; 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 (