296 THE PRINCIPLES OF SCIENCE. 



result. It is apparent that as the number of balls in the 

 box is increased, the absolute probability of any one hypo- 

 thesis concerning the exact proportion of balls is decreased, 

 but the aggregate results of all the hypotheses will assume 

 the character of a wide 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, 

 indeed, utterly 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, especially De Morgan's ' Treatise on Proba- 

 bilities/ in the * Encyclopaedia Metropolitana/ and Mr. Tod- 

 hunter's ' History of the Theory of Probability/ The ab- 

 breviating 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 

 combinations 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 



