Finally the third law establishes that “every movement exactly possessed in common by all the bodies of any system does not alter the particular movements of those different bodies in respect to each other; but those movements continue to take place as if the whole of the system was motionless.” This law “of the independence or of the coexistence of movements” was formulated by Galileo. It is no more a priori than the two preceding ones. How could we be sure, if experience did not show it to us, that a common motion communicated to a system of bodies moving in relation to one another, would change nothing in their particular motions? When his law was made known by Galileo, on all hands there arose a cloud of objections, tending to prove a priori that this proposition was false and absurd. It was only admitted later when, in order to examine it, the logical point of view was set aside for the physical point of view. It was then seen that experience always confirmed this law, and that, if it ceased to operate, the whole economy of the universe would be thrown into utter confusion. For instance, the movement of the translation of the earth in no way affects the mechanical phenomena which take place upon the surface or within the globe. As the law of the independence of motions was unknown when the theory of Copernicus appeared, an objection was put to him which was thought to be drawn from experience. He was told that if the earth moved round the sun all the movements which take place upon it or within it would be modified by the action. Later on when Galileo’s law became known, the fact was explained and the objection disappeared.

Once these three laws are established, mechanics has sufficient foundation. Henceforth the scientific edifice can be constructed by simple logical operations, and without any further reference to the external world. But this purely rational development no more transforms mechanics into an a priori science than the application of analysis deprives geometry of its character as a natural science. What proves this, in one case as in the other, is the possibility of passing from the abstract to the concrete and of applying the results obtained to real cases, merely restoring the elements which science had been compelled to set aside. If it were possible entirely to constitute the science of mechanics according to simple analytical conceptions, we could not imagine how such a science could ever become applicable to the effective study of nature. What guarantees the reality of rational mechanics is precisely its being founded upon some general facts, in a word, upon the data of experience.

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.

[CHAPTER II]
ASTRONOMY

The object of astronomy is the discovery of the laws of the geometrical and mechanical phenomena presented by the celestial bodies; and, by the knowledge of these laws to obtain the precise and rational prevision of the state of our system at any given period whatever. It is in a word, “the application of mathematics to celestial phenomena.”[116]

Mr. H. Spencer has taken occasion of this definition to criticise the place assigned by Comte to astronomy in his classification of the sciences. He makes him contradict himself. He says: you term fundamental sciences the abstract sciences which do not study beings in nature, but the laws which govern phenomena in those beings; by what right is astronomy placed among these sciences, between mathematics and physics? Is not the object of astronomy the study of certain beings in nature? In what does the application of mathematics to celestial phenomena differ from their application to other cases? It appears evident that here Comte introduces into the series of abstract sciences a science which is really concrete, or at least, according to Mr. Spencer’s expression, abstract-concrete.

Comte had foreseen the objection. The answer which he makes throws a strong light upon the sense in which he understands the words “abstract” and “general” as applied to the sciences. He partly accepts the objection. The true astronomical notions, he says, only differ from purely mathematical notions by their special restriction to the celestial case; and this, at first sight, must appear contrary to the essentially abstract nature of the speculations which belong to the first philosophy. But on the other hand, these speculations bear upon the phenomena given in experience, and the order of the abstract sciences should reproduce the real order of dependence of the phenomena. Thus the first of these sciences, mathematics, determines the essential laws of the most general phenomena, which are common to all material beings (form, position, movement). Now, are not the most general phenomena after these, those “of which the the continuous ascendency inevitably dominates the course of all the other phenomena?”[117] In other words, before passing to the study of physical, chemical, biological phenomena, etc., it is indispensable to know the general laws of the milieu in which these phenomena are manifested. Outside of this milieu, they would be impossible, or at any rate, it so conditions them that, were it otherwise, these phenomena would also be different from what they are.