Hence distance T regarded as the sum of n and n is 2 and is real but segment 1 measured directly is imaginary.
Accordingly in order that all distances be tween real points shall be real, it is necessary to confine ourselves to one of the two segments into which the line is separated by the absolute or infinite points, F. and & Each of these segments is infinite in length, i.e., the distance from any point in either segment to either F is infinite, as before shown.
Let z be a point of either segment. Suppose the point to move in accordance with a trans formation that leaves F, and unchanged. We may name velocity of displacement the ratio of the distance traversed to the time consumed. The distance is of course, to be reckoned in I the new way. f now we suppose the point to move with a constant velocity toward F, or F,, it will approach nearer and nearer to F. or F. but will never reach either F. or F, since it would have to travel an infinite distance. If the point be supposed to be endowed with intelli gence, it could never by any motion possible to it or by any experience assure itself of the existence of or F,. If the point refused to assume the existence of F. and F,, its geometry would so far forth be Euclidean in character. It could assume the existence of F1 and F,. It could still choose between the supposition that the F's were coincident and the supposition that they were distinct. The former supposi tion would bring its geometry into the category of Parabolic geometries, like our own Euclidean geometry, while the second supposition would lead to a so-called Hyperbolic theory, such as the geometry of Lobachevsky and Bolyai.
Imaginary Absolute If the absolute elements are imaginary, let them be of the form a + fti and a Pi. The anharmonic ratio of these to a pair (s, se) of real elements is imaginary of the form Hence we have c log (s, a +Pi, se, a =-- c(0 2nir), which is pure imaginary or real according as c is taken to be real or pure imaginary. Hence in order that the distance between two real points may be real when the absolute points are conjugate imaginaries, it is necessary and sufficient to assign to c a pure value, say i c'. The absolute points being imaginary, the range contains no real infinite points, or points at co. Hence the range of straight line returns upon itself like a finite closed curve. The distance between any two of its points is periodic, of period 2ric=--2ttiic'=----27rci. Hence in this sys tem of measurement the length of the straight line is 27re. It is this metric theory that characterizes the so-called Elliptic Geometry of Riemann and Helmholz. The measurement of segments of the straight line of this theory is analogous to the ordinary measurement of arcs of a circle of radius c'.
Extension to the foregoing conclusions respecting the range are readily ex tensible to the pencil If the two absolute lines F, and F. be real and distinct, the angle made with either of them by any other real line of the pencil is infinite and real. If any line rotate in accordance with any transformation leaving F1 and F. unchanged, the line will move for ever without reaching either F1 or F.a fact that profoundly distinguishes this metric sys tem from the ordinary one, for in the latter if a line rotate with any constant finite velocity, no matter how small, it will in course of a finite time return to its initial position. This possi bility of thus returning is owing to the fact, already pointed out, that in ordinary angle measurement the pencil has no real lines at co but, on the contrary, these lines are conjugate imaginaries, viz., the familiar so-called isotropic lines of the pencil. If the absolute lines be conjugate imaginaries other than the isotropic lines of ordinary geometry, the resulting theory of angular measurement is that which belongs to the Elliptic Geometry. In this case, as in that of the range, we must take c to be pure imaginary in order that the angle between two real lines shall be real. If we set c= c'i, the angle between two lines is periodic, of period 2 or c', and the whole angle about the vertex will be 2r c'. In ordinary geometry the whole angle is or ; hence to pass from the elliptic theory of angle measurement to the ordinary system, we must let or c=± i :2. This done, the general expression for the angle between two fines (x'i, and (e es) becomes "x' x' x" where, as the absolute lines are to be the usual isotropic lines, we are to take axx=xi' + and thence SI +eo, a 9 9=ei. + and LI e e = xsixt1 + x', whence the expression for angle becomes e. 9. + x., cos + and this, as should be the case, is the familiar expression for the angle between two lines x + x'iy=0, x". x + x".y=0. We have thus the theorem: in the ordinary Euclidean geometry, the angle between two lines is times the logarithm of the anharmonic ratio of the given pair of lines to the pair of isotropic lines through their common point. This defini tion is due to Laguerre. (Consult Nouvelles annales de Mathiniatique, 1859). If the lines are perpendicular, i.e., have slopes m and 1/m respectively, then the anharmonic ratio in ques tion is 1 (tn, (m i) / m' (i+ (1+ m) = ; hence we have in ordinary Euclidean geometry the following definition of perpendicularity: Two lines, are perpendicular when and only when they are harmonic to the isotropic lines through their common point.