Book V. that EB is of FD: But AE is the same multiple of CF that AB is of CD; therefore EB is the same multiple of FD, that AB is of CD. Therefore, " if one magnitude," &c. Q. E. D. See N. PROP. VI. THEOR. If two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two, the remainders will 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, equimultiples of E, F, be taken from AB, CD; the remainders GB, HD will be either equal to E, F, or equimultiples of them. First, Let GB be equal to E, then will HD be equal to F: Make CK equal to A F; and because AG is the same multiple of E, that CH is of F, and 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; a 1. Ax. 5. wherefore, KH is equal to CD: Take b 2. 5. G K C H B DE F away the common magnitude CH, then the remainder KC is equal to the remainder HD: but KC is equal to F, HD therefore is equal to F. A C K Next, Let GB be a multiple of E; then will HD be the same multiple of F: Make ČK the same multiple of F, that GB is of E: And 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; but AB is the same multiple of E, that CD is of F; therefore, KH is G the same multiple of F that CD is of it; wherefore, KH is equal to CDa. 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 KC is equal to HD; therefore, HD is the same multiple of F, that GB is of Ed: " therefore, two magnitudes," &c. Q. E. D. B H DEF "If, PROP. A. THEOR. Book V. If the first of four magnitudes has to the second the See N. 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; if less, less. Take any equimultiples of each of them, as the doubles of each; then, by def. 5. 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 can be proved to be equal to the fourth, or less than it. Therefore, "if the first," &c. Q. E. D. PROP. B. THEOR. If four magnitudes are proportionals, they are See N. proportionals also when taken inversely. If the magnitude A be to B, as C is to D, then also inversely B is to A, as D to C. E 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 equimulti- GAB ples; and because G is less than E, H is also H C D F less than F; that is, F is greater than H; if, therefore, E be greater that G, F is greater than H: In like manner, if E be equal to G, F may be shown to be equal to H: and if less, less; and E, F, are any equimultiples whatever of B and D, and G, H any whatever of A and C; therefore, as B a 5. Def. 5. Book V. is to A, so is D to C. "If, then, four magnitudes," &c. Q. E. D. See N. b Const. c 3. 5. PROP. C. THEOR. If the first of four magnitudes 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 B the second, that C the third is of the fourth D; then A is to B as C is to D. b a For, of A and C take any equimultiples whatever E and F; and of B and D any equia Hyp. multiples whatever G and H: Then, because " A is the same multiple of B that C is of D; and E is the same multiple of A, that F is A B C D of C; E is also the same multiple of B, that E G F H F is of Dc; therefore E and F are the same multiples 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, 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 G, F is equal to H, or less than H. But E, Fare any equimultiples whatever of A, C, and G, H any equimultiples whatever of B, D. d 5. def. 5. Therefore, A is to B as C is to Da. e B. 5. Next, Let the first A be the same part of the second B, that the third C is of the fourth D; then A is to B as C is to D: For 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 inversely A is to B as C is to D. Therefore, "if the first be the same multiple," &c. Q. E. D. A B C D e Book V. PROP. D. THEOR. If the first be to the second as the third to the fourth, See N. and if the first be a multiple, or 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: First, Let A be a multiple of B; then will C 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, the equimultiples E and F have been taken; A is to E, as C to Fa: But A is equal to E, therefore C is equal to Fb: And F is the same multiple of D, that A is of B. Wherefore, C is the same multiple of D, A B that A is of B. Next, Let the first A be a part of the second B; then will C the third be the same part of the fourth D. Because A is to B, as C is to D; then, inversely, B is to A, as D to C: But A is a part of B, therefore B is a multiple of A; and, by the preceding case, D is the same multiple of C; that is, C is the same part of E C D F D, that A is of B: Therefore, "if the first," &c. Q. E. D. a cor. 4. 5. b A. 5. See the figure at the foot of the preced ing page. c B. 5. PROP. VII. THEOR. Equal magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes. Let A and B be equal magnitudes, and C any other magnitude. A and B have each of them the same ratio to C; and C has the same ratio to each of the magnitudes A and B. Take of A and B any equimultiples whatever D and E, and Book V. of C any multiple whatever F: Then, because D is the same multiple of A, that E is of B, and A is equal a 1. Ax. 5. to B; D is also equal to E: Therefore, if D be greater than F, E is also greater than F; and if equal, equal; if less, less: And D, E are any equimultiples whatever of A, B, and F is any multiple whatever of C. b 5. def. 5. Therefore, as A is to C, so is B to C. See N. Likewise, C has the same ratio to A, that it has to B: For, having made the same construction, D may, in like manner, be shown to be equal to E: Therefore, if F be greater than D, it is also greater than E; and if equal, equal; if less, less: And F is any multiple whatever of C, and D, E are any equimultiples whatever of A, B. Therefore Cis to A, as C is to Bb. Therefore, "equal "magnitudes," &c. Q. E. D. PROP. VIII. THEOR. DA E B C F 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; then AB has a greater ratio to D than BC to D: And D has a greater ratio E to BC than to AB. Fig. 1. A G B If the magnitude, which is not the |