PROPOSITION XXV.—THEOREM. If four magnituules of the same kind are proportionals, the greatest and least of them together are greater than the other two together. LET the four magnitudes a b c d, e, f be proportionals, viz. ab to cd, as e to f; and let a b be the greatest of them, and consequently f the least (v. 14 and A.), a b, together with f, are greater than cd, together with e. Take a g. equal to e, and ch equal to f: then, because as a b is to cd, so is e to f; and that ag is equal to e, and ch equal to f, a b is to cd, as a g to g d ch And because a b the whole, is to the whole cd, as ag is to ch, likewise the remainder b shall be to the remainder hd, as the whole a b is to the whole cd (v. 19): but a b is greater than cd, therefore (v. A.) gb is greater than hd: and because ag is equal to e, and ch to f; ag and f together are equal to ch and e together. If therefore to the unequal a e f magnitudes gb, hd, of which gb is the greater, there be added equal magnitudes, viz. to gb the two a g and f, and ch and e to hd; ab and f together are greater than cd and e. Therefore, if four magnitudes, &c. Q. E. D. PROPOSITION F.—THEOREM. Ratios which are compounded of the same ratios are the same with one another. LET a be to b as d to e; and b to c as e to f: the ratio which is com pounded of the ratios of a to b, and b to c, which by the dea b c finition of compound ratio, is the ratio of a to c, is the same d e f with the ratio of d to f, which, by the same definition, is com pounded of the ratios of d to e, and e to f. Because there are three magnitudes a, b, c, and three others d, e, f, which, taken two and two in order, have the same ratio : ex cquali a is to c, as d to f (v. 22). Next, let a be to b, as e to f, and b to c, as d to e; therefore, exc æquali in proportione perturbata (v. 23), a is to c, as d to f; that a b c is, the ratio of a to c, which is compounded of the ratios of a to de f b, and b to c, is the same with the ratio of d to f, which is com pounded of the ratios of d to e, and e to f. And in like manner the proposition may be demonstrated, whatever be the number of ratios in either case. PROPOSITION G.-THEOREM. If several ratios be the same with several ratios, each to each, the ratio which is compounded of ratios which are the same with the first ratios, each to each, is the same with the ratio compounded of ratios which are the same with the other ratios, each to each. LET a be to b as e to f; and c to d as go to h: and let a be to b as k to l; and to d as 1 to m; then the ratio of k tom, by the definition of compound ratio, is compounded of the ratios of k to 1, and 1 to m, which are the same with the a b C. d. k l m ratios of a to b, and c to d ; and as e to f, so let 0; 8 p; n 0. P the ratio of n p is compounded of the ratios p. which are the same with the ratios of e to f, and g to h : and it is to be shewn that the ratio of k to m. is the same with the ratio of n to p, or that k is to mas n to pa Because k is to l as (a to b, that is, as e to f, that is, as) n to o; and as 1 to m, so is (c to d, and so is g to h, and so is) o to p : ex æquali (v. 22) k is to m as n top. Therefore, if several ratios, &c. Q. E. D. n be to PROPOSITION H.-THEOREM, If a ratio compounded of several ratios be the same with a ratio compounded of any other ratios, and if one of the first ratios, or a ratio. compounded of any of the first, be the same with one of the last ratios; or with the ratio compounded of any of the last ; then the ratio compounded of the remaining ratios of the first, or the remaining ratio of the first, if but one remain, is the same with the ratio compounded of those remaining of the last, or with the remaining ratio of the last. LET the first ratios be those of a to b, b to c, c to d, d to e, and e to f; and let the other ratios be those of g to h, h to k, k to 1, and l to m; also, let the ratio of a to f, which is compounded of (def. of compounded ratio) the first ratios, be the same with the ratio of g to m, which is compounded of the other a b c d e f ratios ; and besides, let the ratio of a to d, which is compounded of the ratios of a to b, b to c, c g. h k 1 m to d, be the same with the ratio of g to k, which is compounded of the ratios of g to h, and h to k: then the ratio compounded of the remaining first ratios, to wit, of the ratios of d to e, and e to f, which compounded ratio is the ratio of d to f, is the same with the ratio of k to m, which is compounded of the remaining ratios of k to 1, and 1 to m of the other ratios. Because, by the hypothesis, a is to d, as.g to k, by inversion (v. B.), d is to a, as k to g; and as a is to f, so is. g therefore (v. 22), ex æquali, d is to f, as k tom. If therefore a ratio which is, &c. Q. E, D. to m; I PROPOSITION K.-THEOREM. If there be any number of ratios, and any number of other ratios, such that the ratio compounded of ratios which are the same with the first ratios, each to each, is the same with the ratio compounded of ratios which are the same, each to each, with the last ratios ; and if one of the first ratios, or the ratio which is compounded of ratios which are the same with several of the first ratios, each to each, be the same with one of the last ratios, or with the ratio compounded of ratios which are the same, each to each, with several of the last ratios ; then the ratio compounded of ratios which are the same with the remaining ratios of the first, each to each, or the remaining ratio of the first, if but one remain, is the same with the ratio compounded of ratios which are the same with those remaining of the last, each to each, or with the remaining ratio of the last. LET the ratios of a to b, c to d, e to f, be the first ratios : and the ratios of g to h, k to 1, m to n, o to p, q, to r, be the other ratios. And let a be to b, as s to t; and c to d'ast to y, and e to f, as v to X. Therefore, by the definition of compound ratio, the ratio of g to x is com s, t, v, x h, k, l. m, n, o, p. y, z, a, b, c, d. y to p; and pounded of the ratios of s to t, t to y, and y to x, which are the same with the ratios of a to b, c to d, e to f, each to each. Also, as g to h, so let у be to z; and k to 1 as z to a, m to n as a to b, o to p as b to c; and q to r as c to d. Therefore by the same definition, the ratio of d is compounded of the ratios of y to z, z to a, a to b, b to c, and c to d, which are the same, each to each, with the ratios of g to h, k to 1, m to n, o to a to r. Therefore, by the hypothesis, s is to x as y to d. Also, let the ratio of a to b, that is, the ratio of s to t, which is one of the first ratios, be the same with the ratio of e to g, which is compounded of the ratios of e to f, and f to g, which, by the hypothesis, are the same with the ratios of g to h and k to 1, two of the other ratios; and let the ratio of to l be that which is compounded of the ratios of h to k, and k to l, which are the same with the remaining first ratios, viz. of c to d, and e to f; also, let the ratio of m to p be that which is compounded of the ratios of m to n, n to o, and o to p, which are the same, each to each, with the remaining other ratios, viz. of m to n, o to p, and 9 Then the ratio of h to 1 is the same with the ratio of m to p, or h is to l as m to p. h, k, 1. s, t, v, x y, z, a, b, c, d. e, f, g. m, n, o, p. to r. And by to di Because e is to f as (g to h, that is, as) y to z; and f is to g as (k to 1, that is, as) z to a: therefore, ex æquali, e is to g as y to a. the hypothesis, a is to b, that is, s is to t, as e to g; wherefore s is to t, as y is to a; and, by inversion, t is to s, as a to y; and s is to x, as у therefore, ex æquali, t is to x, as a to d. Also, because h is to k as (c to d, that is, as) t to v; and k is to l, as (e to f, that is, as) y to x; therefore, ex æquali, h is to l, as t to x. In like manner, it may be demonstrated that m is to p, as a to d: and it has been shewn that t is to x as a to d; therefore (v. 11) h is to l, as m to p. Q. E. D. The propositions G and K are usually, for the sake of brevity, expressed in the same terms with propositions F and H : and therefore it was proper to shew the true meaning of them when they are so expressed; especially since they are very frequently made use of by geometers. EXERCISES ON BOOK V. THEOREMS. 1. If three magnitudes have the same ratio to each other, the sum of the first and third is greater than twice the second. 2. The differences of magnitudes that are continual proportionals are proportionals also. 3. If there be four proportional magnitudes of which the first is a multiple of the second, then the third is the same multiple of the fourth. (Prove also that if the second be a multiple of the first, the fourth is the same multiple of the third.) 4. If the ratio between the first and second of four magnitudes to the second be greater than the ratio between the third and fourth to the fourth, then the first has a greater ratio to the second than the third has to the fourth. 5. If there be four proportional magnitudes, the ratio between the third and fourth is equal to, greater, or less, according as the first and second are equal to, greater, or less than each other. BOOK VI. DEFINITIONS. I. SIMILAR rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals. Definition I. II. Reciprocal figures, viz. triangles and parallelograms, are such as have their sides about two of their angles proportionals in such manner that a side of the first figure is to a side of the other as the remaining side of this other is to the remaining side of the first. III. A straight line is said to be cut in extreme and mean ratio when the whole is to the greater segment as the greater segment is to the less. IV. The altitude of any figure is the straight line drawn from its vertex perpendicular to the base. Definition IV. PROPOSITION I.-THEOREM. Triangles and parallelograms of the same altitude are one to another as their bases. LET the triangles a b c, a cd, and the parallelograms ec, cf, have the same altitude viz. the perpendicular drawn from the point a to bd : then, as the base b c is to the base cd, so is the triangle a b c to the triangle a cd, and the parallelogram ec to the parallelogram c f. Produce bd both ways to the points h, l, and take any number of straight lines bg, gh, each equal to the base bc; and dk kl, any number of them, each equal to the base cd; and join a g, a h, a k, al: then, because cb, bg, g h are all equal, the triangles a hg, agb, a b c, are all equal (i. 38); therefore, whatever multiple the base b c is of the |