660 SCIENCE AND HYPOTHESIS 



The first involuntary, unaccompanied by muscular sensations, and 

 attributed to external objects they are external changes ; the second, 

 of opposite character and attributed to the movements of our own 

 body, are internal changes. 



2. We notice that certain changes of each in these categories may 

 be corrected by a correlative change of the other category. 



3. We distinguish among external changes those that have a corre- 

 lative in the other category which we call displacements; and in 

 the same way we distinguish among the internal changes those which 

 have a correlative in the first category. 



Thus by means of this reciprocity is defined a particular class of 

 phenomena called displacements. The laws of these phenomena are 

 the object of geometry. 



Law of Homogeneity. The first of these laws is the law of homo- 

 geneity. Suppose that by an external change we pass from the aggre- 

 gate of impressions A to the aggregate B, and that then this change 

 a is corrected by a correlative voluntary movement ft, so that we are 

 brought back to the aggregate A. Suppose now that another external 

 change a brings us again from the aggregate A to the aggregate B. 

 Experiment then shows us that this change a', like the change a, may 

 be corrected by a voluntary correlative movement ft', and that this 

 Momevent ft' corresponds to the same muscular sensations as the 

 movement ft which corrected a. 



This fact is usually enunciated as follows : Space is homogeneous 

 and isotropic. We may also say that a movement which is once pro- 

 duced may be repeated a second and a third time, and so on, without 

 any variation of its properties. In the first chapter, in which we 

 discussed the nature of mathematical reasoning, we saw the import- 

 ance that should be attached to the possibility of repeating the same 

 operation indefinitely. The virtue of mathematical reasoning is due 

 to this repetition; by means of the law of homogeneity geometrical 

 facts are apprehended. To be complete, to the law of homogeneity 

 must be added a multitude of other laws, into the details of which I 

 do not propose to enter, but which mathematicians sum up by saying 

 that these displacements form a " group." 



The Non-Euclidean World. If geometrical space were a frame- 

 work imposed on each of our representations considered individually, 

 it would be impossible to represent to ourselves an image without this 

 framework, and we should be quite unable to change our geometry. 

 But this is not the case; geometry is only the summary of the laws 

 by which these images succeed each other. There is nothing, there- 

 fore, to prevent us from imagining a series of representations, similar 

 in every way to our ordinary representations, but succeeding one 

 another according to laws which differ from those to which we are 

 accustomed. We may thus conceive that beings whose education has 



