SPACE 651 



would be preferable, but then we should have to enunciate the axiom 

 explicitly. Other definitions may give rise to no less important reflec- 

 tions, such as, for example, that of the equality of two figures. Two 

 figures are equal when they can be superposed. To superpose them, 

 one of them must be displaced until it coincides with the other. But 

 how must it be displaced? If we asked that question, no doubt we 

 should be told that it ought to be done without deforming it, and as 

 an invariable solid is displaced. The vicious circle would then be 

 evident. As a matter of fact, this definition defines nothing. It has 

 no meaning to a being living in a world in which there are only fluids. 

 If it seems clear to us, it is because we are accustomed to the pro- 

 perties of natural solids which do not much differ from those of the 

 ideal solids, all of whose dimensions are invariable. However, im- 

 perfect as it may be, this definition implies an axiom. The possi- 

 bility of the motion of an invariable figure is not a self-evident truth. 

 At least it is only so in the application to Euclid's postulate, and not 

 as an analytical a priori intuition would be. Moreover, when we study 

 the definitions and the proofs of geometry, we see that we are com- 

 pelled to admit without proof not only the possibility of this motion, 

 but also some of its properties. This first arises in the definition of 

 the straight line. Many defective definitions have been given, but 

 the true one is that which is understood in all the proofs in which the 

 straight line intervenes. " It may happen that the motion of an in- 

 variable figure may be such that all the points of a line belonging to 

 the figure are motionless, while all the points situate outside that line 

 are in motion. Such a line would be called a straight line." "We 

 have deliberately in this enunciation separated the definition from the 

 axiom which it implies. Many proofs such as those of the cases of 

 the equality of triangles, of the possibility of drawing a perpen- 

 dicular from a point to a straight line, assume propositions the 

 enunciations of which are dispensed with, for they necessarily imply 

 that it is possible to move a figure in space in a certain way. 



The Fourth Geometry. Among these explicit axioms there is one 

 which seems to me to deserve some attention, because when we aban- 

 don it we can construct a fourth geometry as coherent as those of 

 Euclid, Lobatschewsky, and Eiemann. To prove that we can always 

 draw a perpendicular at a point A to a straight line A B, we consider 

 a straight line A C movable about the point A, and initially identical 

 with the fixed straight line A B. We then can make it turn about the 

 point A until it lies in A B produced. Thus we assume two propo- 

 sitions first, that such a rotation is possible, and then that it may 

 continue until the two linos lie the one in the other produced. If 

 the first point is conceded and the second rejected, we are led to a 

 series of theorems even stranger than those of Lobatschewsky and 

 Riemann, but equally free from contradiction. I shall give only one 



