But even this verification is needless. I might have rigorously proved that this mean is less than 0.003. To prove this result, I should have had to make a rather long calculation for which there is no room here, and for which I confine myself to citing an article I published in the Revue générale des Sciences, April 15, 1899. The only point to which I wish to call attention is the following: in this calculation, I should have needed only to rest my case on two facts, to wit, that the first and second derivatives of the logarithm remain, in the interval considered, between certain limits.
Hence this important consequence that the property is true not only of the logarithm, but of any continuous function whatever, since the derivatives of every continuous function are limited.
If I was certain beforehand of the result, it is first, because I had often observed analogous facts for other continuous functions; and next, because I made in my mind, in a more or less unconscious and imperfect manner, the reasoning which led me to the preceding inequalities, just as a skilled calculator before finishing his multiplication takes into account what it should come to approximately.
And besides, since what I call my intuition was only an incomplete summary of a piece of true reasoning, it is clear why observation has confirmed my predictions, and why the objective probability has been in agreement with the subjective probability.
As a third example I shall choose the following problem: A number u is taken at random, and n is a given very large integer. What is the probable value of sin nu? This problem has no meaning by itself. To give it one a convention is needed. We shall agree that the probability for the number u to lie between a and a+ is equal to ϕ(a)da; that it is therefore proportional to the infinitely small interval da, and equal to this multiplied by a function ϕ(a) depending only on a. As for this function, I choose it arbitrarily, but I must assume it to be continuous. The value of sin nu remaining the same when u increases by 2π, I may without loss of generality assume that u lies between 0 and 2π, and I shall thus be led to suppose that ϕ(a) is a periodic function whose period is 2π.
The probable value sought is readily expressed by a simple integral, and it is easy to show that this integral is less than
2πMk ⁄ nk,
Mk being the maximum value of the kth derivative of ϕ(u). We see then that if the kth derivative is finite, our probable value will tend toward 0 when n increases indefinitely, and that more rapidly than 1/nk−1.
The probable value of sin nu when n is very large is therefore naught. To define this value I required a convention; but the result remains the same whatever that convention may be. I have imposed upon myself only slight restrictions in assuming that the function ϕ(a) is continuous and periodic, and these hypotheses are so natural that we may ask ourselves how they can be escaped.
Examination of the three preceding examples, so different in all respects, has already given us a glimpse, on the one hand, of the rôle of what philosophers call the principle of sufficient reason, and, on the other hand, of the importance of the fact that certain properties are common to all continuous functions. The study of probability in the physical sciences will lead us to the same result.