Henceforth mechanics, originally an empirical science, becomes rationalised and is placed on a mathematical basis. It can now be developed without further appeal to experience by purely mathematical methods.[166] For this reason it is called rational mechanics. If, however, we consider the mechanics of the solar system, further empirical information must be obtained. The observational data were furnished by Tycho Brahe and Kepler. From this information Newton deduced the law of gravitation. When, therefore, the mutual force acting between bodies is of the inverse-square variety, we obtain a special branch of mathematical mechanics, called celestial mechanics.
From what has been said, we realise that the solutions of these problems of rational and celestial mechanics are of a purely mathematical nature, involving technical difficulties of their own. And so we need not be surprised to find the names of pure mathematicians on every page of the treatises consecrated to these arduous sciences. Thus, in rational mechanics we find the great names of Newton, Euler, d’Alembert, Lagrange and Hamilton, whereas in celestial mechanics we come upon the equally great names of Laplace, Jacobi and Poincaré. We have stressed these points in order to show how mechanics, originally, an empirical science, has since become mathematised.
Now, theoretically at least, it is of no interest to the pure mathematician to know whether these mathematical speculations correspond in any way to the workings of the real world; and regardless of whether Newton’s law is correct or at fault, the solution of the problem of the n bodies acting under Newton’s law of the inverse square would still present considerable interest. But, on the other hand, it would be a grave mistake to assume that these abstruse mathematical problems constitute mere intellectual pastimes, cross-word puzzles of a higher sort. Rational and celestial mechanics claim to correspond to the real workings of the world. The only way to verify the justice of this claim will be continually to check the anticipations of theory by the results of observation.
If this checking up proves harmony to prevail between theory and observation, we may assume that our fundamental principles are correct. If not, three courses are open: Either we may recognise that our fundamental principles are at fault, and attempt to replace them by others; or else we may retain our fundamental principles, and adjust theory to observation by introducing suitable disturbing influences or hypotheses ad hoc; finally, we may assume that nature is irrational, hence is not amenable to mathematical investigation.
It is impossible to give any golden rule in this respect. The hypothesis of Neptune was an illustration of the second procedure, where disturbing influences were introduced while the basic principles were maintained. The Einstein theory is an example of the first procedure, where the fundamental principles have been abandoned. As for the third procedure, it would be adopted only as a last resort, since it would be equivalent to throwing up the sponge. Nevertheless, it is far from certain that we may not have to resign ourselves to this attitude eventually, when our observations deal with phenomena on a microscopic scale.
Reverting to Newtonian science, we may say that until quite recently the fundamental principles of rational mechanics appeared to be in no danger. The most accurate astronomical observations yielded results in full conformity with theoretical previsions, when account was taken of a minimum of disturbing influences and of the inevitable inaccuracy in our knowledge of the planetary masses. We understand, therefore, whence arises the justification for the law of inertia, hence, incidentally, for the classical definitions of time and space.
It was not because stones thrown on the ice appeared to follow straight courses with constant speeds that the law of inertia was deemed to be established, for stones eventually come to a stop. It was because the numerous consequences of the law appeared to be verified indirectly to a high degree of accuracy. Hence, although it is undoubtedly correct to say that the fundamental principles of mechanics were nothing but hazardous generalisations from exceedingly crude observations, their justification appeared to have been established a posteriori, at least until quite recently.
From the principles of mechanics it is possible to deduce a certain very general principle. We refer to the principle of Least Action, alternative forms of which are given by Hamilton’s principle, and by Gauss’ principle of minimum effort. Any one of these general principles comprises all the principles of mechanics in a highly condensed form. Least action enables us to anticipate the following result: Free bodies unsolicited by forces invariably follow geodesics with constant speeds. Even when only partially free, as, for instance, when constrained to move over a fixed surface (without friction), the bodies will invariably follow the geodesics of the surface, also with constant speed.
Here a word of explanation may be necessary. The geodesics of three-dimensional Euclidean space are the familiar Euclidean straight lines; and our previous statement is but a presentation of the law of inertia. But when we consider a curved surface, the straight lines of space cannot lie on its surface. The geodesics of the curved surface are then no longer straight lines in the Euclidean sense, but they are the nearest approach to such lines, i.e., the least curved of all lines traceable on the curved surface. In the case of the sphere, geodesics are great circles. All surfaces have their geodesics more or less capriciously distorted in shape according to the variation in curvature from place to place of the surface. Again, as in the case of three-dimensional space, a geodesic is the shortest line between two points of the surface (when the distance is computed along the surface).[167]
An inextensible thread stretched between two points over a curved surface lies along a geodesic. Of course, in practice, it would be difficult to obtain adherence of the thread to the surface, were the surface concave. But this difficulty can be remedied if we consider a second surface fitting over the original one. The thread would then be stretched in the interstice between the two surfaces; and a body moving over the surface would be given by a small ball rolling without friction in the interstice between the two surfaces. In this way the ball would always remain in contact with the surface.