from the first, is to a magnitude taken from the other; the remainder shall be to the remainder, as the whole to the whole. * Let the whole AB be to the whole CD, as AE, a magnitude taken from AB, to CF, a magnitude taken from CD; the remainder EB shall be to the remainder FD, as the whole AB to the whole CD. Because AB is to CD, as AE to CF, likewise alternately (16. 5.) BA is to AE, as DC to CF; and because, if magni. A tudes, take jointly, be proportionals, they are also proportionals (17. 5.) when taken separately; therefore, as BE is to DF, so is EA to FC; and alternately, as E c BE is to EA, so is DF to FC: but, as AE to CF, so, by the hypothesis, is AB to CD; therefore also BE, the remainder shall be to the remainder DF, as the whole AB to the whole CD: Wherefore, if the whole, &c. Q. E. D. • B D Cor. If the whole be to the whole, as a magnitude 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 proportionals by conversion, that is, the first is to its excess above the second, as the third to its excess above the fourth. Let AB be to BE, as CD to DF; then BA is to A AE, as DC to CF. Because AB is to BE, as CD to DF, by division (17. 5.), AE is to EB, as CF to FD, and by inver- . sion (B. 5.), BE is to EA, as DF to FC. Wherefore, by composition (18. 5.), BA is to AE, as DC is to CF. If, therefore, four, &c. Q. E. D. B D PROP. XX. THEOR. If 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.* Let A, B, C be three magnitudes, and D, E, F other three, which, taken two and two, have the same ratio, viz, as A is to B, * See Note. 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 (8. 5.); therefore A has to B a greater ratio than C has to B; but as D is to E, so is A to B; therefore (13. 5.) D has to E a greater ratio than C to À Ś B; and because B is to C, as E to F, by inversion, D E ' F C is to B, as F is to E; and D was shown to have to E a greater ratio than C to B; therefore D has to E a greater ratio than F to E (Cor. 13. 5.); but the magnitude which has a greater ratio than another to the same magnitude, is the greater of the two (10. 5.); 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 B (7. 5.): 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 to E(11. 5.); and therefore D is equal to F (9. 5.). A B C A B C Next, let A be less than C; D D E F D E F shall be less than F: for C is 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. PROP. XXI. THEOR. If there be three magnitudes, and other three, which have the same ratio taken two and two, but in a cross order, if the first magnitude be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less.* 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 other magnitude, A has to B a greater ratio (8. 5.) than C has to B: but as E to F, so is A to B; therefore (13. 5.) E has to F a greater ratio than C to B: and because B is to C, as D to E, by inversion, C is to Á B C • See Note. B, as E to D: and E was shown to have to Fa D E F Secondly, let A be equal to C, D shall be equal to F. Because A and C are equal, A is (7. 5.) 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 D (11. 5.); and therefore D is equal to F (9. 5.). Next, let A be less than C; D shall be less than F; for C is greater A B C À B C than A, and, as was shown, C is to D E F D E F B, as E to D; and in like manner B is to A, as F to E; therefore F is greater than D, by case first; and therefore D is less than F. Therefore, if there be three, &c. Q. E. D. . PROP. XXII. THEOR. If there be any number of magnitudes, and as many others, 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 “ ex æquo."* .First, let there be three magnitudes A, B, C, and as 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 equimultiples of B, E; as G is to K, so is (4. 5.) H to L. For À B D F the same reason, K is to M, as L See Note. to N: and because there are three GKM H LN 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, | A. B. C. D. viz, as A is to B, so is E to F, and as B to C, so F | E. F. G. H. to G; and as C to 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 same ratio; by the foregoing case, 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. O PROP. XXIII. THEOR. 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 of the others has to the last. N. B. This is usually cited by the words “ ex æquali in proportione perturbata ;" ur,“ 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; 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 equimultiples have (15. 5.); 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 (11. 5.). And because as B is to A B C D E See Note. C, so is D to E, and that H, K are G H L K M N equimultiples of B, D, and L, M, of C, E; as H is to L, so is (4. 5.) K. to M: and it has been shown, that G is to H, as M 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; and if less, less (21. 5.); 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. 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, 1 | A. B. C. D. have the same ratio, viz. A 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 i | E. F. G. H. to 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 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. Q. E. D. PROP. XXIV. THEOR. 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 which the third and sixth together have to the fourth. * H 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 : AG, the first and fifth G together, shall have to C the second, the н 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: and because as AB is to C, so is DE to F; and as C to BG, so F to EH; ex æquali (22. 5.), AB is to BG, as DE to EH: and because these magnitudes are proportionals, they shall likewise be proportionals when taken jointly (18. 5.): Á Č Ď F B! See Note. |