B D D D A Th. If the first of four magnitudes have to the second the same ratio which the third has to the fourth; then, if the first be greater than the second, the third is also greater than the fourth; if equal, equal; and if less, less. Let equimultiples be taken of the four magnitudes A, B, C, D: and because A is to B as C is to D, 2A is to 2B as 2C is B to 2D: therefore, if 2A exceed 2B, 2C will exceed 2D. But if A be greater than A B, 2A must be greater than 2B; and so, 2C cc is greater than 2D; but if 2C exceed 2D, C is greater than D. In like mamer, if the first equal the second, or is less than it, it may be proved that the third equals the fourth, or is less than it. Therefore, if the first of four, &c. Q. E. D. A E A с B Th. If four magnitudes be proportionals, they are proportionals also when taken inversely. If A is to B as C is to D; then, inversely, B is to A as D is to C. The antecedents are A, C, the consequents B, D. Take E, F, equimultiples of the consequents, and G, H, equimultiples of the antecedents. Now, because B contains A as often as D contains C (a), and E, F are equimultiples of B, D; therefore E contains A as often as F contains C (a). But G, H are equimultiples of A, C: therefore, if E be greater than G, F is greater than H; if equal, equal: and if less, less (6). But E, F are equimultiples of B, D, and G, H of A, C: therefore B is to A as D is to C. Therefore, if four magnitudes, &c. Q. E. D. (6) def. 5, 5. G H E C Th. If the first be the same multiple, or part of the second, that the third is of the fourth, the first is to the second as the third is to the fourth. Let A, B, C, D be the magnitudes, in or- A der. And, first, let A, C be equimultiples of B, D; then A is to B as C is to D. Take E, F equimultiples of A, C, and G, H equimultiples of B, D. Then, because A, C are equimultiples of B, D (a); and oE, F equimultiples of A, C; therefore E, F are equimultiples of B, D (b). But G, H are equimultiples of B, D, any whatever therefore, if E exceeds G, F will exceed H; or, if E be equal to G, or less than it, F is equal to H, or less than it. But E, F are equimultiples of the first and third; and G, H of the second and fourth : therefore A is to B as C is to D (c). с H A Next, let A, B, C, D be the terms in order; and let A same, Q. E. D. (d) p. B, 5. C D Th. If the first be to the second as the third is to the fourth, and if the first be a multiple or part of the second, the third is the same multiple or part of the fourth. Let A, B, C, D be four magnitudes, in order; so that A is to B as C is to D. First, if A, C be equimultiples of B, D, take E equal to A: then make F the same multiple of D that A, or E AB is of B. And because A is to B as C is to D; and E, F are taken equimultiples of B,D; therefore A is to E as C is to F (a). But A equals E, therefore C equals F (6); and F, A are equimultiples of D, B: wherefore C is the same multiple of D that A is of B. Again, see the figure, second case of p. C, above. If A, C be equal parts of B, D; because A is to B as C is to D; then, inversely, B is to A as D is to C (c). But A is a part of B, and so B is a multiple of A (d); but by the preceding case, D, B are equimultiples of C, A: therefore C is the same part of D that A is of B. Hence, if the first, &c. Q. E. D. Recite (a) cor. p. 4, 5; (6) p. A, 5; (c) p. B, 5; (d) def. 1 and 2, b. 5. F А 7 Th. Equal magnitudes have the same ratio to a magnitude; and a magnitude has the same ratio to equal magnitudes. The magnitudes A, B, being equal, bave to C the same ratio: and the magnitude C has to A and B the same ratio. Take D, E any equimultiples of A, B, and F any multiple of C: then because D, E are equimultiples of B the equals A, B, they are equal to one another (a). D Therefore if d be greater than F, E is greater than F; if equal, equal; and if less, less (b). And D, E, are any equimultiples of A, B, and F is any multiple of C: therefore A is to C as B is to C. C has also the same ratio to A that it has to B: for D may be shown equal to E, as before: and, it F be greater than D, it is also greater E than E; if equal, equal; and if less, less. And F is any multiple of Q. E. D. (6) def. 5, 5. 8 Th. Of unequal magnitudes the greater has a less ratio to the same than the less has to it; and a magnitude has a less ratio to the less than to the greater. Given A, B, unequal magnitudes of which A is the less ; also C, a magnitude of the same kind as A and B (a); so that the same measuring unit (V), may apply to A, B, C. 1. The ratio of B to C is less than that of A to C. Be. cause B is greater than A, it is also a greater multiple of their common measuring unit; and Č will contain the greater of two multiples a less number of times than it will contain the other: but C is the cousequent, and A and B are its antecedents (c); therefore the ratio of B to C is less À than that of A to C (d). 2. The ratio of C to A is less than that of C to B. Let some measure be applied to C, which will also measure A and B: then because A is less than B, it will contain the measure applied to C, or any multiple of it, a less number of times than B will contain it: but C is a common antecedent, and A and B are its conseguents (d); therefore, the ratio of C to A is less than that of C to B. Wherefore, of unequal magnitudes, &c. Q. E. D. Recite (a) def. A, and 4,5; () def. 2, 1, and B, 5; (c) def. 3,5; (d) def. 7,5. B 9 Th. Magnitudes are equal to each other which have the same ratio to a magnitude; and those are equal magnitudes to which a magnitude has the same ratio. 1. Given the ratio of A to C the same as that of B to C; A is equal to B. Because A and B have a ratio to C, the magnitudes are of the same kind (a), and are measured by the same unit (b): for the same reason A and B are the antecedents and C the consequent of the ratios (c): and because the ratios are equal, A and B are equimultiples of their common measure (d); therefore A and B are equal magnitudes. 2. Given the ratio of C to A equal to that of C to B. A, B, and C are magnitudes of the same kind, and measured by the same unit, as above; and because C has a ratio to A and B, C is the antecedent and A and B are the consequents of the ratios (c); and because the ratios are equal, A and B are equimultiples of their common measure (d): therefore A and B are equal magnitudes. Wherefore, magnitudes are equal, &c. Q. E. D. Recite (a) def. 4, 5; (6) def. B, 5; (c) def. 3 and 7,5; (d) def. A, 5. А 10 Th. That is the less magnitude of two, which has a greater ratio to a third magnitude; and that magnitude is the greater of two, to which a third magnitude has a greater ratio. 1. Given A to C greater than B to C; A is less than B. Because A and B have ratios to C, the magnitudes are of the same kind (a), and are measured by the same unit (b); for the same reason A and B are the antecedents and C is the consequent of the ratios (c); and because C coutains A a greater number of times than it contains B, A is a less multiple of the common measure than B is: therefore A is a less magnitude than B. 2. Given C to B greater than C to A; B is greater than A. A, B and C are magnitudes of the same kind, and measured by the same unit, as above; and because C has ratios to A and B, C is the antecedent, and A and B are the consequents of the ratios (c); and because B contains the unit of measure applied to C, or a muliiple of it, a greater number of times tha A contains the same unit, or the same multiple of it; therefore B is a greater magnitude than A. Wherefore, that is the less magnitude, &c. Q. E. D. Recite (a) def. 4,5; (6) def. B, 5; (c) def. 3 and 7, 5. 11 Th. Ratios that are the same to the same ratio are the same to one another. A D D F M N Given A to B as C to D, and C to Das E to F: then, A is to B as E is to F. The antecedents are A, C, E; the consequents, B, D, F. Take G, H, K equimultiples of A, C, E; and L, M, N equimultiples of B, D, F (a). Then, since A is to B as C is to D; and G, H are equimultiples of A, C, and L, M of B, D; therefore, if G be greater than L, H is greater than M; if equal, equal; and if less, less. Again, since C is to D as E is to F; and H, K are equimultiples of C, E and M, N of D, F; therefore, if H be greater than M, K is greater than N; if equal, equal; and if less, less. But it proves, as above, that if Gobe greater than L, H is greater than M; if equal, equal; and if less, less : therefore, if G be greater than L, K is greater than N; if equal, equal ; and if less, less. And G, K are equimultiples of A, E; and L, N of B, F. Therefore A is to B as E is to F (b). Wherefore, ratios that are the same, &c. Q. E. D. Recite (a) df. 5,5; (6) ax. 1, 1. A B 12 Th. If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so shall the sum of all the antecedents be to the sum of all the consequents. Given the magnitudes A, B; C, D; E, F; GEHtitka 60 that A is to B as C is to D, or as E is to F: then A is to B as A+C+E is to B+D+F. DTake G, H, K equimultiples of A, C, E ; and L, M, N equimultiples of B, D, F. Let M+to N From this arrangement, if G be greater than L, H is greater than M, and K greater than N; if equal, equal : and if less, less (a). Wherefore, if Ğ be greater than L, G+H+K is greater than L+M+N; if equal, equal; and if less, less. But G and G+H+K are equimultiples of A and A+C+E (6); also L and L+M+N are equimultiples of B and B+D+F (6). Therefore, A is to B as A+C+E is to B+D+F. A D F 13 Th. If the first have to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth has to the sixth ; the first shall have to the second a greater ratio than the fifth has to the sixth. Take A, B; C, D; E, F, six magnitudes, G-Hotek two and two in order, of the same kind (a). Then, because A has a ratio to B, C to D, and E to F, the same unit will measure the antecedent and consequent of each couplet (6). M-NAnd because A is to B as C is to D, B contains A as many times, or parts of times, as D contains C; and because C is to D greater than E is to F, D contains C more times, or parts of times, than F contains E (c): but the quotients of B divided by A, and of D divided by C prove equal; therefore the quotient of B by A is greater than that of F by . (d)--that is, the ratio of A to B is greater than the ratio of E to F. Wherefore, if the first has to the second, &c. Q. E. D. (6) def. B, 5; (d) def. 7, 5. Cor. If the order of the couplets were transposed; it might be demonstrated, in the same way, that the ratio of the first to the second is less than that of the fifth to the sixth. |