32. We obtain from secs. 28-30 the formulas for the Loba i chevskian geometry by placing m= and replacing the trig k It is worth noticing that the formulas for the Euclidean trigonometry come out of those in sec. 31 or sec. 32 as limit cases when k = (cf. also sec. 43). B 33. The formulas of sec. 32 may be used to obtain an expression for the angle of parallelism belonging to a distance x. Let BM (Fig. 28) be parallel to CN and BC perpendicular to CN. The figure NCBM may be regarded as the limit of a right triangle ABC in which M N A FIG. 28. BC=x is constant, 4 approaches zero and B approaches II (x). 34. If we substitute in the formulas of sec. 32 the values found in sec. 33, and make certain simple reductions, the formulas of sec. 32 take the following forms: These are the forms found by Lobachevsky, except that he writes A-II(a), B=II (3), where a and 3 are the distances corresponding to the angles of parallelism A and B respectively. We shall make no use of these equations, but have given them to facilitate comparison with Lobachevsky's own work. с A FIG. 29. B 35. The above formulas are for right triangles. We shall now obtain one for oblique triangles. Let ABC (Fig. 29) be any triangle with the angles A, B, and C, and the opposite sides a, b, and c, respectively. Draw BD perpendicular to AC and let BD=h, AD=k. Then cos ma=cos mh cos m(k−b) =cos me cos mb +sin mb sin mk cos mh IX. NON-EUCLIDEAN ANALYTIC GEOMETRY 36. Let OX and OY (Fig. 30) be two axes of coordinates intersecting at right angles and MP and NP the perpendiculars from any point P to OX and OY respectively. We shall take OM = x, ON = y as the coordinates of P. set of coordinates (x, y) To every point P corresponds a single and to any set of coordinates corresponds not more than one point P. But if x and y are assumed arbitrarily there is not necessarily a corresponding point P in the Lobachevskian geometry, since the two perpendiculars at M and N may be parallel or non-intersecting. Between the two sets of coordinates there exist, in either the Riemannian or the Lobachevskian geometry, the relations (sec. 29) whence tan mx-tan mr cos 0, tan my=tan mr sin 0, tan2 mx+tan2 my=tan2 mr. 37. The equation of a straight line may be obtained as follows: Let LK (Fig. 31) be any straight line determined by the parameters p and a, where p is the length of the perpendicular OD from the origin and a the angle made by OD with the positive direction of OX. Let P(x, y) be any point on LK and draw OP. Then in the triangle OPD, where (r, 0) are the polar coordinates of P. Hence (sec. 29) tan mr cos (-a) = tan mp, whence (sec. 36) tan mx cos a +tan my sin a=tan mp, the required equation. 38. The distance between two points may be found as follows: Let P1(x1,y1) and P2(x2, y2) (Fig. 32) be any two points with the polar coordinates (r1, 01) and (r2, 02) respectively. Draw OP1, OP2, and P1P2. Then in the triangle OP1P2 OP1=r1, OP2=r2, 4P20P1=01-02. cos m P1P2 = cos mr1 cos mr2+sin mr1 sin mr2 cos (01 − 0.2) = cos mr1 cos mr2[1+tan mr1 tan mr2 cos (01-02)]. By use of the formulas of sec. 36, this reduces readily to 1+ tan mx1 tan mx2 + tan my tan my 2 cos m PIP2 √1+tan2 mx1+tan2 my1√1+tan2 mx2 +tan2 my2 the required formula. 39. The angle between two lines may be determined as follows: Let PL and PL2 (Fig. 33) be two straight lines intersecting at P. Draw from O the two perpendiculars OD1 and OD2 on PL1 and PL2 respectively, and (as in sec. 37), let OD1 p1, XOD1 =α1, OD2=P2, 4XOD2=α2. Draw OP and place OP=r, 4XOP=0, 40PD1=ß1, 40PD2=ẞ2, and 4L1PL2=4=ñ− (31+ ß2). Now from the right triangles OPD, and OPD2, we have cosp=cosmpi cos mp2 sin (0-a1) sin (0-a2) + cos (0-α1) tan mr = tan mpi, 0=cos mpi cos mp2 cos (0-α1) cos (0-α2) — sin mp, sin mp2 Adding this equation to equation (1), we have - tan2 mr cos = cos mp1 cos mp2 cos (α1− α2)+sin sin cos a1 cos a2+ sin a1 sin a2 +tan mp, tan mp2 √1+tan2 mp1 V1+tan2 mp2 which gives the required angle in terms of the functions which enter into the equations of the lines. 40. The formulas of secs. 36-39 apply to either the Riemannian or the Lobachevskian geometry. It is now convenient to separate the two cases. In the Riemannian geometry, where m = instead of x and y, the new coordinates 1 we will introduce, k' η, and 7, where x = k tan k' y 7=k tan k' (1) |