548 POPULAR SCIENCE MONTHLY 



the position A to the position B I may take several routes. To the 

 first of these routes will correspond a series S of muscular sensations; 

 to a second route will correspond another series 8" of muscular sensa- 

 tions which generally will be completely different, since other muscles 

 will be used. 



How am I led to regard these two series S and S" as corresponding 

 to the same displacement AB? It is because these two series are 

 capable of correcting the same external change. Apart from that, 

 they have nothing in common. 



Let us now consider two external changes : a and /?, which shall be, 

 for instance, the rotation of a sphere half blue, half red, and that of a 

 sphere half yellow, half green; these two changes have nothing in 

 common, since the one is for us the passing of blue into red and the 

 other the passing of yellow into green. Consider, on the other hand, 

 two series of internal changes S and S" ; like the others, they will have 

 nothing in common. And yet I say that a and /? correspond to the 

 same displacement, and that 8 and S" correspond also to the same dis- 

 placement. Why? Simply because S can correct /? as well as a and 

 because a can be corrected by 8" as well as by 8. And then a question 

 suggests itself: If I have ascertained that 8 corrects a and /? and that 

 S" corrects a, am I certain that S" likewise corrects /?? Experiment 

 alone can teach us whether this law is verified. If it were not verified, 

 at least approximately, there would be no geometry, there would be no 

 space, because we should have no more interest in classifying the in- 

 ternal and external changes as I have just done, and, for instance, in 

 distinguishing changes of state from changes of position. 



It is interesting to see what has been the role of experience in all 

 this. It has shown me that a certain law is approximately verified. 

 It has not told me wherefore space is, and that it satisfies the condition 

 in question. I knew in fact, before all experience, that space satisfied 

 this condition or that it would not be ; nor have I any right to say that 

 experience told me that geometry is possible; I very well see that 

 geometry is possible, since it does not imply contradiction; experience 

 only tells me that geometry is useful. 



6. Visual Space 



Although motor impressions have had, as I have just explained, an 

 altogether preponderant influence in the genesis of the notion of space, 

 which never would have taken birth without them, it will not be 

 without interest to examine also the role of visual impressions and to 

 investigate how many dimensions ' visual space ' has, and for that 

 purpose to apply to these impressions the definition of § 3. 



A first difficulty presents itself: consider a red color sensation 

 affecting a certain point of the retina; and on the other hand a blue 

 color sensation affecting the same point of the retina. It is necessary 



