82 POPULAR SCIENCE MONTHLY 



vealed to us by our muscular sensations; but nothing tells us the 

 initial situation; nothing can distinguish it for us from all the other 

 possible situations. This puts well in evidence the essential relativity 

 of space. 



§ 4. Identity of the Different Spaces 



We are therefore led to compare the two continua C and C engen- 

 dered, for instance, one by my first finger D, the other by my second 

 finger D'. These two physical continua both have three dimensions. 

 To each element of the continuum C, or, if you prefer, to each point of 

 the first tactile space, corresponds a series of muscular sensations 2, 

 which carry me from a certain initial situation to a certain final situa- 

 tion. 1 Moreover, the same point of this first space will correspond to 

 2 and to 2 -\- a-, if o- is a series of which we know that it does not make 

 the finger D move. 



Similarly to each element of the continuum C, or to each point of 

 the second tactile space, corresponds a series of sensations 2', and the 

 same point will correspond to 2' and to 2' + o-', if o-' is a series which 

 does not make the finger D' move. 



What makes us distinguish the various series designated o- from 

 those called o-' is that the first do not alter the tactile impressions felt 

 by the finger D and the second preserve those the finger D' feels. 



Now see what we ascertain: in the beginning my finger D' feels a 

 sensation A' ; I make movements which produce muscular sensations S; 

 my finger D feels the impression A; I make movements which produce 

 a series of sensations o- ; my finger D continues to feel the impression A , 

 since this is the characteristic property of the series o-; I then make 

 movements which produce the series S' of muscular sensations, inverse 

 to S in the sense above given to this word. I ascertain then that my 

 finger D' feels anew the impression A'. (It is of course understood 

 that S has been suitably chosen.) 



This means that the series s-f-ff-j-s', preserving the tactile im- 

 pressions of the finger I)' , is one of the series I have called o-'. In- 

 versely, if one takes any series a, s' -+- a -f- s will be one of the series 

 that we call <r. 



Thus if s is suitably chosen, s -+- o- -f- s f will be a series a, and by 

 making a vary in all possible ways, we shall obtain all the possible 

 series a . 



Not yet knowing geometry, we limit ourselves to verifying all that, 

 but here is how those who know geometry would explain the fact. In 

 the beginning my finger D' is at the point M, in contact with the object 

 a, which makes it feel the impression A'. I make the movements cor- 

 responding to the series S; I have said that this series should be suitably 



1 In place of saying that we refer space to axes rigidly bound to our body, 

 perhaps it would be better to say, in conformity to what precedes, that we refer 

 it to axes rigidly bound to the initial situation of our body. 



