From each of these take the common magnitude AE; then the remainder AG is equal to the remainder EB. Then, because AE is the same multiple of CF that AG is of FD, and that AG is equal to EB; [Construction. therefore AE is the same multiple of CF that EB is of FD. [Hypothesis. But AE is the same multiple of CF that Wherefore, if one magnitude &c. Q.E.D. E B If two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two, the remainders shall be either equal to these others, or equimultiples of them. Let the two magnitudes AB, CD be equimultiples of the two E, F; and let AG, CH, taken from the first two, be equimultiples of the same É, F: the remainders GB, HD shall be either equal to E, F, or equimultiples of them. First, let GB be equal to E: HD shall be equal to F. Make CK equal to F. Then, because AG is the same mul tiple of E that CH is of F, [Hyp. and that GB is equal to E, and CK is equal to F; therefore AB is the same multiple of E that KH is of F. But AB is the same multiple of E that CD is of F; [Hypothesis. therefore KH is the same multiple of F that CD is of F; therefore KH is equal to CD. [V. Axiom 1. From each of these take the common magnitude CH; then the remainder CK is equal to the remainder HD. But CK is equal to F; therefore HD is equal to F. [Construction. Next let GB be a multiple of E: HD shall be the same multiple of F. Make CK the same multiple of F that GB is of E. Then, because AG is the same F BDE But AB is the same multiple of E that CD is of F ; [Hyp. therefore KH is the same multiple of F that CD is of F; therefore KH is equal to CD. [V. Axiom 1. From each of these take the common magnitude CH; then the remainder CK is equal to the remainder HD. And because CK is the same multiple of F that GB is of E, [Construction. and that CK is equal to HD; therefore HD is the same multiple of F that GB is of E. Wherefore, if two magnitudes &c. Q.E.D. PROPOSITION A. THEOREM. If the first of four magnitudes have the same ratio to the second that the third has to the fourth, then, if the first be greater than the second, the third shall also be greater than the fourth, and if equal equal, and if less less. Take any equimultiples of each of them, as the doubles of each. Then if the double of the first be greater than the double of the second, the double of the third is greater than the double of the fourth. [V. Definition 5. But if the first be greater than the second, the double of the first is greater than the double of the second; therefore the double of the third is greater than the double of the fourth, and therefore the third is greater than the fourth. In the same manner, if the first be equal to the second, or less than it, the third may be shewn to be equal to the fourth, or less than it. Wherefore, if the first &c. Q.E.D. PROPOSITION B. THEOREM. If four magnitudes be proportionals, they shall also be proportionals when taken inversely. Let A be to B as C is to D: then also, inversely, B shall be to A as D is to C. Take of B and D any equimul tiples whatever E and F; and of A and C any equimultiples First, let E be greater than G, then Then, because A is to B as C is and of B and D the second and therefore H is less than F; [V. Def. 5. that is, F is greater than H. GA -BA H CD F Therefore, if E be greater than G, F is greater than H. In the same manner, if E be equal to G, F may be shewn to be equal to H; and if less, less. But E and F are any equimultiples whatever of B and D, and G and H are any equimultiples whatever of A and C''; therefore B is to A as D is to C. [Construction. [V. Definition 5. Wherefore, if four magnitudes &c. Q.E.D. If the first be the same multiple of the second, or the same part of it, that the third is of the fourth, the first shall be to the second as the third is to the fourth. First, let A be the same multiple of B that C is of D: A shall be to B as C is to D. Take of A and C any equimultiples whatever E and F; and of B and D any equimultiples whatever G and H. Then, because A is the same multiple of B that C is of D; [Construction. [Hypothesis. and that E is the same multiple of A that Fis of C; therefore E is the same multiple of B that Fis of D; [V. 3. that is, Eand Fare equimultiplesof Band D. But G and H are equimultiples of B and D; [Construction. therefore if E be a greater multiple of B than G is of B, F is a greater multiple of D than H is of D; that is, if E be greater than G, F is greater than H. In the same manner, if E be equal to G, F may be shewn to be equal to H; and if less, less. But E and F are any equimultiples A B C D EGF H whatever of A and C, and G and H are any equimultiples whatever of B and D; therefore A is to B as C is to D. [Construction. [V. Definition 5. Next, let A be the same part of B that C is of D: A shall be to B as Cis to D. For, since A is the same part of B that C is of D, therefore B is the same multiple of A that D is of C; therefore, by the preceding case, B is to A as D is to C; therefore, inversely, A is to B as C is to D. Wherefore, if the first &c. Q.E.D. A B C D [V. B. If the first be to the second as the third is to the fourth, and if the first be a multiple, or a part, of the second, the third shall be the same multiple, or the same part, of the fourth. therefore C is the same multiple of D that A is of B. Q Next, let A be a part of B : C shall be the same part of D. For, because A is to B as C is to D; therefore, inversely, B is to A as D is to C. But A is a part of B; that is, B is a multiple of A; [Hypothesis. [V.B. [Hypothesis. therefore, by the preceding case, D is the same multiple of C; that is, C is the same part of D that A is of B. Wherefore, if the first &c. Q.E.D. PROPOSITION 7. THEOREM. Equal magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes. |