CHAP. I.] NATURE AND DESIGN OF THIS WORK. 17 



necessary consequence. The Theory of Probabilities stands, as 

 it has already been remarked (1. 1 2), in equally close relation to 

 Logic and to Arithmetic ; and it is indifferent, so far as results 

 are concerned, whether we regard it as springing out of the lat- 

 ter of these sciences, or as founded in the mutual relations which 

 connect the two together. 



16. There are some circumstances, interesting perhaps to the 

 mathematician, attending the general solutions deduced by the 

 above method, which it may be desirable to notice. 



1st. As the method is independent of the number and the 

 nature of the data, it continues to be applicable when the latter 

 are insufficient to render determinate the value sought. When 

 such is the case, the final expression of the solution will contain 

 terms with arbitrary constant coefficients. To such terms there 

 will correspond terms in the final logical equation (I. 15), the 

 interpretation of which will inform us what new data are re- 

 quisite in order to determine the values of those constants, and 

 thus render the numerical solution complete. If such data are 

 not to be obtained, we can still, by giving to the constants their 

 limiting values and 1, determine the limits within which the 

 probability sought must lie independently of all further expe- 

 rience. When the event whose probability is sought is quite in- 

 dependent of those whose probabilities are given, the limits thus 

 obtained for its value will be and 1, as it is evident that they 

 ought to be, and the interpretation of the constants will only 

 lead to a re-statement of the original problem. 



2ndly. The expression of the final solution will in all cases 

 involve a particular element of quantity, determinable by the so- 

 lution of an algebraic equation. Now when that equation is of 

 an elevated degree, a difficulty may seem to arise as to the se- 

 lection of the proper root. There are, indeed, cases in which 

 both the elements given and the element sought are so obviously 

 restricted by necessary conditions that no choice remains. But 

 in complex instances the discovery of such conditions, by un- 

 assisted force of reasoning, would be hopeless. A distinct me- 

 thod is requisite for this end, a method which might not 

 inappropriately be termed the Calculus of Statistical Conditions. 

 Into the nature of this method I shall not here further enter 



