652 SCIENCE AND HYPOTHESIS 



of these theorems, and I shall not choose the least remarkable of 

 them. A real straight line may be perpendicular to itself. 



Lie's Theorem. The number of axioms implicity introduced into 

 classical proofs is greater than necessary, and it would be interesting 

 to reduce them to a minimum. It may be asked, in the first place, if 

 this reduction is possible if the number of necessary axioms and 

 that of imaginable geometries is not infinite? A theorem due to 

 Sophus Lie is of weighty importance in this discussion. It may be 

 enunciated in the following manner : Suppose the following prem- 

 isses are admitted: (1) space has n dimensions; (2) the movement of 

 an invariable figure is possible; (3) p conditions are necessary to 

 determine the position of this figure in space. 



The number of geometries compatible with these premisses 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 the movement is granted, 

 we can only invent a finite and even a rather restricted number of 

 three-dimensional geometries. 



Riemann's Geometries. However, this result seems contradicted 

 by Eiemann, for that scientist constructs an infinite number of geo- 

 metries, and that to which his name is usually attached is only a par- 

 ticular case of them. All depends, he says, on the manner in which 

 the length of a curve is defined. Now, there is an infinite number 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 movement of a variable figure 

 such as we assume to be possible in Lie's theorem. These geometries 

 of Eiemann, so interesting on various grounds, can never be, there- 

 fore, purely analytical, and would not lend themselves to proofs analo- 

 gous to those of Euclid. 



On the Nature of Axioms. Most mathematicians regard Lobat- 

 schewsky's geometry as a mere logical curiosity. Some of them have, 

 however, gone further. If several geometries are possible, they say, 

 is it certain that our geometry is the one that is true ? Experiment no 

 doubt teaches us that the sum of the angles of a triangle is equal to 

 two right angles, but this is because the triangles we deal with are 

 too small. According to Lobatschewsky, the difference is proportional 

 to the area of the triangle, and will not this become sensible when we 

 operate on much larger triangles, and when our measurements become 

 more accurate? Euclid's geometry would thus be a provisory geo- 

 metry. Now, to discuss this view we must first of all ask ourselves, 

 what is the nature of geometrical axioms? Are they aynthetic 

 a priori intuitions, as Kant affirmed? They would then be imposed 

 upon us with such a force that we could not conceive of the contrary 

 proposition, nor could we build upon it a theoretical edifice. There 

 would be no non-Euclidean geometry. To convince ourselves of this, 



