than E; if equal, equal; and if less, less. Therefore equal magnitudes, &c. And F is any multiple of Q. E. D. (b) 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. 1. Be 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 (b), may apply to A, B, C. The ratio of B to C is less than that of A to C. cause B is greater than A, it is also a greater multiple of their common measuring unit; and C will contain the greater of two multiples a less number of times than it will contain the other: but C is the consequent, and A and B are its antecedents (c); therefore the ratio of B to C is less A B 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 consequents (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; (b) def. 2, 1, and B, 5; (c) def. 3, 5; (d) def. 7, 5. 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 A B C 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 (e); 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. Recite (a) def. 4, 5; (c) def. 3 and 7, 5; (b) def. B, 5; Q. E. D. 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 contains 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 multiple of it, a greater number of times than 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; (b) 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. the consequents, B, D, F. M N The antecedents are A, C, E; 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 G-be 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, Ñ of B, F. Therefore A is to B as E is to F (b). Wherefore, ratios that are the same, &c. Q. E. D. 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 con sequents. Given the magnitudes A, B; C, D; E, F ; G→▬▬▬▬▬▬H▬▬▬▬▬▬▬▬▬ K▬▬▬▬▬▬ so 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. A C E+ D F B M+N Take G, H, K equimultiples of A, C, E; and L, M, N equimultiples of B, D, F. L-+-+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 G 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 (b); also L and L+M+N are equimultiples of B and B+D+F (b). Therefore, A is to B as A+C+E is to B+D+F. Recite (a) def. 5, 5; (b) p. 1, 5. Q. E. D. 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, Gtwo 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 (b). A C+ E B D F L And because A is to B as C is to D, 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 Ê (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. Recite (a) def. 4, 5; (c) def. 3, 5; Q. E. D. (b) 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. 14 Th. If the first have to the second the same ratio which the third has to the fourth; then if the first be greater than the third, the second shall be greater than the fourth; if equal, equal; and if less, less. Given A to B as C to D, and A, C antecedents, B, D consequents. 1. If A be greater than C, the quotient of B divided by A will be less than that of B divided by C (a); but the quotients of B by A and D by C are given equal; for the consequent divided by the antecedent is the ratio (b): therefore the quotient of B divided by C is greater A B than the quotient of D divided by C; and so, B is greater than D. C D 2. If A equals C, the two quotients of B divided by A and by C are equal (c); but the quotients of B by A and D by C are given equal, as above (b): therefore the quotients of B and D by C are equal and so, B is equal to D. 3. If A be less than C, the quotient of B by A is greater than that of B by C (a): but the quotients of B by A and D by C are given equal (b): therefore the quotient of B by C is less than that of D by C; and so, B is less than D. Wherefore, if the first have to the second, &c. Recite (a) p. 8, 13, 5; (b) def. 3, 5; (c) p. 9, 5. Q. E. D. NOTE. The ratio of C to B is here introduced as a medium of comparison between B and D; also the magnitudes are so divided in the diagram that each may be taken greater or less as the case may require. 15 Th. Magnitudes have the same ratio to one another which their equimultiples have. Given two magnitudes C, F; and equimultiples of A them, AB, DE: C is to F as AB is to DE. H Because AB is a multiple of C, C is a part of AB (a): for the same reason F is a part of DE. And because AB, DE are equimultiples of C, F; AB contains the measure C as often as DE contains the measure F (b). Apply the measure C, from A to G, from G to H, and from H to B: apply also the measure F, from D to K, from K to L, and from L to E. Then, because the parts AG, GH, HB, are equal; and the parts DK, KL, LE are equal; AG is to DK as GH is to KL, as HB is to LE (c). Now AG, GH, HB are the antecedents, and DK, KL, LE are the consequents; wherefore AG is to DK as AB is to DE (d): but AG is equal to C and DK to F; therefore, C is to F as AB is to DE. Therefore, magnitudes have the same ratio, &c. Recite (a) def. 1, 2 of 5; (c) p. 14, 5 (case 2); (b) def. A, 5; Q. E. D. 16 Th. If four magnitudes of the same kind be proportionals, they shall be proportionals also when taken alternately. If A, B, C, D be four magnitudes of the same kind, and have A to B as C to D; then, alternately, A:C:: B: D. Take E, F equimultiples of A, B, and G, H equimultiples of C, D. Because E, F are equimultiples of A, B, and that magnitudes have the same ratio which their equimultiples have (a); therefore A is to B as E is to F: but A is to B as C is to D (b); therefore C is to D as E is to F. E Again, because G, H are equimultiples of C, D; therefore C is to D as G is to H (a): but C is to D as E is to F, as above; therefore E is to Fas G is to H (c): and so, the equimultiples are proportionals (d). Wherefore, if É be greater than G, F is greater than H; if equal, equal; and if less, less (e). But E, F are equimultiples of A, B, and G, H of C, D: therefore A is to C as B is to D. Wherefore, if four magnitudes of the same kind, &c. Recite (a) p. 15, 5; (d) def. 6, 5; Q. E. D. (b) hypothesis; (c) p. 11, 5; 17 Th. If the sum of two magnitudes have to one of them the same ratio which the sum of other two has to one of these, the one left of the former two shall have to the other the same ratio which the one left of the latter two has to the other of these. Let AE, EB be two magnitudes, and CF, FD other two: 113 the sum of the former is AE+EB, of the latter CF+FD. Then, since AE+EB : EB :: CF+FD: FD (a); inversely, EB AE÷EB :: FD: CF+FD (b). And since ratio is the quotient of the consequent divided by the antecedent (c) and division is indicated by writing the AE+EB CF+FD divisor under the dividend: AE EB CF FD EB + EB EB FD FD But EB FD AE CF EB FD mer (d); there will remain = FD ; that is, ; take these equals from the for Now draw these equals into line; therefore, EB: AE:: FD: CF; and inversely, AE: EB:: CF: FD (b). Wherefore, if the sum of two magnitudes, &c. Recite (a) hyp. (b) p. B, 5; (d) ax. 3, 1. Q. E. D. (c' def. 3, 5; |