NATUEE 721 



if we take a throw of the dice in which the number of possible cases 

 is 36. That is the first degree of generality. But if we ask, for in- 

 stance, what is the probability that a point within a circle is within 

 the inscribed square, there are as many possible cases as there are 

 points in the circle that is to say, an infinite number. This is 

 the second degree of generality. Generality can be pushed further 

 still. We may ask the probability that a function will satisfy a given 

 condition. There are then as many possible cases as one can imagine 

 different functions. This is the third degree of generality, which we 

 reach, for instance, when we try to find the most probable law after a 

 finite number of observations. Yet we may place ourselves at a quite 

 different point of view. If we were not ignorant there would be no 

 probability, there could only be certainty. But our ignorance cannot 

 be absolute, for then there would be no longer any probability at all. 

 Thus the problems of probability may be classed according to the 

 greater or less depth of this ignorance. In mathematics we may set 

 ourselves problems in probability. What is the probability that the 

 fifth decimal of a logarithm taken at random from a table is a 9? 

 There is no hesitation in answering that this probability is l-10th. 

 Here we possess all the data of the problem. We can calculate our 

 logarithm without having recourse to the table, but we need not give 

 ourselves the trouble. This is the first degree of ignorance. In the 

 physical sciences our ignorance is already greater. The state of a 

 system at a given moment depends on two things its initial state, 

 and the law according to which that state varies. If we know both 

 this law and this initial state, we have a simple mathematical problem 

 to solve, and we fall back upon our first degree of ignorance. Then it 

 often happens that we know the law and do not know the initial 

 state. It may be asked, for instance, what is the present distribution 

 of the minor planets? We know that from all time they have obeyed 

 the laws of Kepler, but we do not know what was their initial dis- 

 tribution. In the kinetic theory of gases we assume that the gaseous 

 molecules follow rectilinear paths and obey the laws of impact and 

 elastic bodies; yet as we know nothing of their initial velocities, we 

 know nothing of their present velocities. The calculus of probabili- 

 ties alone enables us to predict the mean phenomena which will result 

 from a combination of these velocities. This is the second degree of 

 ignorance. Finally it is possible, that not only the initial conditions 

 but the laws themselves are unknown. We then reach the third 

 degree of ignorance, and in general we can no longer affirm anything 

 at all as to the probability of a phenomenon. It often happens that 

 instead of trying to discover an event by means of a more or less im- 

 perfect knowledge of the law, the events may be known, and we want 

 to find the law; or that, instead of deducing effects from causes, we 

 wish to deduce the causes from the effects. Now, these problems are 



