434 PROCEEDINGS OF THE AMERICAN ACADEMY 



separately. Now it is obvious that belief cannot be considered as 

 formed by the superposition of several less beliefs. 



But although belief itself cannot be considered as a quantity, we may 

 submit to mathematical computation those combinations of circum- 

 stances which induce partial belief. The whole mathematical doctrine 

 of probabilities may be considered as founded on the following defini- 

 tion : — 



If of m events one and one only must occur (or have occurred) ; and 

 if an individual is entirely ignorant of any reason why one of these 

 events should occur rather than another, and if n of these events belong 



to a class A, then we call the fraction — the prohahility, for the mind 

 of the individual in question, that the event which will occur, or has oc- 

 curred, belongs to the class A. The probability might equally have 



been expressed by the fraction ; but this would be less conven- 



ient for mathematical computation, although more in accordance with 

 the language of common life. 



The solution of every possible mathematical problem in probabilities 

 consists simply in determining, from the conditions of the question, the 

 values of m and m, or rather of their ratio. 



It will readily be perceived that probability, as thus defined, (and this 

 is equivalent to the usual definition,) is not a quality inherent in the 

 event itself. The latter may be determined by laws as exact as those 

 which regulate the motions of the heavenly bodies. The principle that 

 every event which occurs is the result of law, and neither has or ever had 

 any absolute uncertainty inherent in it, may be regarded as an induction 

 almost as perfect as the laws of motion. 



Suppose that a die is loaded on one side. As long as we are 

 entirely ignorant of any reason why the load should be on one side 

 rather than another, the probability of any one side turning up will 

 still be ^ by definition. If we found the same side of a die to be 

 thrown four times out of five, we should be disposed to inquire 

 whether the frequency with which it was thrown was the result of 

 chance or of law. Expressed in exact philosophical language, our 

 question should be as follows : — 



Were these successive sixes the result of separate and independent 

 causes, conditions, or determining reasons ? 



Or was there a common element in the causes or determining 

 reasons which produced them ? 



