a minimum. It may first be asked whether this reduction is possible, whether the number of necessary axioms and that of imaginable geometries are not infinite.
A theorem of Sophus Lie dominates this whole discussion. It may be thus enunciated:
Suppose the following premises are admitted:
1º Space has n dimensions;
2º The motion of a rigid figure is possible;
3º It requires p conditions to determine the position of this figure in space.
The number of geometries compatible with these premises will be limited.
I may even add that if n is given, a superior limit can be assigned to p.
If therefore the possibility of motion is admitted, there can be invented only a finite (and even a rather small) number of three-dimensional geometries.
Riemann's Geometries.—Yet this result seems contradicted by Riemann, for this savant constructs an infinity of different geometries, and that to which his name is ordinarily given is only a particular case.
All depends, he says, on how the length of a curve is defined. Now, there is an infinity of ways of defining this length, and each of them may be the starting point of a new geometry.
That is perfectly true, but most of these definitions are incompatible with the motion of a rigid figure, which in the theorem of Lie is supposed possible. These geometries of Riemann, in many ways so interesting, could never therefore be other than purely analytic and would not lend themselves to demonstrations analogous to those of Euclid.