The principle, henceforth crystallized, so to speak, is no longer subject to the test of experiment. It is not true or false, it is convenient.
Great advantages have often been found in proceeding in that way, but it is clear that if all the laws had been transformed into principles nothing would be left of science. Every law may be broken up into a principle and a law, but thereby it is very clear that, however far this partition be pushed, there will always remain laws.
Nominalism has therefore limits, and this is what one might fail to recognize if one took to the very letter M. LeRoy's assertions.
A rapid review of the sciences will make us comprehend better what are these limits. The nominalist attitude is justified only when it is convenient; when is it so?
Experiment teaches us relations between bodies; this is the fact in the rough; these relations are extremely complicated. Instead of envisaging directly the relation of the body A and the body B, we introduce between them an intermediary, which is space, and we envisage three distinct relations: that of the body A with the figure A´ of space, that of the body B with the figure B´ of space, that of the two figures A´ and B´ to each other. Why is this detour advantageous? Because the relation of A and B was complicated, but differed little from that of A´ and B´, which is simple; so that this complicated relation may be replaced by the simple relation between A´ and B´ and by two other relations which tell us that the differences between A and A´, on the one hand, between B and B´, on the other hand, are very small. For example, if A and B are two natural solid bodies which are displaced with slight deformation, we envisage two movable rigid figures A´ and B´. The laws of the relative displacement of these figures A´ and B´ will be very simple; they will be those of geometry. And we shall afterward add that the body A, which always differs very little from A´, dilates from the effect of heat and bends from the effect of elasticity. These dilatations and flexions, just because they are very small, will be for our mind relatively easy to study. Just imagine to what complexities of language it would have been necessary to be resigned if we had wished to comprehend in the same enunciation the displacement of the solid, its dilatation and its flexure?
The relation between A and B was a rough law, and was broken up; we now have two laws which express the relations of A and A´, of B and B´, and a principle which expresses that of A´ with B´. It is the aggregate of these principles that is called geometry.
Two other remarks. We have a relation between two bodies A and B, which we have replaced by a relation between two figures A´ and B´; but this same relation between the same two figures A´ and B´ could just as well have replaced advantageously a relation between two other bodies A´´ and B´´, entirely different from A and B. And that in many ways. If the principles of geometry had not been invented, after having studied the relation of A and B, it would be necessary to begin again ab ovo the study of the relation of A´´ and B´´. That is why geometry is so precious. A geometrical relation can advantageously replace a relation which, considered in the rough state, should be regarded as mechanical, it can replace another which should be regarded as optical, etc.