4 o 4 POPULAR SCIENCE MONTHLY 



of the elements E is indistinguishable from any element of the cut. 

 Or else, on the contrary, in each of the series E x , E 2 , '",E n satisfying 

 the first two conditions, there will be an element E indistinguishable 

 from one of the elements of the cut. In the first case we can go from 

 A to B by a continuous route without quitting C and without meeting 

 the cuts; in the second case that is impossible. 



If then for any two elements A and B of the continuum C, it is 

 always the first case which presents itself, we shall say that C remains 

 all in one piece despite the cuts. 



Thus, if we choose the cuts in a certain way, otherwise arbitrary, it 

 may happen either that the continuum remains all in one piece or that 

 it does not remain all in one piece; in this latter hypothesis we shall 

 then say that it is divided by the cuts. 



It will be noticed that all these definitions are constructed in setting 

 out solely from this very simple fact, that two manifolds of impressions 

 sometimes can be discriminated, sometimes can not be. That postu- 

 lated, if, to divide a continuum, it suffices to consider as cuts a certain 

 number of elements all distinguishable from one another, we say that 

 this continuum is of one dimension; if, on the contrary, to divide a 

 continuum, it is necessary to consider as cuts a system of elements 

 themselves forming one or several continua, we shall say that this con- 

 tinuum is of several dimensions. 



If to divide a continuum C, cuts forming one or several continua 

 of one dimension suffice, we shall say that C is a continuum of two 

 dimensions; if cuts suffice which form one or several continua of two 

 dimensions at most, we shall say that C is a continuum of three dimen- 

 sions; and so on. 



To justify this definition it is proper to see whether it is in this 

 way that geometers introduce the notion of three dimensions at the 

 beginning of their works. Now, what do we see ? Usually they begin 

 by defining surfaces as the boundaries of solids or pieces of space, lines 

 as the boundaries of surfaces, points as the boundaries of lines, and 

 they affirm that the same procedure can not be pushed further. 



This is just the idea given above: to divide space, cuts that are 

 called surfaces are necessary; to divide surfaces, cuts that are called 

 lines are necessary ; to divide lines, cuts that are called points are neces- 

 sary; we can go no further, the point can not be divided, so the point 

 is not a continuum. Then lines which can be divided by cuts which 

 are not continua will be continua of one dimension; surfaces which 

 can be divided by continuous cuts of one dimension will be continua 

 of two dimensions; finally space which can be divided by continuous 

 cuts of two dimensions will be a continuum of three dimensions. 



