PROPOSITION XXIII, THEOREM. If there be any number of magnitudes, and as many others, which taken two and two in a cross order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first has to the last of the others. N.B. This is usually cited by the words "ex æquali in proportione perturbatâ;" or "ex æquo perturbato." First, let there be three magnitudes A, B, C, and other three D, E, F, which taken two and two in a cross order have the same ratio, that is, such that A is to B, as E to F; Take of A, B, D any equimultiples whatever G, H, K; and that magnitudes have the same ratio which their equimultiples have; (v. 15.) therefore as A is to B, so is G to H: and for the same reason, as E is to F, so is M to N: therefore as G is to H, so is M to N: (v. 11.) and it has been shewn that G is to H, as M to N: therefore, because there are three magnitudes G, H, L, and other three K, M, N, which have the same ratio taken two and two in a cross order; if G be greater than L, K is greater than N: and if equal, equal; and if less, less: (v. 21.) but G, K are any equimultiples whatever of A, D; (constr.) therefore as A is to C, so is D to F. (v. def. 5.) Next, let there be four magnitudes A, B, C, D, and other four E, F, G, H, which taken two and two in a cross order have the same ratio, viz. A to B, as G to H; B to C, as Fto G; and C to D, as E to F. Then A shall be to D, as E to H. A.B.C.D E.F.G.H Because A, B, C are three magnitudes, and F, G, H other three, which taken two and two in a cross order, have the same ratio; by the first case, A is to C, as F to H; but Cis to D, as E is to F; wherefore again, by the first case, A is to D, as E to H; PROPOSITION XXIV. THEOREM. If the first has to the second the same ratio which the third has to the fourth; and the fifth to the second the same ratio which the sixth has to the fourth; the first and fifth together shall have to the second, the same ratio which the third and sixth together have to the fourth. Let AB the first have to C the second the same ratio which DE the third has to F the fourth; and let BG the fifth have to C the second the same ratio which EH the sixth has to F the fourth. Then AG, the first and fifth together, shall have to C the second, the same ratio which DH, the third and sixth together, has to F the fourth. Because BG is to C, as EH to F; by inversion, C is to BG, as F to EH: (V. B.) and because, as AB is to C, so is DE to F; (hyp.) and as C to BG, so is F to EH; ex æquali, AB is to BG, as DE to EH: (v. 22.) and because these magnitudes are proportionals when taken separately, they are likewise proportionals when taken jointly; (v. 18.) therefore as AG is to GB, so is DH to HE: but as GB to C, so is HE to F: (hyp.) therefore, ex æquali, as AG is to C, so is DH to F. (v. 22.) COR. 1.-If the same hypothesis be made as in the proposition, the excess of the first and fifth shall be to the second, as the excess of the third and sixth to the fourth. The demonstration of this is the same with that of the proposition, if division be used instead of composition. COR. 2.-The proposition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to a fourth magnitude: as is manifest. PROPOSITION XXV. THEOREM. If four magnitudes of the same kind are proportionals, the greatest and least of them together are greater than the other two together. Let the four magnitudes AB, CD, E, F be proportionals, and let AB be the greatest of them, and consequently F the least. (v. 14. and A.) Then AB together with F shall be greater than CD together with E. Take AG equal to E, and CH equal to F. and that AG is equal to E, and CH equal to F, and because AB the whole, is to the whole CD, as AG is to CR, likewise the remainder GB is to the remainder HD, as the whole AB is to the whole CD: (v. 19.) but AB is greater than CD; (hyp.) therefore GB is greater than HD; (v. a.) and because AG is equal to E, and CH to F; AG and F together are equal to CH and E together: (I. ax. 2.) therefore if to the unequal magnitudes GB, HD, of which GB is the greater, there be added equal magnitudes, viz. to GB the two AG and F, and CH and E to HD; AB and F together are greater than CD and E. (1. ax. 4.) PROPOSITION F. THEOREM. Ratios which are compounded of the same ratios, are the same to one another. Let A be to B, as D to E; and B to C, as E to F. Then the ratio which is compounded of the ratios of A to B, and B to C, which, by the definition of compound ratio, is the ratio of A to C, shall be the same with the ratio of D to F, which, by the same definition, is compounded of the ratios of D to E, and E to F. A.B.C 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 æquali, 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: A.B.C therefore, ex æquali in proportione perturbata, (v. 23.) that is, the ratio of A to C, which is compounded of the ratios of A to B, and B to C, is the same with the ratio of D to F, which is compounded 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 to several ratios, each to each; the ratio which is compounded of ratios which are the same to the first ratios, each to each, shall be the same to the ratio compounded of ratios which are the same to the other ratios, each to each. Let A be to B, as E to F; and C to D, as G to H: and let A be to B, as K to L; and C to D, as L to M. Then the ratio of K to M, by the definition of compound ratio, is compounded of the ratios of K to L, and L to M, which are the same with the ratios of A to B and C to D. Again, as E to F, so let N be to O; and as G to H, so let O be to P. Then the ratio of N to P is compounded of the ratios of N to O, and O to 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 M, as N to P. A.B.C.D. K.L.M E.F.G.H. N.O.P Because K is to L, as (A to B, that is, as E to F, that is, as) N to 0: and as L to M, so is (C to D, and so is G to H, and so is) O to P: ex æquali, K is to M, as N to P. (v. 22.) Therefore, if several ratios, &c. Q. E. D. PROPOSITION H. THEOREM. If a ratio which is compounded of several ratios be the same to a ratio which is compounded of several other ratios; and if one of the first ratios, or the ratio which is compounded of several of them, be the same to one of the last ratios, or to the ratio which is compounded of several of them; then the remaining ratio of the first, or, if there be more than one, the ratio compounded of the remaining ratios, shall be the same to the remaining ratio of the last, or, if there be more than one, to the ratio compounded of these remaining ratios. 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 L, and L to M: also, let the ratio of A to F, which is compounded of the first ratios, be the same with the ratio of G to M, which is compounded of the other ratios; and besides, let the ratio of A to D, which is compounded of the ratios of A to B, B to C, C 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, shall be the same with the ratio of K to M, which is compounded of the remaining ratios of K to L, and L to M of the other ratios. A.B.C.D.E.F Because, by the hypothesis, A is to D, as G to K, (hyp.) (v. 22.) Q. E.D. PROPOSITION K. THEOREM. If there be any number of ratios, and any number of other ratios, such, that the ratio which is compounded of ratios which are the same to the first ratios, each to each, is the same to the ratio which is compounded of ratios which are the same, each to each, to the last ratios; and if one of the first ratios, or the ratio which is compounded of ratios which are the same to several of the first ratios, each to each, be the same to one of the last ratios, or to the ratio which is compounded of ratios which are the same, each to each, to several of the last ratios; then the remaining ratio of the first, or, if there be more than one, the ratio which is compounded of ratios which are the same each to each to the remaining ratios of the first, shall be the same to the remaining ratio of the last, or, if there be more than one, to the ratio which is compounded of ratios which are the same each to each to these remaining ratios. 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 L, 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, as Tto V; and E to F, as V to X: therefore, by the definition of compound ratio, the ratio of S to X is compounded of the ratios of S to T, T to V, and V to X, which are the same to the ratios of A to B, C to D, E to F: each to each. Also, as G to H, so let Y be to Z; and K to L, 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 Y to d is compour.ded 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, to the ratios of G to H, K to L, M to N, O to P, and Q 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 to 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 to the ratios of G to H, and K to L, two of the other ratios; and let the ratio of h to be that which is compounded of the ratios of h to k, and k to l, which are the same to the remaining first ratios, viz. of C to D, and E to F; |