F, and therefore KH is equal to CD (5. Ax. 1): Take away the common magnitude CH from both; then the remainder KC is equal to the remainder HD: But KC is equal to F; therefore HD is equal to F. Next, let GB be a multiple of E: HD shall be the same multiple of F. A K H Make CK the same multiple of F that GB is of E: Then, because AG is the same multiple of E that CH is of F, and GB the same multiple of E that CK is of F, therefore AB is the same multiple of E that KH is of F G (5.2): But AB is the same multiple of E that CD is of F; therefore KH is the same multiple of F that CD is of F, and therefore KH is equal to CD: Take away the common magnitude CH from both; therefore the remainder CK is equal to the remainder HD : And because CK is the same multiple of F that GB is of E, 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. PROP. A. Q. E. D. THEOR. BDEF If the first of four magnitudes have to the second the same ratio which the third has to the fourth, then, if the first be greater than the second, the third is also 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 (5. Def. 5), 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: But if the first be greater than the second, the double of the first is greater than the double of the second; therefore also the double of the third is greater than the double of the fourth, and therefore the third is greater than the fourth: And, in like manner, if the first be equal to the second, or less than it, the third may be proved to be equal to the fourth, or less than it. If four magnitudes are proportionals, they are proportionals also when taken inversely. If A be to B as C is to D, then also, inversely, B shall be to A as D to C. and GABE HC D F Take of B, D, any equimultiples whatever, E, F, of A, C, any equimultiples whatever G, H: Then, first, if E be greater than G, G is less than E: And because A is to B as C is to D, and of A and C, the first and third, G and H are equimultiples, and of B and D, the second and fourth, E and F are equimultiples, and that G is less than E, therefore also H is less than F, that is, F is greater than H: If therefore E be greater than G, F is greater than H; and, in like 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 any whatever of A and C; therefore, B is to A as D to C. Wherefore, If four magnitudes &c. PROP. C. THEOR. 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 is 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, C, any equimultiples whatever E, F, and of B, D, any equimultiples whatever G, H: Then, because A is the same multiple of B that C is of D, and that E is the same multiple of A that F is of C, therefore E is the same multiple of B that F is of D (5. 3), that is, EA B C D and F are equimultiples of B and D But G and H are equimultiples of B and D; 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; EGFH that is, if E be greater than G, F is greater than H: And, in like manner, if E be equal to G, F may be shewn to be equal to H, and if less, less: But E, F are any equimultiples whatever of A, C, and G, H are any equimultiples whatever of B, D; therefore A is to B as C is to D. Next, let A be the same part of B that C is of D: A shall be to B, as C is to D. For 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; and, inversely (5. B), A is to B as C is to D. Wherefore, If the first be the same multiple A B C D &c. Q. E.D. PROP. D. THEOR. If the first be to the second as the third 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. Let A be to B as C is to D, and first, let A be a multiple of B: C shall be the same multiple of D. Take E equal to A, and, whatever multiple A or E is of B, make F the same multiple of D: Then, because A is to B as C is to D, and of B and D have been taken equimultiples, E and F, therefore, A is to E as C to F (5. Cor. 4): But A is equal to E; therefore C is equal to F (5. A): And F is the same multiple of D that A is of B; therefore also C is the same multiple of D that A is of B. Next, let A be a part of B: C shall be the the same part of D. For, because A is to B as C is to D, therefore, inversely (5. B), B is to A as D to C: But A is a part of B; therefore B is a multiple of A, and, 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. PROP. VII. THEOR. Equal magnitudes have the same ratio to the same magnitudes: and the same has the same ratio to equal magnitudes. Let A and B be equal magnitudes, and C any other : A and B shall have each the same ratio to C, and C shall have the same ratio to each of A and B. Take of A and B any equimultiples whatever D and E, and of C any multiple whatever F: Then, because D is the same multiple of A that E is of B, and that A is equal to B, therefore DĎ is equal to E: Hence, if D be greater than B F, E will be greater than F, and if equal, equal, and if less, less: But D and E are any equimultiples whatever of A and B, and F is any multiple of C; therefore, A is to C as B is to C. So also C shall have the same ratio to A that it has to B. For the same construction being made, it may be shewn, as before, that D is equal to E: therefore, if F be greater than D, F is also greater than E, and if equal, equal, and if less, less: But F is any multiple whatever of C, and D and E are any equimultiples whatever of A and B; therefore, C is to A as C is to B. Wherefore, Equal magnitudes &c. Q.E.D. PROP. VIII. THEOR. of unequal magnitudes the greater has a greater ratio to the same than the less has: and the same magnitude has a greater ratio to the less than it has to the greater. Let AB, BC be unequal magnitudes, of which AB is the greater, and let D be any magnitude whatever: AB shall have to D a greater ratio than BC has to D, and D shall have to BC a greater ratio than it has to AB. GB LKH D If the magnitude which is not the greater of the two AC, CB, be not less than D, take EF, FG, E the doubles of AC, CB, (fig. 1): But if FA that which is not the greater of the two AC, CB, be less than D (fig. 2), it can be multiplied, so as to become greater than D: Let it be so multiplied, and let the other be multiplied as often; and let EF, FG be the equimultiples thus taken of AC, CB, so that EF, FG, will be each of them greater than D: And in each case, take H the double of D, K its triple, and so on, till we come to that multiple of D which first becomes greater than FG: Let L be that multiple, and K the multiple of D which is next less than L. Then, because L is the multiple of D, which first be |