x.] THE THEORY OF PROBABILITY. 213 



estimate concerning any proposition should lie be we en 

 p and p + dp. The complete probability of the proposition 

 is then the integral taken between the limits I and o, or 

 again . 



Difficulties of the Theory. 



The theory of probability, though undoubtedly true, 

 requires very careful application. Not only is it a branch 

 of mathematics in which oversights are frequently com- 

 mitted, but it is a matter of great difficulty in many cases, 

 to be sure that the formula correctly represents the data 

 of the problem. These difficulties often arise from the 

 logical complexity of the conditions, which might be, 

 perhaps, to some extent cleared up by constantly bearing 

 in mind the system of combinations as developed in the 

 Indirect Logical Method. In the study of probabilities, 

 mathematicians had unconsciously employed logical pro- 

 cesses far in advance of those in possession of logicians, 

 and the Indirect Method is but the full statement of these 

 processes. 



It is very curious how often the most acute and power- 

 ful intellects have gone astray in the calculation of 

 probabilities. Seldom was Pascal mistaken, yet he in- 

 augurated the science with a mistaken solution. 1 Leibnitz 

 fell into the extraordinary blunder of thinking that the 

 number twelve was as probable a result in the throwing 

 of two dice as the number eleven. 2 In not a few cases the 

 false solution first obtained seems more plausible to the 

 present day than the correct one since demonstrated. 

 James Bernoulli candidly records two false solutions of a 

 problem which he at first thought self-evident ; and he 

 adds a warning against the risk of error, especially when 

 we attempt to reason on this subject without a rigid 

 adherence to methodical rules and symbols. Montmort 

 was not free from similar mistakes. D'Alembert con- 

 stantly fell into blunders, and could not perceive, for 

 instance, that the probabilities would be the same when 



1 Montucla, Histoire des Matiiematiques, vol. iii. p. 386. 



* Leibnitz Opera, Dutens' Edition, vol. vi part i. p. 217. Tod- 

 hunter's History of the Theory of Probability, p. 48. To the latter 

 work I am indebted for many of the statements in the text. 



