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 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.

In the succeeding chapters treating of the concrete department of
mathema