CHAP. XX.] PROBLEMS ON CAUSES. 365 



circular polarization in rock-crystal, and that of certain oblique 

 faces in its crystalline structure.* 



These problems are all of a similar character. A certain hypo- 

 thesis is framed, of the various possible consequences of which 

 we are able to assign the probabilities with perfect rigour. Now 

 some actual result of observation being found among those con- 

 sequences, and its hypothetical probability being therefore known, 

 it is required thence to determine the probability of the hypo- 

 thesis assumed, or its contrary. In Mitchell's problem, the hy- 

 pothesis is that of a " random distribution of the stars," the 

 possible and observed consequence, the appearance of a close 

 double star. The very small probability of such a result is held 

 to imply that the probability of the hypothesis is equally small, 

 or, at least, of the same order of smallness. And hence the high 

 and, and as some think, determinate probability of a disposing 

 cause in the stellar arrangements is inferred. Similar remarks 

 apply to the other examples adduced. 



2 1 . The general problem, in whatsoever form it may be pre- 

 sented, admits only of an indefinite solution. Let x represent the 

 proposed hypothesis, y a phenomenon which might occur as one 

 of its possible consequences, and whose calculated probability, on 

 the assumption of the truth of the hypothesis, is/?, and let it be re- 

 quired to determine the probability that if the phenomenon y is 

 observed, the hypothesis x is true. The very data of this pro- 

 blem cannot be expressed without the introduction of an arbi- 

 trary element. We can only write 



Prob. x = , Prob. xy = ap; (1) 



a being perfectly arbitrary, except that it must fall within the 

 limits and 1 inclusive. If then P represent the conditional pro- 

 bability sought, we have 



ff^ ap 

 Prob. y Prob. y ' 



It remains then to determine Prob. y . 



* Edinburgh Review, No. 185, p. 32. This article, though not entirely free 

 from error, is well worthy of attention. 



