x] THE THEORY OF PROBABILITY. 213 



estimate concerning any proposition should lie beween 

 p and ip + dp. Tlie complete probability of the proposition 

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



again i. 



DijfficitUies of the Theory. 



The theory of probability, though undoubtedly true, 

 requires very careful application. JSTot 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 tluit the formula correctly represents the data 

 (jf the problem. These difficulties often arise from the 

 logical complexity of the conditions, which miglit 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 tlie 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 witli a mistaken solution.^ Leibnitz 

 fell into the extraordinary blunder of thinking that the 

 number twelve was as probable a result in the throwing 

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

 false solution first obtained seems more plausible to the 

 ]iresent day than the correct one since demonstrated. 

 James Bernoulli candidly records two false solutions of a 

 ]jroblem which he at first thought self-evident ; and he 

 atid.s a warning against the risk of error, especially when 

 we attempt to reason on this subject witliout a rigid 

 adherence to methodical rules and symbols, Montmort 

 was not free i'lom similar mistakes. D'Alembert con- 

 stantly f(,'ll into Ijlunders, and could not perceive, for 

 instance, that tlie pi-obabilities would be the same when 



' .Moiituclu, Ilidoirc des Maf}u^maH(]ueii, vol. iii. p. 386. 



- Li=ibiiitz Opera, Dutcns' Edition, vol. vi. part i. p. 217. TmU 

 huiifer's tlislori/ of thr Tlwnr)/ of J'TohahiUtij, p. 48. To tlie latter 

 work I am iiKJcLlLd lor iuany of the fctatciiuulsi in the text. 



