SPACE 671 



used other criteria for instance, another finger or the sense of sight 

 but the first criterion is sufficient. I know that if it answers in the 

 affirmative all other criteria will give the same answer. I know 

 it from experiment. I cannot know it a priori. For the same reason 

 I say that touch cannot be exercised at a distance ; that is another way 

 of enunciating the same experimental fact. If I say, on the contrary, 

 that sight is exercised at a distance, it means that the criterion fur- 

 nished by sight may give an affirmative answer while the others reply 

 in the negative. 



To sum up. For each attitude of my body my finger determines a 

 point, and it is that and that only which defines a point 

 in space. To each attitude corresponds in this way a point. But 

 it often happens that the same point corresponds to several different 

 attitudes (in this case we say that our finger has not moved, but the 

 rest of our body has). We distinguish, therefore, among changes of 

 attitude those in which the finger does not move. How are we led to 

 this? It is because we often remark that in these changes the object 

 which is in touch with the finger remains in contact with it. Let us 

 arrange then in the same class all the attitudes which are deduced one 

 from the other by one of the changes that we have thus distinguished. 

 To all these attitudes of the same class will correspond the same point 

 in space. Then to each class will correspond a point, and to each 

 point a class. Yet it may be said that what we get from this experi- 

 ment is not the point, but the class of changes, or, better still, the 

 corresponding class of muscular sensations. Thus, when we say that 

 space has three dimensions, we merely mean that the aggregate of 

 these classes appears to us with the characteristics of a physical con- 

 tinuum of three dimensions. Then if, instead of defining the points 

 in space with the aid of the first finger, I use, for example, another 

 finger, would the results be the same? That is by no means a priori 

 evident. But, as we have seen, experiment has shown us that all our 

 criteria are in agreement, and this enables us to answer in the affirma- 

 tive. If we recur to what we have called displacements, the aggre- 

 gate of which forms, as we have seen, a group, we shall be brought 

 to distinguish those in which a finger does not move, and by what has 

 preceded, those are the displacements which characterize a point in 

 space, and their aggregate will form a sub-group of our group. To 

 each sub-group of this kind, then, will correspond a point in space. 

 We might be tempted to conclude that experiment has taught us the 

 number of dimensions of space; but in reality our experiments have 

 referred not to space, but to our body and its relations with neighbor- 

 ing objects. What is more, our experiments are exceeding crude. In 

 our mind the latent idea of a certain number of groups pre-existed; 

 these are the groups with which Lie's theory is concerned. Which 



