xii.] THE INDUCTIVE OR INVERSE METHOD. 251 



must have been a condition or cause for such events. We 

 have found that the theory of probability, although never 

 yielding a certain result, often enables us to establish an 

 hypothesis beyond the reach of reasonable doubt. There 

 is, however, another method of applying the theory, 

 which possesses for us even greater interest, because it 

 illustrates, in the most complete manner, the theory of 

 inference adopted in this work, which theory indeed it 

 suc2:ested. The problem to be solved is as follows : — 



An event having happened a certain number of times, 

 and failed a certain number of times, required the pro- 

 halility that it vsill happen any given nwiiber of times 

 in the future under the same ciixumstances. 



All the larger planets hitherto discovered move in one 

 direction round the sun ; what is the probability that, if a 

 new planet exterior to Neptune be discovered, it will move 

 in the same direction ? All known permanent gases, ex- 

 cept chlorine, are colourless ; what is the probability that, 

 if some new permanent gas should be discovered, it will 

 be cohnirless ? In the general solution of this problem, we 

 wish to infer the future happening of any event from the 

 number of times that it has already been observed to 

 happen. Now, it is very instructive to find that there is 

 no known process by which we can pass directly from the 

 data to the conclusion. It is always requisite to I'ecede 

 from the data to the probability of some hypothesis, and 

 to make that hypothesis the ground of our inference 

 concerning future events. Mathematicians, in fact, make 

 every hypothesis which is applicable to the question in 

 hand ; they then calculate, by the inverse method, the 

 probability of every such hypothesis according to the 

 data, and the probability that if each hypothesis be true, 

 the required future event will happen. The total pro- 

 bability that the event will haitpen is the sum of the 

 separate probabilities contributed by each distinct hypo- 

 thesis. 



To illustrate more precisely the method of solving the 

 probleui, it is desirable to adopt some concrete mode of 

 representation, and the ballot-box, so often employed by 

 matheiuaticians, will best serve our purpose. Let the 

 liajipi'iiing of any event be represented by the drawing of 

 a unite L)all from a ballot-box, while the failui'e of au 



