b 17. 5. Book V: lya, BA is to AE, as DC is to CF; and because A they are also proportionals when taken sepa- с F B D nitude taken from the first, is to a magnitude taken from the other; the remainder likewise is to the remainder; as the magnitude taken from the first to that taken from the other: The demonstration is contained in the preceding. PROP. E. THEOR. If four magnitudes be proportionals, they are also А С 17.5. visiona, AE is to EB, as CF to FD ; and by in- E B. 5. version , BE is to EA, as DF to FC. Where F * 18. 5. fore, by composition, BA is to AE, as DC is to CF: If, therefore, four, &c. Q. E. D. B D PROP. XX. THEOR. See N. Ir there be three magnitudes, and other three, which, taken two and two, have the same ratio ; if the first be greater than the third, the fourth shall be greater than the sixth, and if equal, equal; and if less, less. A 8. 3. 13, 5. as Let A, B, C, be three magnitudes, and D, E, Fother three, Book V. which, taken two and two, have the same ratio, viz, as A is to B, so is D to E; and as B to C, so is E to F. If A be greater than C, D shall be greater than F; and if equal, equal; and if less, less. Because A is greater than C, and B is any other magnitude, and that the greater has to the same magnitude a greater ratio than the less has to it a; therefore A has to B a greater ratio than C has to B: But as D is to E, so A B с is A to B; therefore b D has to E a greater ratio than C to B; and because B is to C, D E to F, by inversion, C is to B, as F is to E; and D was shown to have to E a greater ratio than Cto B; therefore D has to E a greater c Cor. 13.5. ratio than F to E. But the magnitude which has a greater ratio than another to the same magnitude, is the greater of the twod; D is therefore greater than F. Secondly, Let A be equal to C; D shall be equal to F: Because A and C are equal to one another, A is to B, as C is to Be: But A is to B, as D to E; and C is to B, as F to E; wherefore D is to E, as F tc Ef; f 11. 5. and therefore D is equal to Fs. A B C Next, Let A be less than C; В с D shall be less than F: For Cis D E F D E F greater than · A, and, as was shown in the first case, C is to B, as F to E, and in like manner, B is to A, as E to D; therefore F is greater than D, by the first case; and therefore D is less than F. Therefore, if there be three, &c. Q. E. D. d 10. 5. © 7. 5. PROP. XXI. THEOR. If there be three magnitudes, and other three, See X. Book V.' Let A, B, C be three magnitudes, and D, E, F other three, which have the same ratio, taken two and two, but in a cross order, viz. as A is to B, so is E to F, and as B is to C, so is D to E. If A be greater than C, D shall be greater than F; and if equal, equal; and if less, less. Because A is greater than C, and B is any * 8. 5. other magnitude, A has to B a greater ratio a than C has to B: But as E to F, so is A to B : 13. 5. therefore b E has to F a greater ratio than C to B: And because B is to C, as D to E, by A B C inversion, C is to B, as E to D: And E was shown to have to Fa greater ratio than C to D E F B; therefore E has to f a greater ratio than Cor. 13. 5. E to DC; but the magnitude to which the same has a greater ratio than it has to another, is d 10. 5. the lesser of the twod: F therefore is less than D; that is, D is greater than F. Secondly, Let A be equal to C; D shall be equal to F. e 7. 5. Because A and C are equal, A ise 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 $11. 5. is to F, as E to Df; and 8 9. 5. therefore D is equal to F&. Next, Let A be less than C; A B C Α. Β E F PROP. XXII. THEOR. See N. If there be any number of magnitudes, and as many other, which, taken two and two in order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N. B. This is usually cited by the words “ ex æquali," or "er sequo. First, Let there be three magnitudes A, B, C, and as Book V. many others D, E, F, which, taken two and two, have the same ratio : that is, such that A is to B as D to E; and as B is to C, so is E to F; A shall be to C, as D to F. Take of A and D any equimultiples whatever G and H; and of B and E any equimultiples whatever K and L; and of C and F any whatever M and N: Then because A is to B, as D to E, and that G, H are equimultiples of A, D, and K, L A B C D E F equimultiples of B, E; as G is to K, so isa H to L: For the C K M H L N - 4.5. same reason, K is to M, as L 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, H is greater than N; and if equal, equal; and if less, lessb; and G, H are any equin ultiples whatever of A, D, and M, N are any equimultiples whatever of C, F: Therefore, as A is to C, . 5 Def. 5. so 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 same ratio, viz. as A is to B, so is E to A. B. C. D. F; and as B to C, so F to G; and as C to E. F. G. H. D, so G to H: A shall be to D, as E to H. Because A, B, C, are three magnitudes, and E, F, G, other three, which, taken two and two, have the samne ratio; by the foregoing case, A is to C, as E to G: But Cis 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 nuinber, &c. Q.E.D. h 20. 5. 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, bave the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. N. B. This is usually cited by the words “ ex æquali in proportione perturbata;" or“ ex æquo perturbato." so is st, 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; and as B is to C, so 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 same ratio which their B, so is G to H: And for the A B C D E F is E to F; as therefore G is to G H L K M N + 11. 5. H, so is M to Nb. And because as B is to C, so is D to E, and of B, D, and L, M of C,E; as it has been shown that G is to two and two in a cross order; if G be greater than L, Kiş 21. 5. greater than N: and if equal, equal; and if less, lessd ; and G, K are any equimultiples whatever of A, D, and L, N any whatever of C, F; as, therefore, A is to C, so is D to F. |