to B, so is C to D; Wherefore as A 11. 5. C is to D, sob is E to F: Again, B because G, H are equimultiples of F G C D H C, D, as C is to D, so is G to Ha: but as C is to D, so is E to F. Wherefore, as E is, to F, so 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 14. 5. fourth; and if equal, equal; if less, less. Wherefore, if E be greater than G, Flikewise is greater than H: and if equal, equal; if less, less: And E, F are any equimultiples whatever of A, B; and G, H any whatever of C, D. Thered 5 Def. 5. fore A is to C as B to Dd. If then four magnitudes, &c. Q. E.D. PROP. XVII. THEOR. See N. IF magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately; that is, if two inagnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two has to the other of these. 7 Let AB, BE, CD, DF be the magnitudes taken jointly which are proportionals; that is, as AB to BE, so is CD to DF: they shall also be proportionals taken separately, viz. as AE to EB, so 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 because GH is the same, multiple of AE, that HK is of EB, wherefore GH is the 1. 5. same multiplea of AE, that GK is of AB: But GH is the same multiple of AE, that LM is of CF; wherefore GK is. |