and we have asked first what is a physical continuum and when it may be said to have n dimensions. If we consider different systems of impressions and compare them with one another, we often recognize that two of these systems of impressions are indistinguishable (which is ordinarily expressed in saying that they are too close to one another, and that our senses are too crude, for us to distinguish them) and we ascertain besides that two of these systems can sometimes be discriminated from one another though indistinguishable from a third system. In that case we say the manifold of these systems of impressions forms a physical continuum C. And each of these systems is called an element of the continuum C.
How many dimensions has this continuum? Take first two elements A and B of C, and suppose there exists a series Σ 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 Σ 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 Σ 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 Σ which join A to B cut the cut,