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 and 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.
Yet let no one say: But that proves geometry an experimental science; in separating its principles from laws whence they have been drawn, you artificially separate it itself from the sciences which have given birth to it. The other sciences have likewise principles, but that does not preclude our having to call them experimental.
It must be recognized that it would have been difficult not to make this separation that is pretended to be artificial. We know the role that the kinematics of solid bodies has played in the genesis of geometry; should it then be said that geometry is only a branch of experimental kinematics? But the laws of the rectilinear propagation of light have also contributed to the formation of its principles. Must geometry be regarded both as a branch of kinematics and as a branch of optics? I recall besides that our Euclidean space which is the proper object of geometry has been chosen, for reasons of convenience, from among a certain number of types which preexist in our mind and which are called groups.
If we pass to mechanics, we still see great principles whose origin is analogous, and, as their 'radius of action,' so to speak, is smaller, there is no longer reason to separate them from mechanics proper and to regard this science as deductive.
In physics, finally, the role of the principles is still more diminished. And in fact they are only introduced when it is of advantage. Now they are advantageous precisely because they are few, since each of them very nearly replaces a great number of laws. Therefore it is not of interest to multiply them. Besides an outcome is necessary, and for that it is needful to end by leaving abstraction to take hold of reality.
Such are the limits of nominalism, and they are narrow.
M. LeBoy has insisted, however, and he has put the question under another form.
