How many dimensions has this continuum? Take first two elements _A_ and _B_ of _C_, and suppose there exists a series [Sigma] 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 [Sigma] 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 [Sigma] 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 [Sigma] 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 forming 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 recognize how many dimensions any physical continuum has. It only remains to find a physical continuum which is, so to speak, equivalent to space, of such a sort that to every point of space corresponds an element of this continuum, and that to points of space very near one another correspond indistinguishable elements. Space will have then as many dimensions as this continuum. The intermediation of this physical continuum, capable of representation, is indispensable; because we can not represent space to ourselves, and that for a multitude of reasons. Space is a mathematical continuum, it is infinite, and we can represent to ourselves only physical continua and finite objects. The different elements of space, which we call points, are all alike, and, to apply our definition, it is necessary that we know how to distinguish the elements from one another, at least if they are not too close. Finally absolute space is nonsense, and it is necessary for us to begin by referring space to a system of axes invariably bound to our body (which we must always suppose put back in the initial attitude).