CHAP. I.] NATURE AND DESIGN OF THIS WORK. 15 



the probabilities of any simple events : required the probability of 

 a given compound event, i. e. of an event compounded in a given 

 manner out of the given simple events. The problem can also 

 be solved when the compound event, whose probability is re- 

 quired, is subjected to given conditions, i. e. to conditions de- 

 pendent also in a given manner on the given simple events. 

 Beside this general problem, there exist also particular problems 

 of which the principle of solution is known. Various questions 

 relating to causes and effects can be solved by known methods 

 under the particular hypothesis that the causes are mutually ex- 

 clusive, but apparently not otherwise. Beyond this it is not 

 clear that any advance has been made toward the solution of 

 what may be regarded as the general problem of the science, viz. : 

 Given the probabilities of any events, simple or compound, con- 

 ditioned or unconditioned : required the probability of any other 

 event equally arbitrary in expression and conception. In the 

 statement of this question it is not even postulated that the 

 events whose probabilities are given, and the one whose proba- 

 bility is sought, should involve some common elements, because 

 it is the office of a method to determine whether the data of a 

 problem are sufficient for the end in view, and to indicate, when 

 they are not so, wherein the deficiency consists. 



This problem, in the most unrestricted form of its statement, 

 is resolvable by the method of the present treatise ; or, to speak 

 more precisely, its theoretical solution is completely given, and 

 its practical solution is brought to depend only upon processes 

 purely mathematical, such as the resolution and analysis of equa- 

 tions. The order and character of the general solution may be 

 thus described. 



15. In the first place it is always possible, by the preliminary 

 method of the Calculus of Logic, to express the event whose 

 probability is sought as a logical function of the events whose 

 probabilities are given. The result is of the following character : 

 Suppose that X represents the event whose probability is sought, 

 A, By C, &c. the events whose probabilities are given, those 

 events being either simple or compound. Then the whole rela- 

 tion of the event X to the events A, B, C, &c. is deduced in the 

 form of what mathematicians term a development, consisting, in 



