SPACE 663 



we study the laws according to which these operations are combined 

 we see that they form a group, which has the same structure as that 

 of the movements of invariable solids. Now, we have seen that it IB 

 from the properties of this group that we derive the idea of geomet- 

 rical space and that of three dimensions. We thus understand how 

 these perspectives gave rise to the conception of three dimensions, 

 although each perspective is of only two dimensions, because they 

 succeed each other according to certain laws. Well, in the same way 

 that we draw the perspective of a three-dimensional figure on a plane, 

 so we can draw that of a four-dimensional figure on a canvas of three 

 (or two) dimensions. To a geometer this is but child's play. We can 

 even draw several perspectives of the same figure from several different 

 points of view. We can easily represent to ourselves these perspective, 

 since they are of only three dimensions. Imagine that the different 

 perspectives of one and the same object occur in succession, and 

 that the transition from one to the other is accompanied by muscular 

 sensations. It is understood that we shall consider two of these tran- 

 sitions as two operations of the same nature when they are associated 

 with the same muscular sensations. There is nothing, then, to prevent 

 us from imagining that these operations are combined according to 

 any law we choose for instance, by forming a group with the same 

 structure as that of the movements of an invariable four-dimensional 

 solid. In this there is nothing that we cannot represent to ourselves, 

 and, moreover, these sensations are those which a being would experi- 

 ence who has a retina of two dimensions, and who may be displaced 

 in space of four dimensions. In this sense we may say that we can 

 represent to ourselves the fourth dimension. 



Conclusions. It is seen that experiment plays a considerable role 

 in the genesis of geometry ; but it would be a mistake to conclude from 

 that that geometry is, even in part, an experimental science. If it were 

 experimental, it would only be approximate and provisory. And 

 what a rough approximation it would be! Geometry would be only 

 the study of the movements of solid bodies; but, in reality, it is not 

 concerned with natural solids: its object is certain ideal solids, abso- 

 lutely invariable, which are but a greatly simplified and very remote 

 image of them. The concept of these ideal bodies is entirely mental, 

 and experiment is but the opportunity which enables us to reach the 

 idea. The object of geometry is the study of a particular " group " ; 

 but the general concept of group pre-exists in our minds, at least 

 potentially. It is imposed on us not as a form of our sensitiveness, but 

 as a form of our understanding ; only, from among all possible groups, 

 we must choose one that will be the standard, so to speak, to which 

 we shall refer natural phenomena. 



Experiment guides us in this choice, which it does not impose on 

 us. It tells us not what is the truest, but what is the most convenient 



