any whatever M and N. Then, because A is to B, as D is to E, and that G, H are equimultiples of Á, D, and K, L equimultiples of B, E; as G is to K, so is H to L (a): for the same reason, K is to M, as L is to N: and because there are three magnitudes G, K, M, and other three, H, L, N, which, two and two, have the same ratio; if G be greater than M, I is greater than N; and if equal, equal; if less, less (b) and G, H are any equimultiples whatever of A, D, and M, N are any equimultiples whatever of C, F; therefore, as A is to C, so is D to F (c). Next, let there be four magnitudes A, B, C, D, and other four E, F, G, H, which, two and two, have the same ratio, viz. as A is to B, so is E to F; and as B is to C, so is F to G; and as Cis to D, so is G to H: then shall A be to D, as E is to H. Because A, B, C are three magnitudes, and E, F, G other three, which, taken two and two, have the same ratio; by the foregoing case, A is to C, as E is to G: but C G is to D, as G is to II; wherefore again, by the first case, A is to D, as E is to H: and so on, whatever be the number of magnitudes. SCHOLIUM. This proposition is expressed by the terms, ex æquali, or ex æquo, as explained in the 20th definition. It may be algebraically expressed as follows: THEOREM. If A, B, C, D be any magnitudes, and E, F, G, H be as many others, such that 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 has to the last of the first magnitudes the same ratio which the first has to the last of the others. DEMONSTRATION. 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 is to F; and that B is to C, as D is to E; then shall A be to C, as D is to F. Take of A, B, D any equimultiples whatever G, H, K; and of C, E, F any equimultiples whatever L, M, N; and because G, H are equimultiples of A, B, and that magnitudes have the same ratio which their equimultiples have (a); 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: but as A is to B, so is E to F; as therefore G is to H, so is M to N (6). And because as B is to C, so is D to E, and that H, K are equimultiples of B, D, and L, M of C, E; as H is to L, so is K to M (c): and it has been shown that G is to H, as M is to N: then, 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; if less, less (d); and G, K are any equimultiples whatever of A, D ; and L, N any whatever of C, F; therefore as A is to C, so is D to F (e). (a) V. 15. 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 is to B, as G is to H; B is to C, as F is to G; and C is to D, as E is to F: then shall A be to D, as E is 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 is to H: but C is to D, as E is to F; wherefore again, by the first case, A is to D, as E is to H: and so on, whatever be the number of magnitudes. SCHOLIUM. This proposition is expressed by the terms, ex æquali in proportione perturbata, or ex æquo perturbate, as explained in the 21st definition. Algebraically expressed, it is as follows: THEOREM. If A, B, C, D be any magnitudes, and E, F, G, H as many others, such that THEOREM.-If the first have 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. DEMONSTRATION. Because BG is to C, as EH is to F; by inversion, C is to BG, as F is to EH and because AB is to C, as DE is to F; and C is to BG, as F is to EH; ex æquali, AB is to BG, as DE is to EH (a): and because these magnitudes are proportionals, they shall likewise be proportionals when taken jointly (b); therefore as AG is to BG, so is DH to EH; but as BG is to C, so is EH to F. Therefore, ex æquali, as AG is to C, so is DH to F (a). COROLLARY 1. If the same hypothesis be made as in the proposition, the difference of the first and fifth shall be to the second, as the difference 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." COROLLARY 2. This 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. SCHOLIUM. The foregoing proposition, algebraically expressed, is as follows: THEOREM. If A : B :: C: D, and E: B :: F: D; then A+E : B:: C+ F: Ď. are THEOREM.-If four magnitudes of the same kind proportionals, the greatest and least of them together are greater than the other two together. DEMONSTRATION. Let the four magnitudes AB, CD, E, F be proportionals, viz. AB to CD, as E to F; and let AB be the greatest of them, and consequently_F_the least (a); then shall AB together with F, be greater than CD together with E. B G. D H E F (a) V. A & 14. Take AG equal to E, and CH equal to F: then, because as AB is to CD, so is E to F, and that AG is equal to E, and CH equal to F; AB is to CD, as AG is to CH; and because AB the whole, is to the whole CD, as AG is to CH, likewise the remainder GB shall be to the remainder HD, as the whole AB is to the whole CD (b): but AB is greater than CD, therefore GB is greater than HD (c).; 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 magnitudes GB, HD, of which GB is the greater, there be added equal magnitudes, viz. to GB the two AG and F, and to HD the two CH and E; AB and F together are greater than CD and E. SCHOLIUM. This proposition, may be algebraically expressed as follows: THEOREM. If A: B :: C: D, and if A is the greatest of them, then A+ Dis B + C. |