Book V. a. 8. 5. Let A, B, C be three magnitudes, and D, E, F other three which have the fame ratio taken two and two, but in a cross order, viz. as A is to B, fo is E to F, and as B is to C, fo is D to E. If A be greater than C, D fhall be greater than F; and if equal, equal; and if lefs, lefs. A B C Because A is greater than C, and B is any other magnitude, A has to B, a greater ratio a than C has to B. but as E to F, fo is A to B; therefore b E has to F a greater ratio than C to B. and because B is to C, as D to E, by inverfion, C is to B, as E to D. and E was fhewn to have to Fa greater ratio than C to B; therefore E has to Fa c. Cor.13.5. greater ratio than E to D c. but the magnitude to which the fame has a greater ratio than it has to d. 10. 5. another, is the leffer of the two d. F therefore is lefs than D; that is, D is greater than F. Secondly, Let A be equal to C; D fhall be equal to F. Becaufe A and C are equal, A is to B, as C is to B. but A is to B, as E to F; and C is, to B, as E to D; wherefore E is to F, as E to Df; and therefore D is equal to F. c. 7.5. f. 11. 5. g. 9.5. See N. e TF there be any number of magnitudes, and as many others, which taken two and two in order have the fame ratio; the firft fhall have to the last of the first magnitudes the fame ratio which the first of the other has to the laft. N. B. This is ufually cited by the words "ex aequali, or, ex aequo.”. First, Let there be three magnitudes A, B, C, and as many Book V. others D, E, F, which taken two and two have the fame ratio, that is such that A is to B, as D to E; and as B is to C, fo is E to F. A fhall be to C, as D to F. Take of A and D any equimultiples whatever G and H; and to E, and that G, H are equi- A B C to K, fo is a H to L. for the D E F 3. 4. 5. and if lefs, lefs". and G, H are any equimultiples whatever of A, b. 20. 5. D, and M, N are any equimultiples whatever of C, F. therefore c c. 5. Def. 5. as A is to C, fo is D to F. Next, Let there be four magnitudes A, B, C, D, and other four E, F, G, H, which two and two have the fame ratio, viz. as A is to B, so is E to F; and as B to C, fo F to G; and as C to D, fo G to H. A fhall be to D, as E to H. A. B. C.D. E. F. G.H. Because A, B, C are three magnitudes, and E, F, G other three, which taken two and two have the fame ratio; by the foregoing cafe, A is to C, as E to G. but C is to D, as G is to H; wherefore again, by the first case, A is to D, as E to H. and so on, whatever be the number of magnitudes. Therefore if there be any number, &c. Q. E. D. Book V. PROP. XXIII. THEOR. See N. IF there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the fame ratio; the first shall have to the last of the first magnitudes the fame ratio which the first of the others has to the laft. N. B. This is ufually cited by the words "ex aequali in proportione perturbata, or, ex aequo perturbate." First, Let there be three magnitudes A, B, C, and other three D, E, F, which taken two and two in a' crofs order have the fame ratio, that is fuch that A is to B, as E to F; and as B is to C, fo is D to E. A is to C, as D 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 fame ratio which a. 15. 5. their equimultiples have a; as A is to B, fo is G to H. and for the fame reafon, as E is to F, fo is M to N. b. 11. 5. G. 4. 5. but as A is to B, fo is E to F; as ABC and because as B is to C, fo is D to if G be greater than L, K is greater DEF d. 21. 5. than N; and if equal, equal; and if lefs, lefs d. and G, K are any equimultipies whatever of A, D; and L, N any whatever of C, F; as therefore A is to C, fo is D to F. Next, Let there be four magnitudes A, B, C, D, and other four E, F, G, H, which, taken two and A. B. C.D. two in a cross order, have the fame ratio, viz. AE. F. G.H. to B, as G to H; B to C, as F to G; and C to D, as E to F. A is to D, as E to H. Because A, B, C are three magnitudes, and F, G, H other three, which, taken two and two in a crofs order, have the fame ratio; by the first cafe, A is to C, as F to H. but C is to D, as E is to F; wherefore again, by the first case, A is to D, as E to H. and so on, whatever be the number of magnitudes. Therefore if there be any number, &c. number, &c. Q. E. D. IF Book V. the first has to the fecond the fame ratio which the see N. third has to the fourth; and the fifth to the second the fame ratio which the fixth has to the fourth; the first and fifth together shall have to the second, the famé ratio which the third and fixth together have to the fourth. Let AB the first have to C the fecond the fame ratio, which DE the third has to F the fourth; and let BG the fifth have to C the fecond the fame ratio, which EH the fixth has to F the fourth. AG, the first and fifth together, fhall have to C the fecond the fame ratio, which DH, the third and fixth together, has to F the fourth. Because BG is to C, as EH to F; by inverfion C is to BG, as F to EH. and because as AB is to C, fo is DE to F; and as C to BG, fo F to EH; ex aequalia AB is to BG, as DE to EH. and because these magnitudes are proportionals, they fhall likewife be proportionals when taken joint B AC E H DF a. 22. S ly ; as therefore AG is to GB, fo is DH to HE; but as GB to b. 18. 5 C, fo is HE to F. Therefore, ex aequalia, as AG is to C, fo is DH to F. Wherefore if the firft, &c. QE. D. COR. 1. If the fame Hypothefis be made as in the Propofition, the excess of the first and fifth fhall be to the fecond, as the excess of the third and fixth to the fourth, the Demonftration of this is the Book V. fame with that of the Propofition, if Divifion be used instead of Compofition. COR. 2. The Propofition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to a fecond magnitude the fame ratio that the correfponding one of the second rank has to a fourth magnitude; as is manifeft. IF PROP. XXV. THEOR. F four magnitudes are proportionals, the greatest and leaft of them together are greater than the other two together. 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 a.A,&14.5. confequently F the least 2. AB together with F are greater than CD together with E. b. 19. 5. c. A. 5. See N. B G D H Take AG equal to E, and CH equal to F. then because as AB. to CD, fo is E to F, and that AG is equal to E, and CH equal to F; AB is to CD, as AG to CH. and because AB the whole is to the whole CD, as AG is to CH; likewife the remainder GB fhall be to the remainder HD, as the whole AB is to the whole b CD. but AB is greater than CD, therefore 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 magnitudes GB, HD, of which GB is the greater, A CEF 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. Therefore if four magnitudes, &c. Q. E. D. PROP. F. THEOR. ATIOS which are compounded of the fame ratios, are the fame with one another. RAT |