NATURE 723 



first 10,000 Igarithms that we find in a table. Among these 10,000 

 logarithms I take one at random. What is the probability that its 

 third decimal is an even number? You will say without any hesita- 

 tion that the probability is 1-2, and in fact if you pick out in a table 

 the third decimals in these 10,000 numbers you will find nearly as 

 many even digits as odd. Or, if you prefer it, let us write 10,000 

 numbers corresponding to our 10,000 logarithms, writing down for 

 each of these numbers + 1 if the third decimal of the corresponding 

 logarithm is even, and 1 if odd ; and then let us take the mean of 

 these 10,000 numbers. I do not hesitate to say that the mean of these 

 10,000 units is probably zero, and if I were to calculate it practically, 

 I would verify that it is extremely small. But this verification is 

 needless. I might have rigorously proved that this mean is smaller 

 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 may 

 refer the reader to an article that I published in the Revue generate des 

 Sciences, April 15th, 1899. The only point to which I wish to draw 

 attention is the following. In this calculation I had occasion to rest 

 my case on only two facts namely, that the first and second deriva- 

 tives of the logarithm remain, in the interval considered, between 

 certain limits. Hence our first conclusion is that the property is not 

 only true of the logarithm but of any continuous function what- 

 ever, since the derivatives of every continuous function are limited. 

 If I was certain beforehand of the result, it is because I have often 

 observed analogous facts for other continuous functions; and next, 

 it is because I went through in my mind in a more or less uncon- 

 scious 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 ought to come to approx- 

 imately. And besides, since what I call my intuition was only an 

 incomplete summary of a piece of true reasoning, it is clear that 

 observation has confirmed my predictions, and that the objective and 

 subjective probabilities are in agreement. As a third example I shall 

 choose the following : The number u is taken at random and n 

 is a given very large integer. What is the mean value of sin nuf 

 This problem has no meaning by itself. To give it one, a convention 

 is required namely, we agree that the probability for the number 

 u to lie between a and a + da is <f> (a) da; that it is therefore propor- 

 tional to the infinitely small interval da, and is equal to this multi- 

 plied by a function </> (a), only depending 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 2ir, I may 

 without loss of generality assume that u lies between and 2ir, and I 

 shall thus be led to suppose that <f> (a) is a periodic function whose 

 period is 2 TT . The mean value that we seek is readily expressed by 



