according to their richness. Further, inasmuch as we advanced by a single member, that is, we have made the smallest step possible, we are certain to have omitted no possible group that is poorer than the richest to which we have advanced our operation.
This whole procedure is well known. It yields the entire series of positive numbers—the cardinal numbers. It is to be noted that the concept of magnitude does not appear as yet. What we have obtained is merely the concept of number. The individual members may be chosen quite arbitrarily. They need in no way be equal. Each number represents a quantity type; and it is the sphere of arithmetic to examine these different types in respect to subdivision and combination. If this be done without considering the amount of the number, we call the corresponding science algebra. On the other hand, the extension of formal rules beyond their original application has led to one development of numbers after the other. Thus counting backwards leads to zero and the negative numbers, the square root of the latter to the imaginary numbers. The quantity-type of all the positive numbers is, to be sure, the simplest, though by no means the only possible one. For the purpose of representing other arrangements such as occur among our experiences these new types have proved very useful.
At the same time the numerical series yields a most useful type of arrangement. From its very origin it is arranged in an orderly fashion and it is therefore employed for the purpose of arranging other quantities. Thus we are accustomed to apply the signs of the numerical series to any objects which we desire to use in a definite order, such as the pages of a book, the seats in a theater, as well as countless other groups. We, however, tacitly make the assumption that the arranged groups are to be used in the same sequence in which the natural numbers follow one another. These sequence-numbers represent no magnitudes nor do they represent the only type of arrangement possible. They are, however, the very simplest.
We do not reach the concept of magnitude until we reach the science of time and space. A science of time has not been developed separately. On the contrary, what there is to say about time usually appears for the first time in mechanics. However, it is possible for us to state the fundamental characteristics of time here, so that the want of a distinct science of time will not be felt.
The first and most important property of time (and also of space) is that it is continuous. In other words, any portion of time may be divided at any point. In the numerical series this is not the case; it may be divided only between numbers. The series one to ten has nine places of division, and only nine. A minute or a second, on the other hand, has an unlimited number of possible points of division. In other words, there is nothing in the passage of time preventing us at
