The well-known paradox of the theory of probabilities is that, to all seeming, it can extract knowledge from ignorance and certainty from doubt. The point cannot be better put than by Poincaré in discussing the physical theory of gases, where the doctrine of probability finds an important application. Let me give you his view—partly in paraphrase, partly in translation. “For omniscience,” he says in substance, “chance would not exist. It is but the measure of our ignorance. When we describe an event as accidental we mean no more than that we do not fully comprehend the conditions by which it was brought about. “But is this the full truth of the matter? Are not the laws of chance a source of knowledge? And, stranger still, is it not sometimes easier to generalise (say) about random movements than about movements which obey even a simple law—witness the kinetic theory of gases? And, if this be so, how can chance be the equivalent of ignorance? Ask a physicist to explain what goes on in a gas. He might, perhaps, express his views in some such terms as these: ‘You wish me to tell you about these complex phenomena. If by ill luck I happened to know the laws which govern them, I should be helpless. I should be lost in endless calculations, and could never hope to supply you with an answer to your questions. Fortunately for both of us, I am completely ignorant about the matter; I can, therefore, supply you with an answer at once. This may seem odd. But there is something odder still, namely, that my answer will be right.’” Now, what are the conditions which make it possible thus to extract a correct answer from material apparently so unpromising? They would seem to be a special combination of ignorance and knowledge, the joint effect of which is to justify us in supposing that the particular collection of facts or events with which we are concerned are happening “at random.” If we could calculate the complex causes which determine the fall of a penny, or the collisions of a molecule, we might conceivably deal with pennies or molecules individually; and the calculus of probability might be dispensed with. But we cannot; ignorance, therefore, real or assumed, is thus one of the conditions required to provide us with the kind of chaos to which the doctrine of chances may most fittingly be applied. But there is another condition not less needful, namely, knowledge—the knowledge that no extraneous cause or internal tendency is infecting our chaotic group with some bias or drift whereby its required randomness would be destroyed. Our penny must be symmetrical, and Maxwell’s demons[10] must not meddle with the molecules.