CHANCE. 77 



do not know, but it is difficult not to admit that 

 this probability is represented by a continuous 

 analytical function. The probability that the impulse 

 will be comprised between a and a + e will, then, 

 clearly be equal to the probability that it will be 

 comprised between a + € and a + 2€, provided that e is 

 very small. This is a property common to all 

 analytical functions. Small variations of the function 

 are proportional to small variations of the variable. 



But we have assumed that a very small variation in 

 the impulse is sufficient to change the colour of the 

 section opposite which the needle finally stop.s. 

 From a to a + e is red, from a + e to a + 2€ is black. 

 The probability of each red section is accordingly the 

 same as that of the succeeding black section, and 

 consequently the total probability of red is equal 

 to the total probability of black. 



The datum in the case is the analytical function 

 which represents the probability of a particular 

 initial impulse. But the theorem remains true, what- 

 ever this datum may be, because it depends on a 

 property common to all analytical functions. From 

 this it results finally that we have no longer any need 

 of the datum. 



What has just been said of the case of roulette 

 applies also to the example of the minor planets. 

 The Zodiac may be regarded as an immense roulette 

 board on which the Creator has thrown a very great 

 number of small balls, to which he has imparted 

 different initial impulses, varying, however, according 

 to some sort of law. Their actual distribution is 

 uniform and independent of that law, for the same 

 reason as in the preceding case. Thus we see why 



