THE VALUE OF SCIENCE 83 



chosen, I should so make this choice that these movements carry the 

 finger D to the point originally occupied by the finger D', that is, to 

 the point M; this finger D will thus be in contact with the object a, 

 which will make it feel the impression A. 



I then make the movements corresponding to the series <r; in these 

 movements, by hypothesis, the position of the finger D does not change, 

 this finger therefore remains in contact with the object a and con- 

 tinues to feel the impression A. Finally I make the movements cor- 

 responding to the series 8'. As 8' is inverse to S, these movements 

 carry the finger D' to the point previously occupied by the finger D, 

 that is, to the point M. If, as may be supposed, the object a has not 

 budged, this finger D' will be in contact with this object and will feel 

 anew the impression A'. . . . Q. E. D. 



Let us see the consequences. I consider a series of muscular sensa- 

 tions 2. To this series will correspond a point M of the first tactile 

 space. Now take again the two series s and s', inverses of one another, 

 of which we have just spoken. To the series s -f- 2 + s' will corre- 

 spond a point N of the second tactile space, since to any series of 

 muscular sensations corresponds, as we have said, a point, whether in 

 the first space or in the second. 



I am going to consider the two points N and M, thus defined, as 

 corresponding. What authorizes me so to do? For this correspond- 

 ence to be admissible, it is necessary that if two points M and M r , 

 corresponding in the first space to two series 2 and 2', are identical, 

 so also are the two corresponding points of the second space N and N', 

 that is the two points which correspond to the two series s -j- 2 + s' and 

 s -f- 2' -f- s'. Now we shall see that this condition is fulfilled. 



First a remark. As 8 and S' are inverses of one another, we shall 

 have 8 + S' = 0, and consequently 8 + S' + 2 = 2 + S + S' = 2, or 

 again 2 + 8 + 8' + 2' = 2 + 2' ; but it does not follow that we have 

 8 -f- 2 -f- S' = 2 ; because, though we have used the addition sign to 

 represent the succession of our sensations, it is clear that the order of 

 this succession is not indifferent: we can not, therefore, as in ordinary 

 addition, invert the order of the terms; to use abridged language, our 

 operations are associative, but not commutative. 



That fixed, in order that 2 and 2' should correspond to the same 

 point M = M' of the first space, it is necessary and sufficient for us to 

 have 2' = 2 + o-. We shall then have : 8 + 2' + 8' = 8 + 2 + a + 

 8' = 8 + 2 + 8' + S + a + S'. 



But we have just ascertained that S -f- a -f- 8' was one of the series 

 </. We shall therefore have : 8 + 2' + S' = 8 + 2 + 8' + a', which 

 means that the series S + 2' + S' and 8 + 2 + #' correspond to the 

 same point N = N' of the second space. Q. E. D. 



Our two spaces therefore correspond point for point; they can be 



