In the philosophical study of abstract mathematics, Comte proceeds successively from arithmetical to algebraical calculation, and from the latter to the transcendent analysis or differential and integral calculus. After having stated the manner in which this calculus is presented according to Leibnitz and to Newton, he adopts that of Lagrange, which appears to him the most satisfactory. It is true that at the end of his life his admiration for the author of the _Mécanique analytique_ had greatly diminished. Without here entering into the detail of questions, we will limit ourselves to the indication of a consideration upon the bearings of abstract mathematics, which appears to be of capital importance to Comte. Whether it be a question of ordinary analysis, or especially of transcendental analysis, Comte brings out at once the extreme imperfection of our knowledge, and the extraordinary fecundity of their applications. He can only solve a very small part of the questions which come before us in these sciences. However, “in the same way as in ordinary analysis we have succeeded in utilising to an immense degree a very small amount of fundamental knowledge upon the solution of equations, so, however little advanced geometers may be up to the present time in the science of integrations, they have none the less drawn, from these very few abstract notions the solution of a multitude of questions of the first importance, in geometry, in mechanics, in thermology, etc., etc.”[105] The reason of this is that the least abstract knowledge naturally corresponds to a quantity of concrete researches. The most powerful extension of intellectual means which man has at his disposal for the knowledge of nature consists in his rising to the conception of more and more abstract ideas, which are nevertheless positive. When our knowledge is abstract without being positive, it is “fictitious” or “metaphysical.” When it is positive without being abstract, it lacks generality, and does not become rational. But when, without ceasing to be positive, it can reach to a high degree of abstraction, at the same time it attains the generality, and, along the lines of its furthest extension, the unity which are the end of science.