AK + Hyp. First, let GB be equal to E: HD shall be equal to F. Make CK equal to F: and because AG is the same multiple of E +, that CH is of F, and that GB is equal to E, and CK to F, therefore AB is the same multiple of E, that KH is of F: but AB, by the hypothesis, is the same multiple of E, that CD is of F; therefore KH is the same multiple of F, that CD is of F; wherefore KH is equal to CD: take away the common magnitude CH, then the remainder KC is equal to the remainder HD: but KC is equal + + Constr. to F; therefore HD is equal to F. GH BDEF ⚫ 1 Ax. 5. Next, let GB be a multiple of E: HD shall be the same multiple of F. Make CK the same multiple of A K + Hyp. 2.5. GH BDEF + Hyp. F, that GB is of E: and because AG is the same multiple of Et, 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: but AB is the same multiple of Et, that CD is of F; therefore KH is the same multiple of F, that CD is of F; wherefore KH is equal * to CD: take away CH from both; therefore the remainder KC is equal to the remainder HD: and because GB is the same multiple of E +, that KC is of F, and that KC is equal to HD; † Constr. therefore HD is the same multiple of F, that GB is of E. If, therefore, two magnitudes, &c. Q. E. D. PROPOSITION A. 1 Ax. 5. THEOR.-If the first of four magnitudes has the same See N. ratio to the second 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; if less, less. Take any equimultiples of each of them, as the doubles of each: then, by Def. 5th of this Book, 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; wherefore also the double of the third is greater than the double of the fourth; therefore the third is greater than the fourth: in like manner, if the first be equal to the second, or less than it, the third See N. ↑ Hyp. * 5 Def. 5. + Constr. * 5 Def. 5. See N. + Hyp. + Constr. * 3.5. + Constr. can be proved to be equal to the fourth, or less than it. There fore, if the first, &c. Q. E. D. PROPOSITION B. THEOR.-If four magnitudes are proportionals, they are proportionals also when taken inversely. Let A be to B, as C is to D: then also inversely B shall be to A, as D to C. GABE Take of B and D, any equimultiples whatever E and F ; and of A and C, any equimultiples whatever G and H. First, let E be greater than G, then 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 H is less than F; that is, F is greater than H; if, therefore, E be greater than G, F is greater than H: 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 B and D, and G, H any whatever of A and C; therefore † as B is to A, so is D to C. Therefore, if four magnitudes, &c. Q. E. D. * PROPOSITION C. IICDF THEOR.-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. Let the first A be the same multiple of the second B, that the third C is of the fourth D: A shall be to B as C is to D. Take of A and C, any equimultiples whatever ABCD E and F; and of B and D, any equimultiples EGFI whatever G and 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 is of C, therefore E is the same multiple of B* that F is of D; that is, E 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; that is, if E be greater than G, F is greater than H in like manner, if E be equal to G, or less than it, F may be shewn to be equal to H, or less than it: but E, F are equimultiples †, any whatever, of A, C ; and G, † Constr. H any equimultiples whatever of B, D; therefore † A is to B, † 5 Def. 5. as C is to D. Next, let the first A be the same part of the second B, that the third C is of the fourth D: A shall be to B, as C is to D. A B C 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: wherefore, by the preceding case, B is to A, as D is to C; and therefore inversely * A is to B, as C is to D. Therefore, if the first be the same multiple, &c. Q. E. D. PROPOSITION D. * B. 5. THEOR.-If the first be to the second as the third to the See N. fourth, and if the first be a multiple, or a part of the second; the third is 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 the second, and D the fourth, equimultiples have been taken, E and F*; therefore A is to E, as C to F: but A is equal to E, therefore C is equal * to F and F is the same + multiple of D, that A is of B: therefore C is the same multiple of D that A is of B. + Hyp. A B C D * Cor. 4. 5. to Next let A be a part of B: C shall be the same part of D. Because † A is to B, as C is to D; then inversely, B is A, as D to C: but A is a part † of B, that is, B is a multiple of A; 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. Therefore, if the first, &c. Q. E. D. PROPOSITION VII. THEOR.-Equal magnitudes have the same ratio to the same magnitude: and the same has the same ratio to equal magnitudes. + Constr. A. 5. + Constr. See the last figure. † Hyp. • B. 5. † Hyp. + Constr. † Hyp. + Constr. • 5 Def. 5. Let A and B be equal magnitudes, and C any other: A and B shall each of them have the same ratio to C: and C shall have the same ratio to each of the magnitudes 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 D is equal to E: therefore, if D be greater than F, E is greater than F; and if equal, equal; if less, less; but D, E are any equimultiples of A, B†, and F is any multiple of C; therefore*, as A is to C, so is B to C. Likewise C shall have the same ratio to A, that it has to B. For, having made the same construction, D may, in like manner, be shewn to be equal to E: therefore, if F be greater than D, it is like EB F wise greater than E; and if equal, equal; if less, less: but F is any multiple whatever of C, and D, E are any equimulti * Def. 5. ples whatever of A, B; therefore*, C is to A as C is to B. Therefore, equal magnitudes, &c. Q. E. D. See N. - PROPOSITION VIII. THEOR. 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: 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. 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 E Fig. 1. FA c G B 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. E F Fig. 2. Fig. 3. E + Constr. * 1. 5. FA LKH D GB LK D Then, because L is the multiple of D which is the first that becomes greater than FG, 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, therefore FG is the same multiple of CB *, that EG is of AB; that is, EG and FG are equimultiples of AB and CB: and since it was shewn, that FG is not less than K, and, by the construction, EF is greater than D, therefore the whole EG is greater than K and D together but K together with D is equal to L; therefore EG is greater than L: but FG is not greater † than : GB + Constr, + Constr. L: and EG, FG were proved to be equimultiples of AB, BC; and L is at multiple of D; therefore * AB has to D a greater † Constr. ratio, than BC has to D. * 7 Def. 5. Also, D shall have to BC a greater ratio than it has to AB. For having made the same construction, it may be shewn, in like manner, that L is greater than FG, but that it is not greater than EG: and L is a † multiple of D; and FG, EG † Constr. were proved to be equimultiples of CB, AB; therefore D has to CB a greater ratio * than it has to AB. unequal magnitudes, &c. Q. E. D. PROPOSITION IX. Wherefore, of two7 Def. 5. THEOR.-Magnitudes which have the same ratio to the same See N. 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: A shall be equal to B. For, if they are not equal, one of them must be greater than the other; let A be the greater: then, by what was shewn in the preceding proposition, there are some equimultiples of A and B, and some multiple of C, such, that the I |