722 SCIENCE AND HYPOTHESIS 



classified as probability of causes, and are the most interesting of all 

 from their scientific applications. I play at ecarte with a gentleman 

 whom I know to be perfectly honest. What is the chance that he 

 turns up the king? It is 1-8. This is a problem of the probability 

 of effects. I play with a gentleman whom I do not know. He has 

 dealt ten times, and he has turned the king up six times. What is the 

 chance that he is a sharper ? This is a problem in the probability of 

 causes. It may be said that it is the essential problem of the experi- 

 mental method. I have observed n values of x and the corresponding 

 values of y. I have found that the ratio of the latter to the former 

 is practically constant. There is the event; what is the cause? Is it 

 probable that there is a general law according to which y would be 

 proportional to x, and that small divergences are due to errors of 

 observation? This is the type of question that we are ever asking, 

 and which we unconsciously solve whenever we are engaged in scientific 

 work. I am now going to pass in review these different categories of 

 problems by discussing in succession what I have called subjective and 

 objective probability. 



II. Probability in Mathematics. The impossibility of squaring 

 the circle was shown in 1885, but before that date all geometers con- 

 sidered this impossibility as so " probable " that the Academic des 

 Sciences rejected without examination the, alas! too numerous mem- 

 oirs on this subject that a few unhappy madmen sent in every year. 

 Was the Academic wrong? Evidently not, and it knew perfectly well 

 that by acting in this manner it did not run the least risk of stifling 

 a discovery of moment. The Academic could not have proved that it 

 was right, but it knew quite well that its instinct did not deceive it. 

 If you had asked the Academicians, they would have answered : " We 

 have compared the probability that an unknown scientist should have 

 found out what has been vainly sought for so long, with the proba- 

 bility that there is one madman the more on the earth, and the latter 

 has appeared to us the greater." These are very good reasons, but 

 there is nothing mathematical about them; they are purely psycho- 

 logical. If you had pressed them further, they would have added: 

 " Why do you expect a particular value of a transcendental function 

 to be an algebraical number; if * be the root of an algebraical equa- 

 tion, why do you expect this root to be a period of the function sin 2x, 

 and why is it not the same with the other roots of the same equation ?" 

 To sum up, they would have invoked the principle of sufficient reason 

 in its vaguest form. Yet what information could they draw from it ? 

 At most a rule of conduct for the employment of their time, which 

 would be more usefully spent at their ordinary work than in reading 

 a lucubration that inspired in them a legitimate distrust. But what 

 I called above objective probability has nothing in common with this 

 first problem. It is otherwise with the second. Let us consider the 



