84 POPULAR SCIENCE MONTHLY 



i transformed ' one into the other ; they are isomorphic. How are we 

 led to conclude thence that they are identical? 



Consider the two series a and S -\- <x -\- S' = a. I have said that 

 often, but not always, the series o- preserves the tactile impression A 

 felt by the finger D; and similarly it often happens, but not always, 

 that the series </ preserves the tactile impression A' felt by the ringer 

 D'. Now I ascertain that it happens very often (that is, much more 

 often than what I have just called ' often ') that when the series o- has 

 preserved the impression A of the finger D, the series a preserves at the 

 same time the impression A' of the finger D' ; and, inversely, that if 

 the first impression is altered, the second is likewise. That happens 

 very often, but not always. 



We interpret this experimental fact by saying that the unknown 

 object a which gives the impression A to the finger D is identical with 

 the unknown object a' which gives the impression A' to the finger D'. 

 And in fact when the first object moves, which the disappearance of the 

 impression A tells us, the second likewise moves, since the impression 

 A' disappears likewise. When the first object remains motionless, the 

 second remains motionless. If these two objects are identical, as the 

 first is at the point M of the first space and the second at the point N 

 of the second space, these two points are identical. This is how we 

 are led to regard these two spaces as identical; or better this is wbat 

 we mean when we say that they are identical. 



What we have just said of the identity of the two tactile spaces 

 makes unnecessary our discussing the question of the identity of tactile 

 space and visual space, which could be treated in the same way. 



§ 5. Space and Empiricism 



It seems that I am about to be led to conclusions in conformity with 

 empiristic ideas. I have, in fact, sought to put in evidence the role of 

 experience and to analyze the experimental facts which intervene in the 

 genesis of space of three dimensions. But whatever may be the im- 

 portance of these facts, there is one thing we must not forget and to 

 which besides I have more than once called attention. These experi- 

 mental facts are often verified but not always. That evidently does 

 not mean that space has often three dimensions, but not always. 



I know well that it is easy to save oneself and that, if the facts do 

 not verify, it will be easily explained by saying that the exterior objects 

 have moved. If experience succeeds, we say that it teaches us about 

 space ; if it does not succeed, we hie to exterior objects which we accuse 

 of having moved; in other words, if it does not succeed, it is given a 

 fillip. 



These fillips are legitimate; I do not refuse to admit them; but 

 they suffice to tell us that the properties of space are not experimental 

 truths, properly so called. If we had wished to verify other laws, we 



