644 SCIENCE AND HYPOTHESIS 



them, or we cannot; just as in Feehner's experiments the weight of 

 10 grammes could be distinguished from the ; weight of 12 grammes, 

 but not from the weight of 11 grammes. This is all that is required 

 to construct the continuum of several dimensions. 



Let us call one of these aggregates of sensations an element. It 

 will be in a measure analogous to the point of the mathematicians, 

 but will not be, however, the same thing. We cannot say that our 

 element has no size, for we cannot distinguish it from its immediate 

 neighbors, and it is thus surrounded by a kind of fog. If the astro- 

 nomical comparison may be allowed, our " elements " would be like 

 nebulae, whereas the mathematical points would be like stars. 



If this be granted, a system of elements will form a continuum if 

 we can pass from any one of them to any other by a series of consecu- 

 tive elements such that each cannot be distinguished from its prede- 

 cessor. This linear series is to the line of the mathematician what the 

 isolated element was to the point. 



Before going further, I must explain what is meant by a cut. Let us 

 consider a continuum C, and remove from it certain of its elements, 

 which for a moment we shall regard as no longer belonging to the 

 continuum. We shall call the aggregate of elements thus removed a 

 cut. By means of this cut, the continuum C will be subdivided into 

 several distinct continuums; the aggregate of elements which remain 

 will cease to form a single continuum. There will then be on C two 

 elements, A and B, which we must look upon as belonging to two dis- 

 tinct continuums; and we see that this must be so, because it will be 

 impossible to find a linear series of consecutive elements of C (each 

 of the elements indistinguishable from the preceding, the first being 

 A and the last B), unless one of the elements of this series is indis- 

 tinguishable from one of the elements of the cut. 



It may happen, on the contrary, that the cut may not be sufficient to 

 subdivide the continuum C. To classify the physical continuums, we 

 must first of all ascertain the nature of the cuts which must be made 

 in order to subdivide them. If a physical continuum, C, may be 

 subdivided by a cut reducing to a finite number of elements, all dis- 

 tinguishable the one from the other (and therefore forming neither 

 one continuum nor several continuums), we shall call C a continuum 

 of one dimension. If, on the contrary, C can only be subdivided by 

 cuts which are themselves continuums, we shall say that C is of 

 several dimensions; if the cuts are continuums of one dimension, then 

 we shall say that C has two dimensions ; if cuts of two dimensions are 

 sufficient, we shall say that C is of three dimensions, and so on. Thus 

 the notion of the physical continuum of several dimensions is defined, 

 thanks to the very simple fact, that two aggregates of sensations may 

 be distinguishable or indistinguishable. 



The Ufathematicdl Continuum of Several Dimensions. The con- 



