But may it not be that though abstract and concrete mathematics took their rise at the same time, the one afterwards developed more rapidly than the other; and has ever since remained in advance of it? No: and again we call M. Comte himself as witness. Fortunately for his argument he has said nothing respecting the early stages of the concrete and abstract divisions after their divergence from a common root; otherwise the advent of Algebra long after the Greek geometry had reached a high development, would have been an inconvenient fact for him to deal with. But passing over this, and limiting ourselves to his own statements, we find, at the opening of the next chapter, the admission, that “the historical development of the abstract portion of mathematical science has, since the time of Descartes, been for the most part _determined_ by that of the concrete.” Further on we read respecting algebraic functions that “most functions were concrete in their origin—even those which are at present the most purely {18} abstract; and the ancients discovered only through geometrical definitions elementary algebraic properties of functions to which a numerical value was not attached till long afterwards, rendering abstract to us what was concrete to the old geometers.” How do these statements tally with his doctrine? Again, having divided the calculus into algebraic and arithmetical, M. Comte admits, as perforce he must, that the algebraic is more general than the arithmetical; yet he will not say that algebra preceded arithmetic in point of time. And again, having divided the calculus of functions into the calculus of direct functions (common algebra) and the calculus of indirect functions (transcendental analysis), he is obliged to speak of this last as possessing a higher generality than the first; yet it is far more modern. Indeed, by implication, M. Comte himself confesses this incongruity; for he says:—“It might seem that the transcendental analysis ought to be studied before the ordinary, as it provides the equations which the other has to resolve. But though the transcendental _is logically independent of the ordinary_, it is best to follow the usual method of study, taking the ordinary first.” In all these cases, then, as well as at the close of the section where he predicts that mathematicians will in time “create procedures of a _wider generality_,” M. Comte makes admissions that are diametrically opposed to the alleged law.