Comte could assuredly not foresee the controversies which to-day bear upon the principles of mechanics and which have been summed up by Mr. Poincaré in an article upon Hertz’s mechanical theories.[115] Mr. Poincaré says that the principles of Dynamics have been stated in many ways, but nobody sufficiently distinguished between what is definition, what is experimental truth, and what is mathematical theorem. Mr. Poincaré is satisfied neither with the “classical” conception of mechanics, whose insufficiency has been shown by Hertz, nor with the conception with which Hertz wishes to replace it. In any case it is a high philosophical lesson to see the classical system of analytical mechanics--a system constructed with such admirable accuracy, and made by Laplace to arise altogether, as Comte says, out of a single fundamental law,--to see it after a century labouring under grave difficulties, not unconnected with the progress of physics. Might not this be an argument in support of the theory of d’Alembert and of Comte on the nature of concrete mathematics? Geometry and mechanics would only differ from the other natural sciences by the precision of the relations between the phenomena of which they treat, by the facility which they have for dealing with these relations by means of calculus and analysis, and, consequently, by assuming an entirely rational and deductive form. For the extraordinary power of the instrument should not hide from us the nature of the sciences which make use of it. These, like the others, bear upon natural phenomena. Only, as these phenomena are the most simple, the most general and the most closely allied of all, these sciences are also those which respond in the best way to the positive definition of science. They have “very easily and very quickly replaced empirical statement by rational prevision.” They are composed of laws and not of facts. But, conforming in this again to the positive definition of science, they are empirical in their origin, and they remain relative in the course of their development. Thus positive philosophy, having reached the full consciousness of itself, reacts upon the conception of the sciences which have most contributed to its formation. When the philosophy is universally accepted the idea that a science can be _a priori_, that is both absolute and immutable, will have disappeared. Precisely because it is the most perfect type of a positive science, mathematics will no longer claim these characteristics, and its ancient connection with metaphysics will be finally severed. FOOTNOTES: [102] Cours, I, 101. [103] Cours, I, 121-4. [104] Cours, I, 112 [105] Cours, I, 256. [106] Cours, I, 195-7. [107] Cours, I, 286-7. [108] Elements de philosophie, I, p. 132-3. [109] Cours, I, 298. sq. [110] Cours, i, 314-16. [111] Cours, i, 383-4. [112] Cours, i. 322-3. [113] Cours, i. 422. sq. [114] Cours, i, 455-463. [115] Revue générale des sciences pures et appliquées, 30 septembre 1897.