of the elements E is indistinguishable from any element of the cut. Or else, on the contrary, in each of the series , , ⋅⋅⋅, satisfying the first two conditions, there will be an element indistinguishable from one of the elements of the cut. In the first case we can go from to by a continuous route without quitting and without meeting the cuts; in the second case that is impossible.
If then for any two elements and of the continuum it is always the first case which presents itself, we shall say that 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 postulated, 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 continuum is of several dimensions.
If to divide a continuum cuts forming one or several continua of one dimension suffice, we shall say that is a continuum of two dimensions; if cuts suffice which form one or several continua of two dimensions at most, we shall say that is a continuum of three dimensions; 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 necessary; 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.
