47. Normality. 47.1 A point-track will be said to be 'normal' to the moments of the time-system in the space of which it is a point.
A matrix is said to be 'normal' to the moments which are normal to any of the point-tracks which it contains.
Consider an event-particle and a matrix which contains . Let , ... be the collinear set of time-systems whose points lie in or are parallel to the matrix . Let , ... be the moments of the time-systems ... which contain . Then the levels , ... in which respectively and , and , etc., intersect are identical, and the event-particle is the sole event-particle forming the intersection of and . Also intersects each of these moments , and , and , etc., in rects , etc., respectively. The level and the matrix are said to be mutually 'normal.' It will be noted that any two time-systems, and , determine one level and one matrix which are mutually normal and each contain a given event-particle. Corresponding to any level containing there is one matrix normal to it at ; and corresponding to any matrix containing there is one level normal to it at .
If and be a level and a matrix normal to each other, then the rects in will be called normal to the rects and point-tracks in . A pair of rects which are normal to each other will also be called 'perpendicular' or 'at right-angles.' Two point-tracks can never be normal to each other since no point-track lies on a level. Parallels to normals are themselves normal.
47.2 Continuing the notation of 47.1 we note that the matrix includes straight lines , etc., of the spaces of , etc., and intersects the moments , etc., in rects , etc., which respectively occupy , etc. The rect contains and is normal to every rect lying in . Let ′ be any rect containing and lying in . Then ′ and are mutually normal and both lie in the moment .
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.
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.
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.
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).
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 .
Now assume that it is possible to find one pair of mutually normal rects, and ′ intersecting each other at , and respectively intersecting ′ at and ′, where . Through draw ″ parallel to ′ and intersecting in ″; and through draw ′ parallel to and intersecting in ″.
Then from 48.1, and . Thus ′ and ″ denote the same event-particle. Now . Hence by case (i) of 48.4, . Thus lengths on and are comparable. We need not here consider the theorems, either assumed as independent laws of nature or deduced from previous assumptions, by which we know that the rectangular pair ( and ′) exist, that ′ and ″ coincide and do not lie on opposite sides of , and that .
48.6 Again if and are rects which are not comomental and do not lie in parallel moments, their measurements are still comparable. For two intersecting moments, and , exist, of which contains and contains .
Thus any rect ′ in the level common to and has its measurements comparable both to those on and to those on ; and thus, by the transitiveness of congruence, the measurements on and are comparable. By this procedure the employment of cases (ii) and (iii) of 48.4 is rendered unnecessary. Accordingly these cases become theorems instead of being definitions of congruence as contemplated in their original enunciation. If they had been taken as definitions, the deduction of 48.5 would still be possible. But since the figure would now lie in a matrix, one of and ′ would be a point-track and the other a rect. No very obvious principle then exists by which we could know of the existence of the pair ( and ′) such that apart from the assumption of the theorem which we want to prove.
48.7 Cases (iv) and (v) of 48.4 deal with the comparability of time-measurements in different time-systems. The same remarks as those in 48.6 apply; namely that the method of 48.5 could be applied, if independently we could convince ourselves that the requisite pair, and ′ (one a point-track and one a rect), exist.
This comparability of time-measurements will be achieved by another method which depends on the fact that relative velocity is equal and opposite. The explanation of this method must be reserved for the next chapter.
[7]This definition of congruence is given by Profs. E. B. Wilson and G. N. Lewis in their valuable memoir, 'The Space-Time Manifold of Relativity,' Proc. of the Amer. Acad. of Arts and Sciences, vol. XLVIII, 1912.