The rect ′ occupies one straight line in the space of ; name this straight line ′. Then the straight lines and ′ will be said to be 'normal' to each other. This definition of the normality of straight lines can be given in general terms thus: Two straight lines in the same space are said to be normal to each other when they are respectively occupied by normal rects lying in the same moment of the corresponding time-system. 47.3 Continuing the notation of 47.2 let ′ be the level containing and ′; this level lies in and contains . Let ′ be the matrix normal to ′ at . Then ′ intersects in a rect ″ which is normal both to and to ′. Thus at an event-particle in a level pairs of mutually normal rects, ′ and ″, exist, one of them chosen arbitrarily; and at an event-particle in a moment triads of mutually normal rects, and ′ and ″, exist, with the usual conditions as to freedom of choice. The correspondence between a momentary space and the time-less space of the same time-system enables us immediately to extend these theorems to pairs of normal straight lines in a plane and to triads of intersecting mutually normal straight lines in three dimensions. 48. Congruence. 48.1 Congruence is founded on the notion of repetition, namely in some sense congruent geometric elements repeat each other. Repetition embodies the principle of uniformity. Now we have found repetition to be a leading characteristic of parallelism; accordingly a close connection may be divined to exist between congruence and parallelism. Furthermore we have just elaborated in outline the principles of normality, pointing out how the property has its origin in the interplay of the relations of extension and cogredience. But—as we know from experience—a leading property of normality is symmetry, namely, symmetry round the normal. Now symmetry is merely another name for a certain sort of repetition; accordingly congruence and normality should be connected. We are thus led to look for an expression of the nature of congruence in terms of parallelism and normality, in particular in terms of repetition properties associated with them. 48.2 Congruence, in so far as it is derived from parallelism, is defined by the statements that (i) the opposite sides of parallelograms are congruent to each other, and (ii) routes on the same rect, or on the same point-track, which are congruent to the same route are congruent to each other[7]. Also the general law holds that two routes which (as thus defined) are congruent to a third route, are congruent to each other. This law is a substantial theorem as to parallelism, and not a mere consequence of definitions.