PROPOSITION VIII. (Eucl. v. 9.) 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 and B have the same ratio to C. A would have a greater ratio to C than B has to C; which is not the case. And if A were less than B, V. 7. B would have a greater ratio to C than A has to C; V. 7. which is not the case. A = B. Next, let C have the same ratio to A that C has to B. Then must A = B. For we can show, as before, that A cannot be greater or less than B. .. A = B. Q. E. D. PROPOSITION IX. (Eucl. v. 10.) That magnitude, which has a greater ratio than another has to the same magnitude, is the greater of the two; and that magnitude, to which the same has a greater ratio than it has to another magnitude, is the less of the two. Let A have to C a greater ratio than B has to C. For if A were equal to B, then would ▲ have the same ratio to C that B has to C; which is not the case. V. 8. And if A were less than B, then would A have to C a ratio less than that which B has to C; which is not the case. V. 7. .. A is greater than B. Next, let C have a greater ratio to B than it has to A. Then must B be less than A. For if B were equal to A, then would C have the same ratio to B which it has to A; which is not the case. V. 8. B a And if B were greater than A, then C would have to ratio less than that which C has to A; which is not the case. . B is less than A. V. 7. Q. E. D. PROPOSITION X. (Eucl. v. 12.) If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so must 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, A to B as C to D and as E is to F... Then must A be to B as A, C, E... together is to B, D, F...together. Take of A, C, E,...any equimultiples mA, mC, mE... and of B, D, F...any equimultiples nB, nD, nF... Then A is to B as C is to D and as E is to F... .. if mA be greater than nB, mC is greater than nD, and mE is greater than nF...; and if equal, equal; if less, less. V. 4. ..if mA be greater than nB, mA, mC, mE...together are greater than nB, nD, nF...together; and if equal, equal; if less, less. Now mA and mA, mC, mE...together are equimultiples of A and A, C, E...together. V. 1. And nB and nB, nD, nF...together are equimultiples of B and B, D, F...together. .. A is to B as A, C, E...together is to B, D, F...together. PROPOSITION XI. (Eucl. v. 15.) V. Def. 5. Q. E. D. Magnitudes have the same ratio to one another which their equimultiples have. Let A be the same multiple of C that B is of D. Then must C be to D as A to B. Divide A into magnitudes E, F, G,...each equal to C, and B into magnitudes H, K, L,...each equal to D, the number of the magnitudes being the same in both cases, because A and B are equimultiples of C and D. Then ... E, F, G.........are all equal, and H, K, L... .are all equal. .. E is to H, as F to K, as G to L... .. E is to together, V. 6 Has E, F, G...together is to H, K, L... that is, E is to H as A to B ; and. EC, and H : .. C is to D as A to B. D, V. 10 Q. E. D. SECTION IV. On Proportion by Inversion, Alternation, and Separation. PROPOSITION XII. (Eucl. v. B.) If four magnitudes be proportionals, they must also be proportionals when taken inversely. Let A be to B as C is to D. Then inversely B must be to A as D is to C. Take of A and C any equimultiples mA and mC, Then A is to B as C is to D, .. if mA be greater than nB, mC is greater than nD; and f equal, equal; if less, less. V. 4. Hence, if nB be greater than mA, nD is greater than mC; and if equal, equal; if less, less. .. B is to A as D is to C. V. Def. 5. E. D. PROPOSITION XIII. (Eucl. v. 13.) If the first has to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth; the first must also have to the second a greater ratio than the fifth has to the sixth. Let A have to B the same ratio that C has to D, but C to D a greater ratio than E has to F. Then must A have to B a greater ratio than E has to F. For C has to D a greater ratio than E has to F, we can find such equimultiples of C and E, suppose mCand mE, and such equimultiples of D and F, suppose nD and nF, that mC is greater than nD, but mE not greater than nF. Then A is to B as C is to D, And mE is not greater than nF. A has to B a greater ratio than E has to F. V. Def. 7. Hyp. V. 4. V. Def. 7. Q. E. D. PROPOSITION XIV. (Eucl. v. 14.) 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 must be greater than the fourth; and if equal, equal; and if less, less. Let A have the same ratio to B that C has to D. Then if A be greater than C, B must be greater than D. and B is any other magnitude, .. A has a greater ratio to B than C has to B. V. 7. But A is to B as C is to D. V. 13. V. 9. .. C has a greater ratio to D, than C has to B. .. B is greater than D. Similarly it may be shown that if A be less than C, B must be less than D; and that if A be equal to C, B must be equal to D. Q. E. D. PROPOSITION XV. (Eucl. v. 16.) If four magnitudes of the same kind be proportionals, they must also be proportionals when taken alternately. Let A, B, C, D be four magnitudes of the same kind, and let A be to B as C is to D. Then alternately A must be to C as B is to D. Take of A and B any equimultiples mA and mB, and of C and D any equimultiples nC and nD. Then mA is to mB as A is to B, A is to B, V. 11. ..m4 is to mB as C is to D. V. 5. But nC is to nD as C is to D; and .. mA is to mB as nC is to nD. If.. mA be greater than nC, mB is greater than nD; V. 11. V. 5. and if equal, equal; if less, less. .. A is to C as B is to D. V. 14. V. Def. 5. QE D. |