In the second place, because of their being numbers, they express, as differences of magnitude, differences in what is sensuous only. The system of apparent magnitude—and it is in the heavenly system that magnitude appears most purely and freely, liberated from what is qualitative—must correspond to them. But these living number-spheres are themselves systems composed of many elements—both of the magnitude of distance and of velocity and mass. No one of these elements, taken as a succession of simple numbers, can be likened to the system of heavenly spheres, for the series corresponding to this system can, as to its members, contain nothing else than the system of all these moments. Now if the Platonic numbers were also the elements of each system such as this, it would not be only this element which would have to be taken into account, for the relationship of moments which become distinguished in movement has to be conceived of as a whole, and is the true object of interest and reason. What we have to do is to give briefly the main points as matter of history; we have the most thorough treatment of it given us by Böckh “On the Constitution of the World-Soul in the Timæus of Plato,” in the third volume of the Studies of Daub and Creuzer (p. 26 _et seq._). The fundamental series is very simple: “God first took one part out of the whole; then the second, the double of the first; the third is one and a half times as many as the second, or three times the first; the next is double the second; the fifth is three times the third; the sixth is eight times the first; the seventh is twenty-seven times greater than the first.” Hence the series is: 1; 2; 3; 4 = 2²; 9 = 3²; 8 = 2³; 27 = 3³. “Then God filled up the double and triple intervals” (the relations 1 : 2 and 1 : 3) “by again abstracting portions from the whole. These parts he placed in the intervals in such a way that in each interval there were two means, the one exceeding and exceeded by the extremes in the same ratio, the other being that kind of mean which by an equal number exceeds and is exceeded by the extremes.” That is, the first is a constant geometric relationship, and the other is an arithmetical. The first mean, brought about through the quadration, is thus in the relation 1 : 2, for example, the proportion 1 : √̅2 : 2; the other is in the same relation, the number 1½. Hereby new relations arise which are again in a specially given and more difficult method inserted into that first, but this is done in such a way that everywhere something has been left out, and the last relation of number to number is 256 : 243, or 2^8 : 3^5.