158 THE PRINCIPLES OF SCIENCE. 



combinations, and the effect of laws or logical conditions 

 is to destroy any one or more of these combinations. 

 Now we may make selections from eight things in 2 8 or 

 256 ways ; so that we have no less than 256 different cases 

 to treat, and the complete solution is at least fifty times 

 as troublesome as with two terms. Many series of com- 

 binations, indeed, are contradictory, as in the simpler 

 problem, and may be passed over. The test of consistency 

 is that each of the letters A, B, C, a, b, c shall appear 

 somewhere in the series of combinations ; but I have 

 not been able to discover any mode of calculating the 

 number of cases in which inconsistency would happen. 

 The logical complexity of the problem is so great that 

 the ordinary modes of calculating numbers of combinations 

 in mathematical science fail to give any aid, and ex- 

 haustive examination of the combinations in detail is 

 the only method applicable. 



My mode of solving the problem was as follows : 

 Having written out the whole of the 256 series of com- 

 binations, I examined them separately and struck out 

 such as did not fulfil the test of consistency. I then chose 

 some common form of proposition involving two or three 

 terms, and varied it in every possible manner, both by 

 the circular interchange of letters (A, B, C into B, C, A 

 and then C, A, B), and by the substitution for any one or 

 more of the terms of the corresponding negative terms. 

 For instance, the proposition AB = ABC can be first varied 

 by circular interchange, so as to give BC = BCA and then 

 CA = CAB. Each of these three can then be thrown 

 into eight varieties by negative change. Thus AB = ABO 

 gives aB = aBC, A& = A6C, AB = ABc, ab = a&C, and so on. 

 Thus there may possibly exist no less than twenty-four 

 varieties of the law having the general form AB = ABC, 

 meaning that whatever has the properties of A and B has 

 those also of C. It by 110 means follows that some of the 



