THE conception of a "limit" is one of which the importance in mathematics has been found continually greater than had been thought. The whole of the differential and integral calculus, indeed practically everything in higher mathematics, depends upon limits. Formerly, it was supposed that infinitesimals were involved in the foundations of these subjects, but Weierstrass showed that this is an error: wherever infinitesimals were thought to occur, what really occurs is a set of finite quantities having zero for their lower limit. It used to be thought that "limit" was an essentially quantitative notion, namely, the notion of a quantity to which others approached nearer and nearer, so that among those others there would be some differing by less than any assigned quantity. But in fact the notion of "limit" is a purely ordinal notion, not involving quantity at all (except by accident when the series concerned happens to be quantitative). A given point on a line may be the limit of a set of points on the line, without its being necessary to bring in co-ordinates or measurement or anything quantitative. The cardinal number is the limit (in the order of magnitude) of the cardinal numbers 1, 2, 3, ... , ..., although the numerical difference between and a finite cardinal is constant and infinite: from a quantitative point of view, finite numbers get no nearer to as they grow larger. What makes the limit of the finite numbers is the fact that, in the series, it comes immediately after them, which is an ordinal fact, not a quantitative fact. [Pg 97] There are various forms of the notion of "limit," of increasing complexity. The simplest and most fundamental form, from which the rest are derived, has been already defined, but we will here repeat the definitions which lead to it, in a general form in which they do not demand that the relation concerned shall be serial. The definitions are as follows:— The "minima" of a class with respect to a relation are those members of and the field of (if any) to which no member of has the relation . The "maxima" with respect to are the minima with respect to the converse of . The "sequents" of a class with respect to a relation are the minima of the "successors" of , and the "successors" of are those members of the field of to which every member of the common part of and the field of has the relation . The "precedents" with respect to are the sequents with respect to the converse of . The "upper limits" of with respect to are the sequents provided has no maximum; but if has a maximum, it has no upper limits. The "lower limits" with respect to are the upper limits with respect to the converse of . Whenever has connexity, a class can have at most one maximum, one minimum, one sequent, etc. Thus, in the cases we are concerned with in practice, we can speak of "the limit" (if any).