In the second chapter of his Cours de Philosophic Positive, M. Comte says:—­“Our problem is, then, to find the one rational order, amongst a host of possible systems.” ...  “This order is determined by the degree of simplicity, or, what comes to the same thing, of generality of their phenomena.”  And the arrangement he deduces runs thus:  Mathematics, Astronomy, Physics, Chemistry, Physiology, Social Physics.  This he asserts to be “the true filiation of the sciences.”  He asserts further, that the principle of progression from a greater to a less degree of generality, “which gives this order to the whole body of science, arranges the parts of each science.”  And, finally, he asserts that the gradations thus established a priori among the sciences, and the parts of each science, “is in essential conformity with the order which has spontaneously taken place among the branches of natural philosophy;” or, in other words—­corresponds with the order of historic development.

Let us compare these assertions with the facts.  That there may be perfect fairness, let us make no choice, but take as the field for our comparison, the succeeding section treating of the first science—­Mathematics; and let us use none but M. Comte’s own facts, and his own admissions.  Confining ourselves to this one science, of course our 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.”