A.— MATHEMATICAL AND PHYSICAL SCIENCES 31 



ing to all the possible motions of the gas which would have the same 

 temperature. Later, almost as a concession to human frailty, Gibbs 

 introduces the micro-canonical ensemble, composed of much fewer 

 specimens because they all have exactly the same energy. This is usually 

 welcomed by the beginner because it seems closer to his familiar 

 mechanics, but with more experience he will realise that the gap is still 

 so great that he is really no better off, and he may as well accept the more 

 general idea at once. 



With the old mechanics all this involved ideas which for many readers 

 were distinctly hard to accept. The principle of probability, embodied 

 in the averaging over the ensemble, was frankly laid on top of the logical 

 principles of Newtonian mechanics, and to anyone believing that prob- 

 ability would ultimately be brought down to the old logic the association 

 was most repellent. But we can now see that Gibbs was a prophet far 

 ahead of his time— and indeed, to be frank, far ahead of his own knowledge 

 — for the new mechanics accommodates the ensemble very much more 

 easily than did the old. The new mechanics has shown us that it is 

 impossible to know how the individual molecules are moving, because 

 when one undertakes an experiment to see, that experiment automatically 

 alters the condition of the gas and so fails to tell what was wanted, the 

 state of the molecules without the experiment. In the old days one used 

 to feel that the validity of Gibbs's idea would be spoilt by some skilful 

 experimenter who would really observe the motions of the individual 

 molecules and would therefore rule out the legitimacy of averaging over 

 the whole ensemble, but we now know that there is no danger of this. 

 The real gas in the vessel is not merely one specimen of the ensemble, 

 unrecognisable only because of our clumsiness ; it is itself the whole of 

 the ensemble. We used to think of the gas as either in the state A, or in 

 the state B, or in C, but according to the new physics we have to think 

 of it as in all the states A and B and C. The distinction is typical of the 

 change we must make in our habits of thought, and most of us resist 

 this change strongly, for we find we can hardly help asking : ' But which 

 state was it really in ? ' As I have said, we used to be ashamed of 

 ignorance, but we must now realise that this ignorance is one of the 

 things that makes the world possible. The principle of probability, 

 which used to be loosely superposed on the old logical principle, is now 

 with the new mechanics fully united with it in a higher synthesis. 



Before leaving Gibbs I would like to refer to one thing in his book, 

 where I think he has not even yet come into his own. He considers 

 various types of ensemble of increasing generality. In the micro- 

 canonical the members all have the same energy. Now we never know 

 the exact energy of the gas in a vessel, so that a better idea is the wider 

 one of a gas at a given temperature which therefore has a certain range 

 of admissible energies. This is represented by Gibbs's canonical 

 ensemble, and it is the main one that he uses. In both these the number 

 of atoms in the ensemble is constant. But in the last chapter of his book 

 Gibbs introduces a still wider ensemble. He calls the ones with a 

 constant number of atoms petits ensembles, which I shall translate as petty 

 ensembles, and regards them as parts of a grand ensemble in which the 

 total number of atoms is not fixed. He uses the idea to some extent in 



