PROPOSITION 8.-Theorem. Of two unequal magnitudes, the greater has a greater ratio to any other magnitude than the less has; and the same magnitude has a greater ratio to the less of two other magnitudes, than it has to the greater. Let AB, BC be two unequal magnitudes, of which AB is the greater, and let D be any other magnitude. E Then AB shall have a greater ratio to D than BC has to D; and D shall have a greater ratio to BC than it has to AB. Fig. 1. Fig. 2. E F Fig. 3. A G B L K D Construction. If the magnitude which is not the greater of the two AC, CB, be not less than D, take EF, FG, the doubles of AC, CB (as in fig. 1); but if that which is not the greater of the two AC, CB, be less than D (as in fig. 2 and 3), this magnitude can be multiplied, so as to become greater than D, whether it be AC, or CB. Let it be multiplied until it become greater than D, and let the other be multiplied as often; and let EF be the multiple thus taken of AC, and FG the same multiple of CB; therefore EF and FG are each of them greater than D; and in every one of the cases, take H the double of D, K its triple, and so on, till the multiple of D be that which first 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. Demonstration. Because L is the multiple of D, which is the first that becomes greater than FG, 1. The next preceding multiple K is not greater than FG; that is, FG is not less than K; and since EF is the same multiple of AC, that FG is of CB (constr.); therefore FG is the same multiple of CB, that EG is of AB (V. 1); that is, 2. EG and FG are equimultiples of AB and CB; and since it was shown, that FG is not less than K, and, by the construction, EF is greater than D; therefore 3. The whole EG is greater than K and D together; but K together with D is equal to L (constr.); therefore 4. EG is greater than L; but FG is not greater than L (constr.); and EG, FG were proved to be equimultiples of AB, BC; and L is a multiple of D (constr.); therefore 5. AB has to D a greater ratio than BC has to D (V. Def. 7). Also D shall have to BC a greater ratio than it has to AB. For having made the same construction, it may be shown in like manner that, 1. L is greater than FG, but is not greater than EG; and L is a multiple of D (constr.); and FG, EG were proved to be equimultiples of CB, AB; therefore 2. D has to CB a greater ratio than it has to AB (V. Def. 7). Wherefore, of two unequal magnitudes, &c. Q.E.D. PROPOSITION 9.-Theorem. Magnitudes which have the same ratio to the same magnitude are equal to one another; and those to which the same magnitude has the same ratio are equal to one another. Let A, B have each of them the same ratio to C. C Then A shall be equal to B. Construction. For, if they are not equal, one of them must be greater than the other; let A be the greater; then, by what was shown in the preceding 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 these multiples be taken; and let D, E be the equimultiples of A, B, and F the multiple of C, such that D may be greater than F, but E not greater than F. Α. D F. B E Demonstration. Because A is to C as B is to C (hyp.); and of A, B, are taken equimultiples D, E, and of C is taken a multiple F; and that D is greater than F; therefore 1. E is also greater than F (V. Def. 5); but E is not greater than F (constr.), which is impossible; therefore 2. A and B are not unequal; that is, they are equal. Next. Let C have the same ratio to each of the magnitudes A and B. Then A shall be equal to B. Construction. For, if they are not equal, one of them must be greater than the other; let A be the greater; therefore, as was shown in Prop. 8, 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 not greater than D; let these multiples be taken. Demonstration. Because C is to B as C is to A (hyp.), and that F the multiple of the first, is greater than E the multiple of the second, therefore 1. F the multiple of the third, is greater than D the multiple of the fourth (V. Def. 5); but F is not greater than D (hyp.), which is impossible; therefore 2. A is equal to B. Wherefore, magnitudes which, &c. Q.E.D. PROPOSITION 10.-Theorem. That magnitude which has a greater ratio than 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 less of the two. M Let A have to C a greater ratio than B has to C. A D с B E Construction. For, because A has a greater ratio to C, than B has to C, there are some equimultiples of A and B, and some multiple of C such (V. Def. 7), 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 the equimultiples of A, B, and F the multiple of C; such, that D is greater than F, but E is not greater than F; therefore D is greater than E. F Demonstration. Because D and E are equimultiples of A and B, and that D is greater than E; therefore 1. A is greater than B (V. Ax. 4). Next. Let C have a greater ratio to B than it has to A. Then B shall be less than A. Construction. For there is some multiple F of C (V. Def. 7), and some equimultiples E and D of B and A such, that F is greater than E, but not greater than D; let them be taken; therefore E is less than D. Demonstration. Because E and D are equimultiples of B and A, and that E is less than D, therefore 1. B is less than A (V. Ax. 4). Therefore, that magnitude, &c. Q.E.D. K PROPOSITION 11.—Theorem. Ratios that are the same to the same ratio, are the same to one another. G H A B D L M Construction. Take of A, C, E, any equimultiples whatever G, H, K; and of B, D, F, any equímultiples whatever L, M, N. E F N Demonstration. Because A is to B as C to D, and G, H are taken equimultiples of A, C, and L, M of B, D ; 1. If G be greater than L, H is greater than M; and if equal, equal; and if less, less (V. Def. 5). Again, because C is to D as E is to F, and H, K are taken equimultiples of C, E, and M, N of D, F; 2. 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 B 3. 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 4. As A is to B, so is E to F (V. Def. 5). Wherefore, ratios that, &c. Q.E.D. PROPOSITION 12.-Theorem. 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. Then as A is to B, so shall A, C, E together, be to B, D, F together. G H K A с E D M F N Construction. Take of A, C, E any equimultiples whatever G, H, K; and of B, D, F, any equímultiples whatever, L, M, N. Demonstration. 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 B, D, F; therefore, if G be greater than L, His greater than M, and K greater than N; and if equal, equal; and if less, less (V. Def. 5); wherefore 1. 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; |