244 THE PRINCIPLES OF SCIENCE. 



of mathematics in which positive blunders are frequently 

 committed, but it is a matter of great difficulty in many 

 cases, to be sure that the formulae correctly represent 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. 8 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.* In not a few cases the 

 false solution first obtained seems more plausible to the 

 present day than the correct one since demonstrated. 

 James Bernouilli candidly records two false solutions of 

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

 adds an express warning against the risk of error, especially 

 when we attempt to reason on this subject without a rigid 

 adherence to the methodical rules and symbols. * Mont- 

 mort was not free from similar mistakes/ and as to 

 D'Alembert, great though his reputation was, and perhaps 

 is, he constantly fell into blunders which must diminish 

 the weight of his opinions. 2 He could not perceive, for 



8 Montucla, ' Histoire des Mathe'inatiques,' vol. iii. p. 386. 

 1 Leibnitz ' Opera,' Dutens' Edition, vol. vi. part i. p. 217. Todhunter's 

 ' History of the Theory of Probability,' p. 48. 



u Todhunter, pp. 67-69. * Ibid. p. 63. y Ibid. p. 100. 



* Tl)i<l. pp. 258-59, 286. 



