Riemann.
To prove that a perpendicular may always be erected at a point A to a straight AB, we consider a straight AC movable around the point A and initially coincident with the fixed straight AB; and we make it turn about the point A until it comes into the prolongation of AB.
Thus two propositions are presupposed: First, that such a rotation is possible, and next that it may be continued until the two straights come into the prolongation one of the other.
If the first point is admitted and the second rejected, we are led to a series of theorems even stranger than those of Lobachevski and Riemann, but equally exempt from contradiction.
I shall cite only one of these theorems and that not the most singular: A real straight may be perpendicular to itself.
Lie's Theorem.—The number of axioms implicitly introduced in the classic demonstrations is greater than necessary, and it would be interesting to reduce it to 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,