TEE VALUE OF SCIENCE 551 



can say is that experience has taught us that it is convenient to at- 

 tribute three dimensions to space. 



But visual space is only one part of space, and in even the notion 

 of this space there is something artificial, as I have explained at the 

 beginning. The real space is motor space and this it is that we shall 

 examine in the following chapter. 



Chapter IV. Space and its Three Dimensions 



§ 1. The Group of Displacements 



Let us sum up briefly the results obtained. We proposed to in- 

 vestigate what was meant in saying that space has three dimensions 

 and we have asked first what is a physical continuum and when it 

 may be said to have n dimensions. If we consider different systems 

 of impressions and compare them with one another, we often recognize 

 that two of these systems of impressions are indistinguishable (which 

 is ordinarily expressed in saying that they are too close to one another, 

 and that our senses are too crude, for us to distinguish them) and we 

 ascertain besides that two of these systems can sometimes be discrimi- 

 nated from one another though indistinguishable from a third system. 

 In that case we say the manifold of these systems of impressions forms 

 a physical continuum C. And each of these systems is called an 

 element of the continuum C. 



How many dimensions has this continuum? Take first two ele- 

 ments A and B of C, and suppose there exists a series 2 of elements, 

 all belonging to the continuum C, of such a sort that A and B are the 

 two extreme terms of this series and that each term of the series is 

 indistinguishable from the preceding. If such a series 2 can be found, 

 we say that A and B are joined to one another; and if any two elements 

 of C are joined to one another, we say that C is all of one piece. 



Now take on the continuum C a certain number of elements in a 

 way altogether arbitrary. The aggregate of these elements will be 

 called a cut. Among the various series 2 which join A to B, we shall 

 distinguish those of which an element is indistinguishable from one of 

 the elements of the cut (we shall say that these are they which cut the 

 cut) and those of which all the elements are distinguishable from all 

 those of the cut. If all the series 2 which join A to B cut the cut, we 

 shall say that A and B are separated by the cut, and that the cut 

 divides C. If we can not find on C two elements which are separated 

 by the cut, we shall say that the cut does not divide C. 



These definitions laid down, if the continuum C can be divided by 

 cuts which do not themselves form a continuum, this continuum C has 

 only one dimension ; in the contrary case it has several. If a cut form- 

 ing a continuum of 1 dimension suffices to divide C, C will have 2 

 dimensions; if a cut forming a continuum of 2 dimensions suffices, C 

 will have 3 dimensions, etc. Thanks to these definitions, we can always 



