PROP. IX. THEOR. Magnitudes which have the same ratio to the same magnitude are equal to one another; and those to which the same magnitude has the same ratio are equal to one another. For if not, let A be greater than B; then because A is greater than B, two numbers, m and n, may be found, as in the last proposition, such that mA shall exceed nC, while mB does not exceed nC. But because A: C :: B: C; and if mA exceed nC, mB must also exceed nC (def. 5. 5.): and it is also shewn that mB does not exceed nC, which is impossible. Therefore A is not greater than B; and in the same way it is demonstrated that B is not greater than A.; therefore A is equal to B. Next, let C: A: C: B, A=B. For by inversion (A. 5.) A: C:: B: C; and therefore, by the first case, A=B. PROP. X. THEOR. ·That magnitude, which has a greater ratio than another has to the same magnitude, is the greatest of the two: And that magnitude, to which the same has a greater ratio than it has to another magnitude, is the least of the two. If the ratio of A to C be greater than that of B to C, A is greater than B. Because A: C7B: C, two numbers m and n may be found, such that mA7nC, and mB≤nC (def. 7. 5.). Therefore also mA7mB, and A7B (Ax. 4. 5.). : be Again, let C B7C: A; BA. For two numbers, m and n may found, such that mC7nB, and mCnA (def. 7. 5.). Therefore, since nB is less, and nA greater than the same magnitude mC, nBnA, and therefore B/A. PROP. XI. THEOR. Ratios tnat are equal to the same ratio are equal to one another. If A : B :: C: D; and also C: D:: E: F; then A: B::E: F. Take mA, mC, mE, any equimultiples of A, C, and E; and nB, nD, nF, any equimultiples of B, D, and F. Because A: B:: C: D, if mA 7nB, mC7nD (def. 5. 5.); but if mC7nD, mE7nF (def. 5. 5.), because C: D :: E F; therefore if mA7nB, mE7nF. In the same manner, if mA= nB, mE=nF; and if mA/nB, mE/nF. Now, mA, mE are any equimultiples whatever of A and E; and nB, nF any whatever of B and F; therefore A: B:: E: F (def. 5. 5.). PROP. XII. THEOR. If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so are all the antecedents, taken together, to all the consequents. If A B C D, and C: D:: E: F; then also, A : B :: A+C+E: B+D+F. Take mA, mC, mE any equimultiples of A, C, and E; and nB, nD, nF, any equimultiples of B, D, and F. Then, because A: B:: C: D, if mA 7mB, nC7nD (def. 5. 5.); and when mC7nD, mE 7nF, because C : D E F. Therefore, if mA 7nB, mA+mC+mE7nB+nD+nF: In the same manner, if mA=nB, mA+mC+mE=nB+nD+nF; and if mA/ nB, mA+mC+mE/nB+nD+nF. Now, mA+mC+mE=m(A+C+ E) (Cor. 1. 5.), so that mA and mA+mC+mE are any equimultiples of A, and of A+C+E. And for the same reason nB, and nB+nD+nF are any equimultiples of B, and of B+D+F; therefore (def. 5. 5.) A : B :: A+C+E: B+D+F. PROP. XIII. THEOR. 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 has also to the second a greater ratio than the fifth has to the sixth. : : If A B C D; but C: D7E: F; then also, A: B7E: F. Because C D7E: F, there are two numbers m and n, such that mC7 nD, but mEnF (def. 7. 5.). Now, if mC 7 nD, mA7nB, because A: B :: CD. Therefore mA7nB, and mE/nF, wherefore, A: B7E: F (def. 7. 5.). PROP. XIV. THEOR. If the first have to the second the same ratio which the third has to the fourth, and if the first be greater than the third, the second shall be greater than the fourth; if equal, equal; and if less, less. If A : B :: C: D; then if A7C, B7D; if A=C, B=D; and if A Ż C, BZD. First, let A7C; then A: B7C: B (8. 5.), but A: B:: C: D, therefore C D7C: B (13. 5.), and therefore B7D (10. 5.). : In the same manner, it is proved, that if A=C, B=D; and if A/C, BZD. PROP. XV. THEOR. Magnitudes have the same ratio to one another which their equimultiples have. If A and B be two magnitudes, and m any number, A: B.: mA : mB. Because A: B:: A : B (7. 5.) ; A:B::A+A:B+B (12. 5.), or A: B:: 2A 2B. And in the same manner, since A : B :: 2A : 2B, A : B :: A+2A: B+2B (12. 5.), or A: B:: 3A 3B; and so on, for all the equimultiples of A and B. : PROP. XVI. THEOR. If four magnitudes of the same kind be proportionals, they will also be proportionals when taken alternately. If A: B:: C: D, then alternately, A: C1.: B : D. Take mA, mB any equimultiples of A and B, and C, nD any equimul tiples of C and D. Then (15. 5.) A : B :: mA : mB; now A: B:: C: D, therefore (11. 5.) C: D :: mA; mB. But C: D:: nC: nD (15. 5.); therefore mA: mB:: nC : nD (11. 5.): wherefore if mA7nC, mB 7nD (14. 5.); if mAnC, mB=nD, or if mA /nC, mB /nD; therefore (def. 5. 5.) A: C: B : D. PROP. XVII. THEOR. If magnitudes, taken jointly, be proportionals, they will also be proportionals when taken separately; that is, if the first, together with the second, have to the second the same ratio which the third, together with the fourth, has to the fourth, the first will have to the second the same ratio which the third has to the fourth. If A+B : B :: C+D: D, then by division A : B :: C : D. Take mA and nB any multiples of A and B, by the numbers m and n; and first, let mA 7nB: to each of them add mB, then mA +mB7mB+nB. But mA+mB=m(A+B) (Cor. 1. 5.), and mB+nB=(m+n)B (2. Cor. 2. 5.), therefore m(A+B)7(m+n)B. And because A+B: B:: C+D: D, if m(A+B) 7 (m+n)B, m(C+D) 7(m+n)D, or mC+mD7mD+nD, that is, taking mD from both, mC7 nD. Therefore, when mA is greater than nB, mC is greater than nD. În like manner it is demonstrated, that if mA=nB, mC=nD, and if mA /nB, that mDnD; therefore A: B:: C: D (def. 5. 5.). If magnitudes, taken separately, be proportionals, they will also be proportionals when taken jointly, that is, if the first be to the second as the third to the fourth, the first and second together will be to the second as the third and fourth together to the fourth. : : If A B C D, then, by composition, A+B : B :: C+D: D. Take m(A+B), and nB any multiples whatever of A+B and B; and first, let m be greater than n. Then, because A+B is also greater than B, m(A+B) 7nB. For the same reason, m(C+D)7nD. In this case, therefore, that is, when m7n, m(A+B) is greater than nB, and m(C+D) is greater than nD. And in the same manner it may be proved, that when m=n, m(A+B) is greater than nB, and m(C+D) greater than nD. Next, let mn, or n 7 m, then m(A+B) may be greater than nB, or may be equal to it, or may be less; first, let m(A+B) be greater than nB; then also, mA+mB7nB; take mB, which is less than nB, from both, and mA 7nB-mB, or mA7(n—m)B (6.5.). But if mA 7 (n—m)B, mC7(n−m) D, because A: B:: C: D. Now, (n-m)D=nD-mD (6. 5.), therefore mC7nD-mD, and adding mD to both, mC+mD7nD, that is (1. 5.), m(C+D)7nD. If, therefore, m(A+B) 7nB, m(C+D)7nD. In the same manner it will be proved, that if m(A+B)=nB, m(C+D) =nD; and if m(A+B) ZnB, m(C+D)/nD; therefore (def. 5. 5.), A+ B: B::C+D : D. PROP. XIX. THEOR. If a whole magnitude be to a whole, as a magnitude taken from the first is to a magnitude taken from the other; the remainder will be to the remainder as the whole to the whole. If A B : C: D, and if C be less than A, A-C: B-D :: A: B. Because A: B: C: D, alternately (16. 5.), A : C:: B: D; and therefore by division (17. 5.) A-C: C:: B-D: D. Wherefore, again alternately, A-C: B-D :: C: D; but A: B:: C: D, therefore (11. 5.) A -C: B-D :: A: D. 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. If A B C D, by conversion, A: A-B:: C: C-D. : For, since A: B:: Č: D, by division (17. 5.), A-B: B:: C-D: D, and inversely (A. 5.) B: A-B:: D: C-D; therefore, by composition (18. 5.), A: A-B:: C: C-D. COR. In the same way, it may be proved that A: A+B :: C: C+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 is greater than the sixth; if equal, equal; and if less, less. If there be three magnitudes, A, B, and C, and other three D, E, and F; and if A: B : D: E; and also B: C:: E: F, then if A7C, D7F; if A=C, D = F; and if A/C, D ZF. A, B, C, First, let A7C; then A: B7C: B (8. 5.). But A: B:: D: E, therefore also D E7C: E(13.5.). Now B: C:: E: F, and inversely (A. 5.), C: B:: F: E; and it has been shewn that D: E7C: B, therefore D: E7F: E (13. 5.), and consequently D7F (10. 5.). : Next, let A=C; then A: B:: C: B (7. 5.), but A: B:: D: E; therefore, CBD E, but C: B:: F: E, therefore, D : E : : F : E (11. 5.), and D F (9. 5.). Lastly, let A/C. Then C7A, and because, as was already shewn, C: B:: F: E, and B: A:: E: D; therefore, by the first case, if C7A, F7D, that is, if A/C, D/F. 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 is greater than the sixth; if equal, equal; and if less, less. If there be three magnitudes, A, B, C, and other three, D, E, and F, such that A: B: E: F, and B: C:: D: E; if A7C,D7F; if A=C, D=F; and if A/C, D/F. First, let: A 7C. Then A: B7C: B (8. 5.), but A: BE: F, therefore E: F7C: B (13.5.). Now, B: C:: D: E, and inversely, C: B:: E: D; therefore, E F7E: D (13. 5.), wherefore, D7F (10. 5.). : : A, B, C, D, E, F. Next, let A=C. Then (7. 5.) A: B:: C: B; but A: B:: E: F, therefore, C B:: E F (11. 5.); but B: C:: D: E, and inversely, C: BED, therefore (11. 5.), E: F:: E: D, and, consequently, D=F (9. 5.). Lastly, let A/C. Then C7A, and, as was already proved, C:B:: E: D; and B A :: F: E, therefore, by this first case, since C7A, F7 D, that is, DF. 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 will have to the last of the first magnitudes, the same ratio which the first of the other has to the last.* First, let there be three magnitudes, A, B, C, and other three, D, E, F, which, taken two and two, in order, have the same ratio, viz. A : B :: D : E, and B: C::E:F; then A : C : : D : F. A, B, C, D, E, F, mA, Take of A and D any equimultiples whatever, mA, mD; and of B and D any whatever, nB, nand of C and F any whatever, qC, qF. Because A: B:: D: E, mA : nB : : mD: nE (4. 5.); and for the same reason, nB: qC:: nE qF. Therefore (20. 5.) according as mA is greater than qC, equal to it, or less, mD is greater than qF, equal to it, or less; but mA, mD are any equimultiples of A and D; and qC, qF are any equimultiples of C and F; therefore :: D: F. Again, let there be four magnitudes, and other four which, taken two RB, qC, mD, nE, qF. (def. 5. 5.), A : C *N. B. This proposition is usually cited by the words "ex æquali," or 66 ex æquo." |