From all this rather long discussion on the subject of postulates and axioms we see that the axioms or postulates of geometry are most certainly not imposed upon us a priori in any unique manner. We may vary them in many ways and, as regards real space, our only reason for selecting one system of postulates rather than another (hence one type of geometry in preference to another) is because it happens to be in better agreement with the facts of observation when solid bodies and light rays are taken into consideration. Our choice is thus dictated by motives of a pragmatic nature; and the Kantians were most decidedly in the wrong when they assumed that the axioms of geometry constituted a priori synthetic judgments transcending reason and experience. [Pg 39] CHAPTER III RIEMANN’S DISCOVERIES AND CONGRUENCE THE procedure of presentation of non-Euclidean geometry which we have followed to this point hinges on the parallel postulate, hence on the definition of the straight line. In many respects, a much deeper method of investigation was that pursued by Riemann, founded on the concept of congruence. By congruence we mean the equality of two distances and more generally of two volumes in space. As we have explained elsewhere, the two methods lead to the same results. Indeed, once a metrical geometry has been defined, whether by the method of postulates or by any other means, a corresponding definition of a straight line and of equal or congruent distances is entailed thereby. Thus, with Euclidean geometry, congruent lengths at different places are exemplified by the lengths spanned by a material rod transported from one place to another. Congruent or rigid objects having thus been defined, a straight line is given by the axis of rotation of a material body, two of whose points are fixed, or again by the shortest distance between two points measured with our rigid rod. Nevertheless, although the various methods of presentation are equivalent, it may be of advantage to make the definition of congruence fundamental rather than that of the straight line. Such was the procedure followed by Riemann.