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.
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.
When we revert to experience for an understanding of congruence, we find it exemplified in the rigid bodies of nature, whose geometrical dispositions yield, more or less precisely, the results of pure Euclidean geometry. If we idealise congruence, as thus defined, and express it mathematically, we may say that perfectly rigid bodies are those whose measurements would yield Euclidean results with absolute precision. But though the mathematician has thereby eliminated from his definitions the inaccuracies attendant on physical measurements, his understanding of congruence reduces to a mere idealised copy of the behaviour of special bodies found in nature. While he has thus obtained a possible mathematical definition of congruent bodies (that given in nature), it remains to be seen whether other types of congruence would not also be rationally possible. His aim must therefore be to define congruence mathematically, without appealing to experience.
When, however, we discard the empirical criterion which prompted us to define material bodies as rigid, we find that a unique mathematical definition of rigidity eludes us. For to say that a body remains rigid or congruent with itself during displacement means that the spatial distance between its extremities remains ever the same. But our only means of disclosing this fact is by measuring the body with an admittedly rigid rod at successive intervals of time and noting the continued identity in our numerical results. Hence it follows that the value of these results would be nullified were we to cast any doubt on the maintenance of the rigidity of our measuring rod. And how could we ever justify its rigidity unless we were to compare it with some other rod regarded as rigid, and so on ad infinitum? From all this it appears that a body can be regarded as rigid only with respect to our measuring rod; and in order to ascribe any significance to rigidity we must first admit that our measuring rod is rigid by definition or by convention. We have no other means of establishing this rigidity.
We should reach the same conclusions were we to compare two lengths
and