no general theory of molecular action. Some approximations have been made towards such a theory. Joule and Clausius have measured the velocity of gaseous atoms, or even determined the average distance between the collisions of atom and atom. Thomson has approximated to the number of atoms in a given bulk of substance. Rankine has formed some reasonable hypotheses as to the actual constitution of atoms. It would be a mistake to suppose that these ingenious results of theory and experiment form any appreciable approach to a complete solution of molecular motions. There is every reason to believe, judging from the spectra of the elements, their atomic weights and other data, that chemical atoms are very complicated structures. An atom of pure iron is probably a far more complicated system than that of the planets and their satellites. A compound atom may perhaps be compared with a stellar system, each star a minor system in itself. The smallest particle of solid substance will consist of a great number of such stellar systems united in regular order, each bounded by the other, communicating with it in some manner yet wholly incomprehensible. What are our mathematical powers in comparison with this problem?
After two centuries of continuous labour, the most gifted men have succeeded in calculating the mutual effects of three bodies each upon the other, under the simple hypothesis of the law of gravity. Concerning these calculations we must further remember that they are purely approximate, and that the methods would not apply where four or more bodies are acting, and all produce considerable effects upon each other. There is reason to believe that each constituent of a chemical atom goes through an orbit in the millionth part of the twinkling of an eye. In each revolution it is successively or simultaneously under the influence of many other constituents, or possibly comes into collision with them. It is no exaggeration to say that mathematicians have the least notion of the way in which they could successfully attack so difficult a problem of forces and motions. As Herschel has remarked,14 each of these particles is for ever solving differential equations, which, if written out in full, might belt the earth.
Some of the most extensive calculations ever made were those required for the reduction of the measurements executed in the course of the Trigonometrical Survey of Great Britain. The calculations arising out of the principal triangulation occupied twenty calculators during three or four years, in the course of which the computers had to solve simultaneous equations involving seventy-seven unknown quantities. The reduction of the levellings required the solution of a system of ninety-one equations. But these vast calculations present no approach whatever to what would be requisite for the complete treatment of any one physical problem. The motion of glaciers is supposed to be moderately well understood in the present day. A glacier is a viscid, slowly yielding mass, neither absolutely solid nor absolutely rigid, but it is expressly remarked by Forbes,15 that not even an approximate solution of the mathematical conditions of such a moving mass can yet be possible. “Every one knows,” he says, “that such problems are beyond the compass of exact mathematics;” but though mathematicians may know this, they do not often enough impress that knowledge on other people.
The problems which are solved in our mathematical books consist of a small selection of those which happen from peculiar conditions to be solvable. But the very simplest problem in appearance will often give rise to impracticable calculations. Mr. Todhunter16 seems to blame Condorcet, because in one of his memoirs he mentions a problem to solve which would require a great and impracticable number of successive integrations. Now, if our mathematical sciences are to cope with the problems which await solution, we must be prepared to effect an unlimited number of successive integrations; yet at present, and almost beyond doubt for ever, the probability that an integration taken haphazard will come within our powers is exceedingly small.