comparisons must be between its several parts. M. Comte says, that
the parts of each science must be arranged in the order of their
decreasing generality; and that this order of decreasing generality
agrees with the order of historical development. Our inquiry must be,
then, whether the history of mathematics confirms this statement.

Carrying out his principle, M. Comte divides Mathematics into "Abstract
Mathematics, or the Calculus (taking the word in its most extended
sense) and Concrete Mathematics, which is composed of General Geometry
and of Rational Mechanics." The subject-matter of the first of these is
_number_; the subject-matter of the second includes _space_, _time_,
_motion_, _force_. The one possesses the highest possible degree of
generality; for all things whatever admit of enumeration. The others are
less general; seeing that there are endless phenomena that are not
cognisable either by general geometry or rational mechanics. In
conformity with the alleged law, therefore, the evolution of the
calculus must throughout have preceded the evolution of the concrete
sub-sciences. Now somewhat awkwardly for him, the first remark M. Comte
makes bearing upon this point is, that "from an historical point of
view, mathematical analysis _appears to have risen out of_ the
contemplation of geometrical and mechanical facts." True, he goes on to
say that, "it is not the less independent of these sciences logically
speaking;" for that "analytical ideas are, above all others, universal,
abstract, and simple; and geometrical conceptions are necessarily
founded on them."

We will not take advantage of this last passage to charge M. Comte with
teaching, after the fashion of Hegel, that there can be thought without
things thought of. We are content simply to compare the two assertions,
that analysis arose out of the contemplation of geometrical and
mechanical facts, and that geometrical conceptions are founded upon
analytical ones. Literally interpreted they exactly cancel each other.
Interpreted, however, in a liberal sense, they imply, what we believe to
be demonstrable, that the two had _a simultaneous origin_. The passage
is either nonsense, or it is an admission that abstract and concrete
mathematics are coeval. Thus, at the very first step, the alleged
congruity between the order of generality and the order of evolution
does not hold good.

But may it not be that though abstract and concrete mathematics took
their