552 POPULAR SCIENCE MONTHLY 



recognize how many dimensions any physical continuum has. It only 

 remains to find a physical continuum which is, so to speak, equivalent 

 to space, of such a sort that to every point of space corresponds an 

 element of this continuum, and that to points of space very near one 

 another correspond indistinguishable elements. Space will have then 

 as many dimensions as this continuum. 



The intermediation of this physical continuum, capable of repre- 

 sentation, is indispensable; because we can not represent space to our- 

 selves, and that for a multitude of reasons. Space is a mathematical 

 continuum, it is infinite, and we can represent to ourselves only physical 

 continua and finite objects. The different elements of space, which we 

 call points, are all alike, and, to apply our definition, it is necessary 

 that we know how to distinguish the elements from one another, at 

 least if they are not too close. Finally absolute space is nonsense, and 

 it is necessary for us to begin by referring space to a system of axes 

 invariably bound to our body (which we must always suppose put 

 back in the initial attitude). 



Then I have sought to form with our visual sensations a physical 

 continuum equivalent to space; that certainly is easy and this example 

 is particularly appropriate for the discussion of the number of dimen- 

 sions; this discussion has enabled us to see in what measure it is 

 allowable to say that ' visual space ' has three dimensions. Only this 

 solution is incomplete and artificial. I have explained why, and it is 

 not on visual space, but on motor space that it is necessary to bring 

 our efforts to bear. I have then recalled what is the origin of the 

 distinction we make between changes of position and changes of state. 

 Among the changes which occur in our impressions, we distinguish, 

 first the internal changes, voluntary and accompanied by muscular 

 sensations, and the external changes, having opposite characteristics. 

 We ascertain that it may happen that an external change may be cor- 

 rected by an internal change which reestablishes the primitive sensa- 

 tions. The external changes capable of being corrected by an internal 

 change are called changes of 'position, those not capable of it are called 

 changes of state. The internal changes capable of correcting an 

 external change are called displacements of the whole body; the others 

 are called changes of attitude. 



Now let a and ft be two external changes, a and ft' two internal 

 changes. Suppose that a may be corrected either by a or by ft', and 

 that a can correct either a or ft; experience tells us then that ft' can 

 likewise correct ft. In this case we say that a and ft correspond to the 

 same displacement and also that a and ft' correspond to the same dis- 

 placement. That postulated, we can imagine a physical continuum 

 which we shall call the continuum or group of displacements and which 

 we shall define in the following manner. The elements of this con- 

 tinuum shall be the internal changes capable of correcting an external 



