But congruence, as thus expressed in terms of parallelism, merely establishes the congruent relation among straight routes on rects belonging to one parallel family, or on point-tracks belonging to one parallel family. For such routes in any one parallel family a system of numerical measurement can be established, of which the details need not be here elaborated. But no principle of comparison has yet been established between the lengths of two routes belonging to different parallel families of rects or belonging to different parallel families of point-tracks. When we can determine equal lengths on any two rects, whether parallel or no, the general principles for space-measurement will have been determined; and when we can determine equal lapses [i.e. lengths] of time on any two point-tracks, whether parallel or no, the general principles for time-measurement will have been determined. 48.3 Congruence as between different parallel families results from the following definition founded on the repetition property [i.e. symmetry] of normality : Let and be a pair of mutually normal rects intersecting at , or be a rect and point-track intersecting at [either or being the rect] and mutually normal, and let be the middle event-particle of the straight route intervening between the event-particles and , then the straight routes and are congruent to each other. From the symmetry of normality either both pairs of particles, namely () and (), are joined by rects, or both pairs are joined by point-tracks, or both pairs by null-tracks. As in the analogous case of congruence derived from parallelism, the transitiveness of congruence expresses a substantial law of nature and not a mere deduction from the terms of the definition. Fig. 12. 48.4 The isosceles triangle of 48.3 must lie either on a level or on a matrix. If it lies on a level, all the straight routes of the figure must lie on rects. But on a matrix a pair of normals cannot be of the same denomination, i.e. not both rects nor both point-tracks. Thus five cases remain over for consideration. These cases are diagrammatically symbolised by the annexed figures where continuous lines represent rects, and dotted lines represent point-tracks. Fig. 13. Evidently case (i) is the only case in which the triangle lies on a level: the triangles in the remaining four cases lie on matrices. The relations between the diagrams (ii) and (v) can best be seen by combining them into one figure as in (vi), and the relations between (iii) and (iv) by combining them into one figure as in (vii). Fig. 14. 48.5 Case (i) of 48.4 enables us to complete the congruence theory for spatial measurements. Let and be any two co-momental rects intersecting in the event-particle . Let be any particle on , and let ′ be the rect through parallel to .