A, B, C, First, let A7C. Then A: B>C: B (8. 5.), but A: B:: E: F, therefore F: F> C: B (13. 5.). Now, B:C:: D: E, and inversely, C: B:: E: D; therefore, E: F7E: D (13.5.), wherefore, D>F (10.5.). 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: B:: E: D, therefore (11.5.), E: F: : E: D, and, consequently, D=F (9.5.). Lastly, let A/C. Then C>A, and, as was already proved, C: В :: E:D:and B: A:: F: E, therefore, by the first case, since C>A, F>D, that is, DL F. 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 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, B, C, Take of A and D any equimultiples whatever, mA, mD; and of Band Dany whatever, B, 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:: E:gF. Therefore (20. 5.), according as mA is greater than gC, equal to it, or less, mD is greater than gF, equal to it, or less; but mA, mD are any equimultiples of A and D; and gC, qF are any equimultiples of C and F; therefore (def. 5. 5.), A:C:: D: F. mD, nE, 9F. Again, let there he four magnitudes, and other four which, taken two and two in order, have the same ratio, viz. A: B:: E: F; B: C::F:G;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 forego ing case, A: C::E:G. And because also A, B, C, D, E, F, G, H. C:D::G: H, by that same case, A:D :: E: H. In the same 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 aquali, or ex æquo. 3 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 C, E, nF. Because A: 15:: E: F, and because also A: B:: mA:mB (15.5.), and E: F:: nE: nF; 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 mA>nC,mD>nF (21.5.); if mA=nC,mD=nF; and if mA<nC, A, B, C, D, E, F, mA, MB, nC, mD, nE, nF. mD<nF. Now, mA and mD are any equimultiples of A and D, and nC, nF any equimultiples of C and F; therefore, A:C:: 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:: 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 A, B, C, D, E, F, G, H, 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: D:: E: H. In the same manner may the demonstration be extended to any number of magnitudes. Therefore, &c. Q. E. D. 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:: F: D, 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+F:F. And again by hypothesis, E: B:: F: 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 perturbata; or, ea æque 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. Let A: B::C: D; then if A>B, A+B:A-B::C+D:C-D; or if A<B A+B:B-A::C+D:D-С. For, if A>B, then because A: B:: C: D, by division (17. 5.), A-B:B::C-D: D, and by inversion (A. 5.), B:A-B::D:C-D. But, by composition (18. 5.), A+B:B::C+D: D, therefore, ex æquali (22. 5.), A+B:A-B::C+D:C-D. In the same manner, if B>A, it is proved, that A+B:B-A::C+D:D-C. Therefore, &c. Q. E. D. 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, 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 aquali inversely (23. 5.), A: C::D: F. In the same manner may the proposition be demonstrat. ed, whatever be the number of ratios. Therefore, &c. Q. E. D. Two sides of one figure are said to be reciprocally proportional to two sides of another, when one of the sides of the first is to one of the sides of the second, as the remaining side of the second is to the remaining side of the first. III. A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment, as the greater segment is to the less. IV. The altitude of any figure is the straight line drawn from its vertex perpendicular to the base. Triangles and parallelograms, of the same altitude, are one to another as their bases. Let the triangles ABC, ACD, and the parallelograms EC, CF have the same altitude, viz. the perpendicular drawn from the point A to BD: Then, as the base BC, is to the base CD, so is the triangle ABC to the triangle ACD, and the parallelogram EC to the parallelogram CF. Produce BD both ways to the points H, L, and take any number of straight lines BG, GH, each equal to the base BC; and DK, KL, any number of them, each equal to the base CD; and join AG, AH, AK, AL. Then, because CB, BG, GH are all equal, the triangles AHG, AGB, ABC are all equal (38.1.): Therefore, whatever multiple the base HC is of the base BC, the same multiple is the triangle AHC of the triangle ABC. For the same reason, whatever the base LC is of the base CD, the same multiple is the triangle ALC of the triangle ADC. But if the base HC be equal to the base CL, the triangle AHC is also equal to the triangle ALC (38. 1.): and if the base HC be greater than EAF the base CL, likewise the HGBCDK triangle AHC is greater L than the triangle ALC; and if less, less. Therefore, since there are four magnitudes, viz. the two bases BC, CD, and the two triangles ABC, ACD; and of the base BC and the triangle ABC, the first and third, any equimultiples whatever have been taken, viz. the base HC, and the triangle AHC; and of the base CD and triangle ACD, the second and fourth, have been taken any equimultiples whatever, viz. the base CL and triangle ALC; and since it has been shown, that if the base HC be greater than the base CL, the triangle AHC is greater than the triangle ALC; and if equal, equal; and if less, less: Therefore (def. 5.5.), as the base BC is to the base CD, so is the triangle ABC to the triangle ACD. And because the parallelogram CE is double of the triangle ABC (41.1.), and the parallelogram CF double of the triangle ACD, and because magnitudes have the same ratio which their equimultiples have (15.5.); as the triangle ABC is to the triangle ACD, so is the parallelogram EC to the parallelogram CF. And because it has been shewn, that, as the base BC is to the base CD, so is the triangle ABC to the triangle ACD; and as the triangle ABC to the triangle ACD, so is the parallelogram EC to the parallelogram CF: therefore, as the base BC is to the base CD, so is (11.5.) the parallelogram EC to the parallelogram CF. Wherefore triangles, &c. Q. E. D. COR. From this it is plain, that triangles and parallelograms that have equal altitudes, are to one another as their bases. Let the figures be placed so as to have their bases in the same straight line; and having drawn perpendiculars from the vertices of the triangles to the bases, the straight line which joins the vertices is parallel to that in which their bases are (33. 1.), because the perpendiculars are both equal and parallel to one another. Then, if the |