That postulated, if _A_ and _B_ are two distinguishable elements of a continuum _C_, a series of elements may be found, E_{1}, E_{2}, ..., E_{_n_}, all belonging to this same continuum _C_ and such that each of them is indistinguishable from the preceding, that E_{1} is indistinguishable from _A_, and E_{_n_} indistinguishable from _B_. Therefore we can go from _A_ to _B_ by a continuous route and without quitting _C_. If this condition is fulfilled for any two elements _A_ and _B_ of the continuum _C_, we may say that this continuum _C_ is all in one piece. Now let us distinguish certain of the elements of _C_ which may either be all distinguishable from one another, or themselves form one or several continua. The assemblage of the elements thus chosen arbitrarily among all those of _C_ will form what I shall call the _cut_ or the _cuts_. Take on _C_ any two elements _A_ and _B_. Either we can also find a series of elements E_{1}, E_{2}, ..., E_{_n_}, such: (1) that they all belong to _C_; (2) that each of them is indistinguishable from the following, E_{1} indistinguishable from _A_ and E_{_n_} from _B_; (3) _and besides that none of the elements _E_ is indistinguishable from any element of the cut_. Or else, on the contrary, in each of the series E_{1}, 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 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 _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 dimensions_; and so on.