out what has been vainly sought for so long, with the probability that there is one madman the more on the earth; the second appears to us the greater." These are very good reasons, but there is nothing mathematical about them; they are purely psychological.
And if you had pressed them further they would have added: "Why do you suppose a particular value of a transcendental function to be an algebraic number; and if π were a root of an algebraic equation, why do you suppose this root to be a period of the function sin 2x, and not the same about the other roots of this same equation?" To sum up, they would have invoked the principle of sufficient reason in its vaguest form.
But what could they deduce from it? At most a rule of conduct for the employment of their time, more usefully spent at their ordinary work than in reading a lucubration that inspired in them a legitimate distrust. But what I call above objective probability has nothing in common with this first problem.
It is otherwise with the second problem.
Consider the first 10,000 logarithms 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 not hesitate to answer 1/2; and in fact if you pick out in a table the third decimals of these 10,000 numbers, you will find nearly as many even digits as odd.
Or if you prefer, let us write 10,000 numbers corresponding to our 10,000 logarithms, each of these numbers being +1 if the third decimal of the corresponding logarithm is even, and −1 if odd. Then take the mean of these 10,000 numbers.
I do not hesitate