In this chapter we come to the second important metrical idea in the theory of sets, that of Content. The potency of a set is a metrical relation between one set and another in which the individuals of the sets are regarded as indistinguishable. The content of a set is one in which the individuals are regarded as having a characteristic by means of which they become of varying importance which must be taken into account; it is determined by the relative position of these individuals but is independent of their actual situations in the fundamental region.

The idea is a natural one when we start with intervals instead of points or numbers. The distinguishing characteristic is here apparent, the length of the individual intervals. The potency of a set of intervals is instinctively felt to be an affair of minor importance; what interests ua more is the relation of the intervals to the linear continuum, not regarded as a collection of points (a one-dimensional variety), but as a whole, capable of division into parts comparable by means of finite numbers with itself, a variety of zero dimensions.

Recalling the description and properties of a perfect set dense nowhere, given in Chap. III, we recognise that the parts into which the continuum ‘may be divided are not exclusively segments.