438 PROCEEDINGS OF THE AMERICAN ACADExMY 



which I have called facility ; if so, the objection is well taken against 

 the general principle. It does not, however, affect Micliell's result. 



The second objection is also aimed at what seems to me not only a 

 mathematical, but a common-sense definition. If I correctly under- 

 stand it, Professor Forbes attempts to sustain it by showing that it 

 follows from the assumption alluded to in his objection, that a uniform 

 distribution of the stars is that which would be most probable as the 

 result of random scattering. This is true, but the most jjrobahle result 

 of a trial may be almost infinitely improbable. We shall consider this 

 more fully hereafter. 



In the first conclusion denied by Professor Forbes, it is not made 

 clear whether is meant (I.) the a priori probability that such an event 

 would occur as the result of chance ; or (2.) the a posteriori probability 

 that, having occurred, it was the result of chance ; or between the first 

 and third propositions in the method of reasoning cited above. Let p 

 be the a priori probability of the first proposition, I the a priori prob- 

 ability that the resulting pi'oposition would result from some law, or of 

 the second proposition. Then by the fundamental theorem of the prob- 

 abilities of causes, the probability that the observed contiguity is the 



result of law is - — —-j, and the probability that it is the result of chance 



is .-J—- Now, as above remarked, it is tacitly assumed by nearly all 

 writers, that I, though a small fraction, is very great compared with p. 

 Now p is, to a certain extent, capable of being expressed in exact num- 



bers, but I is not ; therefore r-i — is not. Professor Forbes is therefore 



l-\-p 



correct if he refers to the second of the above meanings, as was re- 

 marked by Professor Boole in a subsequent number of the Philosoph- 

 ical Journal. At the same time, however, we may, by the aid of nu- 

 merical calculation, make an approximate estimate of the probability 

 of the proposition used in its second meaning. 



The second proposition is a demonstrable mathematical certainty, un- 

 less it be held that it is infinitely imjyrobable that two stars should be 

 infinitely near as the result of law, which I apprehend no one will 

 maintain. In fact, p will become infinitely small, while I remains finite, 



SO that ,—, — will become infinitely small. 



The third proposition does not follow at all from Michell's argument. 

 A certain calculable amount of irregularity, or grouping, is to be ex- 

 pected as the result of a random distribution. If the amount of group- 



