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 . Fig. 15. 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. CHAPTER XIII MOTION