The intellectual progress involved in the conception of the reciprocal relation of entire _series_ or groups to one another, as distinguished from the conception of the relations between _individual_ things, is of the utmost importance and in the most expressive manner characterizes the difference between modern scientific thought and ancient thought. Ancient geometry, for example, knew only the cases of the acute, right, and obtuse angled triangle, and treated them separately, while the modern geometrician represents the side of the triangle as starting from the angle zero and traversing the entire field of possible angles. Accordingly, unlike his colleague of old, he does not ask for the particular principles bearing upon these particular cases, but he asks in what continuous relation do the sides and angles stand to one another, and he lets the particular cases develop from out of one another. In this way he attains a much profounder and more effectual insight into the whole of the existing relations. It is in mathematics in especial that the introduction of the concept of continuity and of the function concept arising from it has exercised an extraordinarily deep influence. The so-called _Higher Analysis_, or _Infinitesimal Analysis_, was the first result of this radical advance, and the _Theory of Functions_, in the most general sense, was the later result. This progress rests on the fact that the magnitudes appearing in the mathematical formulas were no longer regarded as certain definite values (or values to be arbitrarily determined), but as _variable_, that is, values which may range through all possible quantities. If we represent the relation between two things by the formula B = f(A), expressed in spoken language by B _is a function_ of A, then in the old conception A and B are each individual things, while in the modern conception A and B represent an inexhaustible series of possibilities embracing every conceivable individual case that may be co-ordinated with a corresponding case. Herein lies the essential advantage of the concept of continuity. It is true that it also introduces into calculation the above-mentioned contradictions which crop up in the ever-recurring discussions concerning the infinitely great and the infinitely small. The system introduced by Leibnitz of calculating with _differentials_, that is, with infinitely small quantities, which in most relations, however, still preserve the character of finite quantities from which they are considered to have been derived, has proved to be as fruitful of practical results as it is difficult of intellectual mastery. We can best conceive of these differentials as the expression of the law of the threshold, which law gave rise to, or made possible, the relation between the continuous and the discrete.