than h. In like manner, if e be equal to g; or less, f is equal to h, or less than it. But e f are equimultiples, any whatever, of a, c, and g, h any equimultiples whatever of b, d. Therefore a is to b, as c is to d (v. def. 5). Next, let the first a be the same part of the second b, that the third c is of the fourth d: 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 (v. B.) a is to b as c is to d. Therefore, if the first be the same multiple, &c. Q. E. D. PROPOSITION D.—THEOREM. If the first be to the second as the third to the fourth, 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; and first let a be a multiple of b: c is the same multiple of d. Take e equal to a, and whatever multiple a ore is of b, make f the same multiple of d. Then, because a is to b as c is to d; and of b a b c d the second, and d the fourth, equimultiples have been taken e and f; a is to e, as c to f (v. iv. cor.). But a is equal to e, therefore c is equal to f (v. A): and f is the same multiple of d that a is of b. Wherefore c is the same multiple of d that a is of b. Next, let the first a be a part of the second b; c the third is the same part of the fourth d. Because a is to b as c is to d; then inversely, b is (v. B.) 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 d that a is of b. Therefore, if the first, &c. Q. E. D. PROPOSITION VII.—THEOREM. Equal magnitudes have the same ratio to the same magnitude, and the same has the same ratio to equal magnitudes. 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;d is (v. ax. 1) equal to e. Therefore, if d be greater than f. e is greater than f; and if equal, d a equal; if less, less. And d, e are any equimultiples of e b a, b, and f is any multiple of c. Therefore (v. def. 5), as a is to c, so is b to c. f 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 shewn equal to e. Therefore, if f be greater than d, it is likewise 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 c is to a as c is to b (v. def. 5). Therefore, equal magnitudes, &c. Q. E. D. a PROPOSITION VIII.-THEOREM. 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 a b, bc be unequal magnitudes, of which a b is the greater, and let d be any magnitude whatever: a b has a greater ratio to d than bc to do and d has a greater ratio to bc than unto a b. e If the magnitude which is not the greater of the two ac, cb, be not less than d, take ef, fg, the doubles of a c, cb, as in fig. 1. But if that which is not the greater of the two ac, cb, be less than d (as in figs. 2 and 3) this magnitude can be multiplied, so as to become greater than d, whether it be ac or cb. Let it be multiplied until it become greater than d, and let the other be multiplied as 8 b often; and let ef be the multiple thus taken of a c, and fg the same multiple of ch. Therefore ef and 1 k h dfg 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 1 be that multiple of d which is first greater than fg, and k the multiple of d which is next less than 1. fig. 1. Then, because 1 is the multiple of d which is the 1 og 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 a c that fg is of e cb; fg is the same multiple of cb that eg is of a b (i. 5); wherefore eg, and fg are equimultiples of a bf and cb And it was shewn, that fg was not less than k, and, by the construction, ef is greater than d ; therefore the whole eg is greater f than k and d together. But k, together with d, is equal to 1; therefore eg is greater than l; but b CH fg is not greater than l; and eg, fg, are equimultiples of ab, bc, 1 k £ d and 1 is a multiple of d; therefore g b (v. def. 7) ab has to d a greater 1 k d ratio than bc has to d. Also d has to be a greater ratio than it has to a b. For, having made the same construction, it may be shewn, in like manner, that 1 is greater than fg, but that it is not greater than eg: fig. 2. tiple of d; and fg; eg are equimultiples of cb, a b; therefore d has to cb a greater ratio (v. def. 7) than it has to a b. Wherefore, of unequal magnitudes, &c. Q. E; D. fig. 3. PROPOSITION IX.-THEOREM. 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, b have each of them the same ratio to c; a is equal to b. For, if they are not equal, one of them is 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 multiple of a is d greater than the multiple of c, but the multiple of b is not greater than that of c. Let such multiples be taken, and let d, e be the equimultiples of a, b, and f the multiple of c, so that d may be greater than f, and e not greater than f. But, because a is to c as b is to c, and of a, b are taken equimultiples d, e, and of c is taken a multiple f; and that d is greater than f; b e shall also be greater than f (v. def. 5); but is not greater than f; which is impossible; a therefore and b are not unequal; that is, they are equal. a e Next, let c have the same ratio to each of the magnitudes a and b; a is equal to b. For, if they are not, one of them is greater than the other ; let a be the greater ; therefore as was shewn in Prop. 8th, there is some multiple f of c, and some equimultiples e and d, of b and a such, that f is greater than e, and not greater than d; but because o is to b as c is to a, and that f, the multiple of the first, is greater than e, the multiple of the second ; f, the multiple of the third, is greater than d, the multiple of the fourth (v. def. 5). But f is not greater than d, which is impossible. Therefore a is equal to b. Wherefore, magnitudes, &c. Q. E. D. a a PROPOSITION X.-THEOREM. That magnitude which has a greater ratio than another has unto the same magnitude is the greater of the two. And that magnitude to which the same has a greater ratio than it has unto another magnitude is the lesser of the two. LET have to c a greater ratio than b has to c; a is greater than b: for because a has a greater ratio to c than b has to C, there are (v. def. 7) some equimultiples of a and b, and d some multiple of c, such that the multiple of is greater than the multiple of c, but the multiple of b is not greater than it : let them be taken, and let d, e be equimultiples of a, b, and f a multiple of c, such that d is greater than f, but e is not greater than f: therec cl fore d is greater than e: and because d and é are equimultiples of a and b, and d is greater than e ; therefore a is (v. 4 ax.) greater than b. Next, let c have a greater ratio to b than it has to a ; b is less than a : for (v. def. 7) there is some multiple f of c, and some equimultiples e and d of b and a, such that f is greater than e, but is not greater than d; e therefore is less than d ; and because e and d are equimultiples of b and a, therefore b is (v. 4 ax.) less than a. That magnitude, therefore, &c. Q. E. D. f PROPOSITION XI.—THEOREM. Ratios that are the same to the same ratio are the same to one another. LET a be to b as c is to d; and as c to d so let e be to f; a is to b as e to f. Take of a, ce, any equimultiples whatever g, h, k; and of b, d, f, any equimultiples whatever l, m, n. Therefore, since a is to b, as c to do and h are taken equimultiples of a, c, and l, m, of b, d ; if g be greater than 1, h is greater than m; and if equal, equal ; and if less, less (v. def. 5). Again, because c is to d as e is to f, and h, k are taken equimultiples of c, e; and m, n, of d, f; if h be greater than m, k is greater than n; and if equal, equal; and if less, less : but if g be greater than 1, it has been shewn that his greater than m: and if equal, equal ; and if less, less; therefore if g be greater than 1, k is greater than n; and if equal, h k og b d f 1 m equal ; and if less, less : and g, k are any equimultiples whatever of a, e; and 1, n any whatever of b, f: therefore, as a is to b, so is e to f (v. def. 5). Wherefore, ratios, &c. Q. E. D. PROPOSITION XII.-THEOREM. If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so shall all the antecedents taken together be to all the con sequents. LET any number of magnitudes a, b, c, d, e, f be proportionals; that is, as a is to b, so is c to d, and e to f. As a is to b, so shall a, C, e together be to b, d, f together. Take of a, c, e any equimultiples whatever g, h, k; and of b, d, f any equimultiples whatever l, m, n : then, because a is to b, as c is to do and as e to f; and that g, h, k are equimultiples of a, c, e, and l, m, n, equimultiples of b, d, f; if g be greater than 1, h is greater than m, and k greater than n; and if equal, equal ; and if less, less (v. def. 5). Wherefore, if g be greater than 1, then g, h, k together are greater than l, m, n together"; and if equal, equal ; and if less, less. And g, and g, h, k, to8 h k a b d f 1 gether are any equimultiples of a, and a, c, e together; because if there be any number of magnitudes equimultiples of as many, each of each, whatever multiple one of them is of its part, the same multiple is the whole of the whole (v. 1). For the same reason 1, and l, m, n are any equimultiples of b, and b, d, f: as therefore a is to b, so are a, c, e, together to b, d, f together. Wherefore, if any number, &c. Q. E. D. |