Hence the importance of Descartes’ fine mathematical discovery, and also of the invention of differential and integral calculus, which may be considered as the complement to Descartes’ fundamental idea concerning the general analytical representation of natural phenomena. It is only, says Comte, since the invention of the calculus, that Descartes’ discovery has been understood and applied to the whole of its extent. Not only does this calculus procure an “admirable facility” for the search after the natural laws of all the phenomena; but, thanks to their extreme generality, the differential formulæ can express each determined phenomenon in a single equation, however varied the subjects may be in which it is considered. Thus, a single differential equation gives the tangents of all curves, another expresses the mathematical law of every variety in motion, etc. Infinitesimal analysis, especially in the conception of Leibnitz, has therefore not only furnished a general process for the indirect formation of equations which it would have been impossible to discover directly, but in the eyes of the philosopher it has another and a no less precious advantage. It has allowed us to consider, in the mathematical study of natural phenomena, a new order of more general laws. These laws are constantly the same for each phenomenon, in whatever objects we study it, and only change when passing from one phenomenon to another “where we have been able moreover, in comparing these variations, to rise sometimes, by a still more general view, to a _positive_ comparison between several classes of various phenomena, according to the analogies presented by the differential expressions of their mathematical laws.”[106] Comte cannot contemplate this immense range of transcendent analysis without enthusiasm. He calls it “the highest thought to which the human mind has attained up to the present time.” The highest, because being the most profoundly abstract among all the positive notions, this thought reduces the most comprehensive range of concrete phenomena to rational unity. As the consideration of analytical geometry suggested to Descartes the idea of “universal mathematics,” which lies at the basis of his method, so we can think that philosophical reflection upon transcendental analysis led Comte to the idea of those “encyclopædic laws,” which hold such an important place in his general theory of nature. For these encyclopædic laws, analogous as they are to the differential formulæ spoken of by Comte, are equally verifiable in orders of otherwise irreducible phenomena, and allow us to conceive them as convergent. III.