CHAP. XX.] PROBLEMS ON CAUSES. 367 



Hence, secondly, if the probability of the phenomenon is very 

 small when the hypothesis is assumed, the probability of the hy- 

 pothesis is very small when the phenomenon is observed, unless 

 either the a priori probability a of the hypothesis is large, or the 

 probability of the phenomenon upon any other hypothesis small. 

 The formula (5) admits of exact verification in various cases, 

 as when c = 0, or a = 1, or a = 0. But it is evident that it does 

 not, unless there be means for determining the values of a and c, 

 yield a definite value of P. Any solutions which profess to ac- 

 complish this object, either are erroneous in principle, or involve 

 a tacit assumption respecting the above arbitrary elements. Mr. 

 De Morgan's solution of Laplace's problem Concerning the ex- 

 istence of a determining cause of the narrow limits within which 

 the inclinations of the planetary orbits to the plane of the ecliptic 

 are confined, appears to me to be of the latter description. Having 

 found a probability^ = .00000012, that the sum of the incli- 

 nations would be less than 92 were all degrees of inclination 

 equally probable in each orbit, this able writer remarks : " If 

 there be a reason for the inclinations being as described, the 

 probability of the event is 1. Consequently, it is 1 : .00000012 

 (i. e. 1 :/?), that there was a necessary cause in the formation of 

 the solar system for the inclinations being what they are." Now 

 this result is what the equation (5) would really give, if, assigning 



to p the above value, we should assume c = 1, a = -. For we 



2 



should thus find, 



P = 



.-. I -Pi Pi ili p. (6) 



But P representing the probability, a posteriori, that all 

 inclinations are equally probable, 1 - P is the probability, d pos- 

 teriori, that such is not the case, or, adopting Mr. De Morgan's 

 alternative, that a determining cause exists. The equation (6), 

 therefore, agrees with Mr. De Morgan's result. 



22. Are we, however, justified in assigning to a and c parti- 

 cular values? I am strongly disposed to think that we are not. 



