ABSTRACT
In the last chapter of Part III, we agreed provisionally to call a series continuous if it had a term between any two. This definition usually satisfied Leibniz,† and would have been generally thought sufficient until the revolutionary discoveries of Cantor. Nevertheless there was reason to surmise, before the time of Cantor, that a higher order of continuity is possible. For, ever since the discovery of incommensurables in Geometry-a discovery of which is the proof set forth in the tenth Book of Euclid-it was probable that space had continuity of a higher order than that of the rational numbers, which, nevertheless, have the kind of continuity defined in Part III. The kind which belongs to the rational numbers, and consists in having a term between any two, we have agreed to call compactness; and to avoid confusion, I shall never again speak of this kind as continuity. But that other kind of continuity, which was seen to belong to space, was treated, as Cantor
remarks,* as a kind of religious dogma, and was exempted from that conceptual analysis which is requisite to its comprehension. Indeed it was often held to show, especially by philosophers, that any subject-matter possessing it was not validly analysable into elements. Cantor has shown that this view is mistaken, by a precise definition of the kind of continuity which must belong to space. This definition, if it is to be explanatory of space, must, as he rightly urges,† be effected without any appeal to space. We find, accordingly, in his final definition, only ordinal notions of a general kind, which can be fully exemplified in Arithmetic. The proof that the notion so defined is precisely the kind of continuity belonging to space, must be postponed to Part VI. Cantor has given his definition in two forms, of which the earlier is not purely ordinal, but involves also either number or quantity. In the present chapter, I wish to translate this earlier definition into language as simple and untechnical as possible, and then to show how series which are continuous in this sense occur in Arithmetic, and generally in the theory of any progression whatever. The later definition will be given in the following chapter.
