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, 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.