If the mathematical sciences have long been the only sciences properly so called, and if to-day they are still more advanced than any others, it is because the geometrical and mechanical phenomena are indeed the simplest of all, and those which are most naturally connected among themselves. The period during which they could be studied by observation could therefore be very short, so short that it is even not absurd to maintain that it never existed, and that, in this case, rational knowledge was not preceded by the empirical establishment of facts. But the difference between mathematics and the other sciences none the less remains one of degree and not of kind. The Science of Mathematics is in advance of the other sciences; but all work on common ground. In a word, like all other sciences it is a natural science. This endeavour to present the whole of the sciences as homogeneous, that is to say, to avoid two distinct classes being formed of mathematics on the one hand, and of the sciences of nature on the other, had already been attempted before Comte. This endeavour imposed itself, so to speak, upon modern philosophers, from the time when Descartes sought for a universal method for science conceived as a whole. Comte, who saw very well the defect in the Cartesian conception, in which the ascendency of mathematics was still too much felt, did not, however, deny that his own conception proceeded from that of Descartes. In another form, the idea of the homogeneity of the sciences is also found in Leibnitz and even in Kant. Does not the _Critique de la raison pure_ show that mathematics on the one hand, and physics on the other, equally rest upon principles which are synthetic _a priori_? In the _Prolégomenes à toute métaphysique future_ just as the chapter corresponding to _l’esthétique transcendentale_ is entitled “How are pure mathematics possible _a priori_?” so the chapter corresponding to the _Logique transcendentale_ bears as its title “How are pure physics _possible a priori_?” On another plan Comte’s theory is parallel to Kant’s. Here as there mathematics as well as physics rests upon synthetic principles--“superior to experience,” says Kant--proceeding from experience, says Comte. The latter, it is true, did not know Kant’s theory, and, had he known it he would not have accepted it. But the analogy of tendency subsists none the less beneath the diversity of doctrines. The immediate antecedent of Comte’s theory is found in d’Alembert. The author of the _Discours préliminaire_ had said, “We will divide the science of nature into physics and mathematics.” II. Every science has its origin in the art corresponding to it. Mathematics arose out of the art of measuring magnitudes. Indeed this art would be very rudimentary if we only practised direct measurement. Among the magnitudes which interest us there are very few which we can measure thus. Consequently the human mind had to seek some indirect way of determining magnitudes.