THE THEOR Y OF PROBABILITIES. 5 



impossibility of dispensing with such an admission, we are led 

 to recognize the principle, that when an event is expected rather 

 than another, we believe it will occur more frequently on the 

 long run. And thus we perceive that we are in the habit of 

 forming judgments as to the comparative frequency of recur- 

 rence of different possible results of similar trials. These 

 judgments are founded, not on the fortuitous and varying cir- 

 cumstances of each trial, but on those which are permanent on 

 what is called the nature of the case. They involve the funda- 

 mental axiom, that on the long run, the action of fortuitous 

 causes disappears. Associated with this axiom is the idea of an 

 average among discordant results, &c. 



I conceive this axiom to be an a priori truth, supplied by 

 the mind itself, which is ever endeavouring to introduce order 

 and regularity among the objects of its perceptions. 



9. With a view to conciseness, I omit several interesting 

 points which here present themselves namely, the connection 

 between the axiom just stated, and the inductive principle ; the 

 real utility of Bernoulli's theorem ; and what seems to me to be 

 the true definition of probability, founded on a reference to the 

 ratios developed on the long run. 



I proceed to illustrate what has been said by a few passages 

 from Laplace's " Essai Philosophique sur les Probabilites" 



10. It seems obvious that no mathematical deduction from 

 premises which do not relate to laws of nature, can establish 

 such laws. Yet it is beyond doubt that Laplace thought Ber- 

 nouilli's theorem afforded a demonstration of a general law of 

 nature, extending even to the moral world. 



At p. xlii. of the Essay, prefixed as an Introduction to the 

 third edition of the Th^orie des Probabilites, after giving some 

 account of the theorem of James Bernouilli, Laplace proceeds: 

 " On peut tirer du the"oreme precedent cette consequence qui 

 doit 6tre regardee comme urie loi generale, savoir que les rap- 

 ports des effets de la nature, sont & fort peu pres constants, 

 quand ces effets sont considered en grand nombre....Je n'excepte 

 pas de la loi pre'ce'dente, les effets dus aux causes morales." 



It appears not to have occurred to Laplace, that this theorem 

 is founded on the mental phenomenon of expectation. But it is 



