THE INDUCTIVE OR INVERSE METHOD. 295 



General Solution of the Inverse Problem. 



In the instance of the inverse method described in the 

 last section, a very few balls were supposed to be hi the 

 ballot-box for the purpose of simplifying the calculation. 

 In order that our solution may apply to natural phe- 

 nomena, we must render our hypothesis as little arbitrary 

 as possible. Having no d priori knowledge of the con- 

 ditions of the phenomena in question, there is no limit 

 to the variety of hypotheses which might be suggested. 

 Mathematicians have therefore had recourse to the most 

 extensive suppositions which can be made, namely, that 

 the ballot-box contains an infinite number of balls ; they 

 have thus varied the proportion of white balls to black 

 balls continuously, from the smallest to the greatest 

 possible proportion, and estimated the aggregate proba- 

 bility which results from this comprehensive supposition. 



To explain their procedure, let us imagine that, instead 

 of an infinite number, the ballot-box contained a large 

 finite number of balls, say 1000. Then the number of 

 white balls might be i or 2 or 3 or 4, and so on, up 

 to 999. Supposing that three white and one black ball 

 have been drawn from the urn as before, there is a certain 

 very small probability that this would have occurred in 

 the case of a box containing one white and 999 black 

 balls ; there is also a small probability that from such a 

 box the next ball would be white. Compound these 

 probabilities, and we have the probability that the next 

 ball really will be white, in consequence of the ex- 

 istence of that proportion of balls. If there be two 

 white and 998 black bans m the box, the probability 

 is greater, and will increase until the balls are supposed 

 to be in the proportion of those drawn. Now 999 different 

 hypotheses are possible, and the calculation is to be made 

 for each of these, and their aggregate taken as the final 



