Take mA and nB any multiples of A and B, by the numbers m and n; and first let mA7nB: to each of them add mB, then mA+mB7 mB+nB. But mA+mB=m(A+B) (Cor. 1. 5.), and mB+nB= (m+n)B (2. Cor. 2. 5.), therefore m(A+B7(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, mC7nD. Therefore, when mA is greater than nB, mC is greater than nD. In like manner,it is demonstrated,that if mA=nB, mC=nD, and if mAZnB, that mDnD; therefore A: B:: C: D (def. 5. 5.). Therefore, &c. Q. E. D. PROP. XVIII. THEOR. 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 n7m, 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 mA7nB-mB, or mA7(nm)B(6. 5). But if mA7 (n- m) B, mC7 (n--m)D, because A: B::C:D Now, (n−m)D=nDmD (6. 5.), therefore, mC7nD-mD, and adding mD to both, mC+m D7D, that is,(1, 5.),m(C+D)7n.D. 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)≤nB,m(C+D) nD; therefore (def.5.5.), A+B: B::C+D: D. Therefore, &c. Q. E. 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, there fore (11. 5.) A-C: B-D:: A: B. Therefore, &c. Q. E. D. COR. A-C: B-D:: C: D. PROP, D. 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. If ABC: D, by conversion, A: A-B:: C: C-D. For, since A: B: CD, 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. Therefore, &c. Q. E. 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 EF, then if A7C, D7F; if AC, DF and if ALC, DLF. : A, B, D, D. E, F, First, let A7C; then A: B7C: B (8. 5.). But A: B::D: E, therefore also D: E7C: B (13. 5.). Now B: C::E: F,and inversely (A 5.), C: B :: F: E; and it has been shewn that D: EC : 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, C: B:: D: E, but C B:: F: E, therefore, D: E:: F: E(11. 5.), and DF (9. 5.). Lastly, let AC. Then C7A, and because, as was already shewn, C: B::F: E, and BA :: E:D; therefore, by the first case, if C7 A, F7D, that is, if AC, D4F. Therefore, &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 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 AC, D4F. A, B, C, F. First, let A7C. Then A: B7C: B (8.5.), but A: BE: F, therefore E: F7C B (13.5). Now, B: C:: D: E, and inversely, C: BED; therefore, E: F7E: D (13. 5.), wherefore, D7F (10. 5.). Next, let A=C. Then (7. 5.) A: B:: C: B; but A: BE: F, therefore, C: B:: E: F(11. 5.); but B: C:: D: E, and inversely, C: B:: E: D, therefore (11. 5.), E : F : : E : D, and, conse quently, D=F (9. 5.). Lastly, let AC. Then CA, and, as was already proved, C : B :: E: D; and B: A :: F: E, therefore, by the first case, since C7A F7D, that is, DF. Therefore, &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 will have to the last of the first magnitudes, the same ratio which the first of the others 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: BD: E, and B : C:: E: F; then A: C:: D: F. A, B, C, Take of A and D any equimultiples whatever, mA, mD; and of B and D any whatever, nB, nE: and 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: F. 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, mĎ are any equimultiples of A and D; and qC, qF are any and F ; therefore (def. 5. 5.), A: C:: D: F. D, E, F, mA, nB, qC, mD, nE. qF. equimultiples of C Again, let there be four magnitudes, and other four which, taken two and two in order, have the same ratio, viz. A: B :: E: F; B: CFG; C: D::G: H, then A: D:: E: H. For since A, B, C are three magnitudes, and E, F, G other three, which, taken two and two have the same ratio, by the foregoA:CE: G. And because also In the same ing case, CD::G: H, by that same case, A: DE: H. manner is the demonstration extended to any number of magnitudes. Therefore, &c. Q. E. D. *N. B. This proposition is usually cited by the words "ex æquali,” or “ex æque:”” Q 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 will have to the last of the first magnitudes the same ratio which the first of the others has to the last.* First, let there be three magnitudes, A. B, C, and other three, D, E, and F, which, taken two and two in a cross order, have the same ratio, viz. A: B : : E ; F, and B : C : : D : E, then A : C : : D : F. Take of A, B. and D, any equimultiples mA, mB, mD; and of C, E, F any equimultiples nC, nE, nF. : Because A: B:: E: F, and because also A : B :: mA: mB (15.5.), and EF nEnF; therefore, mA mB:: nE : nF (11. 5.). Again, because B: C:: D: E, mB : nC :: mD : nE (4. 5.); and it has been just shewn that mA : mB::nE: nF; therefore if mA7nC, mD 7nF (21. 5.); if mA=nC, mD=nF; and if mAnC, mDnF. Now, mA and mD are any equimulti A, B, C, D, E, F, mA, mB, nC, mD, nE, nF. ples of A and D, and nC, nF any equimultiples of C and F; therefore, AC: D: F (def. 5. 5.). Next, Let there be four magnitudes, A, B, C, and D, and other four, E, F, G, and H, which taken two and two, in a cross order, have the same ratio, viz. A: B:: G: H; B C:: A, B C, F: G, and C: D:: E: F, then A: D:: E : H. For, since 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: C:: F: H But C D E F, therefore, again, by the first case, A: DE: H. In the same manner, may the demonstration be extended to any number of magnitudes. Therefore, &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 ratio which the third and sixth together, have to the fourth. Let A B C : D, and also E: B:: F: D, then A+E : B :: C+F: D. : Because E: B:: FD, by inversion, B: E:: D: F. But by hypothesis, A B C D, therefore, ex æquali (22, 5.), A: E::C: F: and by composition (18. 5.), A+E: E::C+FF. And again by hypothesis, E BF: D, therefore, ex æquali (22. 5.), A+E: B:: C+F: D Therefore &c. Q. E. D. *N. B. This proposition is usually cited by the words "ex æquali in proportione pertur "bata ;" or, 66 ex æquo inversely." PROP. E. THEOR. If four magnitudes be proportionals, the sum of the first two is to their difference as the sum of the other two to their difference. 1 Let A B C D; then if A7B, A-BA-BC+D: C-D; or if AZB A+B B-A::C+D: D-C. For, if A7B, then because A: B:: C: D, by division (17. 5.), B: A-BD: C-D. But, by composition (18. 5.), In the same manner, if B7A, it is proved, that A+B: B~A::C+D: D−C. Q. E. D. Therefore, &c. PROP. F. THEOR. Ratios which are compounded of equal ratios, are equal to one another. Let the ratios of A to B, and of B to C, which compound the ratio of A to C, be equal, each to each, to the ratios of D to E, and E to F, which compound the ratio of D to F, A: C:: D : F. For, first, if the ratio of A to B be equal to that of D to E, and the ratio of B to C equal to that of E to F, ex æquali (22. 5.), A : C : : D : F. A, B, C, D, E, F. And next, if the ratio of A to B be equal to that of E to F, and the ratio of B to C equal to that of D to E, ex æquali inversely (23. 5.), A: C:: D: F. In the same manner may the proposition be demonstrated, whatever be the number of ratios. Therefore, &c. Q. E. D. |