2SG THE PEIXCIPLES OF SCIENCE. [CHAP. 



apparent that as the number of balls in the box is increased, 

 the absolute probability of any one hypothesis concerning 

 the exact proportion of balls is decreased, but the aggregate 

 results of all the hypotheses will assume the character of 

 a wider average. 



When we take the step of supposing the balls within 

 the urn to be infinite in number, the possible proportions 

 of white and black balls also become infinite, and the 

 'probability of any one proportion actually existing is 

 infinitely small. Hence the final result that the next ball 

 drawn will be white is really the sum of an infinite 

 number of infinitely small quantities. It might seem 

 impossible to calculate out a problem having an infinite 

 number of hypotheses, but the wonderful resources of the 

 integral calculus enable this to be done with far greater 

 facility than if we supposed any large finite number of 

 balls, and then actually computed the results. I will not 

 attempt to describe the processes by which Laplace finally 

 accomplished the complete solution of the problem. They 

 are to be found described in several English works, espe- 

 cially De Morgan's Treatise on Probabilities, in the Ency- 

 clopaedia Metropolitan^ and Mr. Todhunter's History of 

 the Tlieory of Probability. The abbreviating power of 

 mathematical analysis was never more strikingly shown. 

 But I may add that though the integral calculus is 

 employed as a means of summing infinitely numerous 

 results, we in no way abandon the principles of com- 

 binations already treated. We calculate the values of 

 infinitely numerous factorials, not, however, obtaining 

 their actual products, which would lead to an infinite 

 number of figures, but obtaining the final answer to the 

 problem by devices which can only be comprehended 

 after study of the integral calculus. 



It must be allowed that the hypothesis adopted by 

 Laplace is in some degree arbitrary, so that there was some 

 opening for the doubt which Boole has cast upon it. 1 

 But it may be replied, (i) that the supposition of an 

 infinite number of balls treated in the manner of Laplace 

 is less arbitrary and more comprehensive than any other 

 that can be suggested. (2) The result does not differ 



1 Laws of Thought, pp. 368-375. 



