138 THE PRINCIPLES OF SCIENCE. [chap. 



For instance, the proposition AB = ABC can be first 

 varied liy 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 = ABC 

 gives «B = flBC, Ah = AhC, AB = ABc, ah = ahC, 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 no means follows 

 that some of the varieties may not be equivalent to others ; 

 and trial shows, in fact, that AB = ABC is exactly the 

 same in meaning as Ac = Ahc or Be = Bca. Thus the law 

 in question has but eight varieties of distinct logical mean- 

 ing. I now ascertain by actual deductive reasoning which 

 of the 256 series of combinations result from each of these 

 distinct laws, and mark them off as soon as found. I then 

 proceed to some other form of law, for instance A = ABC, 

 meaning that whatever has the qualities of A has those 

 also of B and C. I find that it admits of twenty-four 

 variations, all of which are found to be logically distinct ; 

 the combinations being worked out, I am able to mark off 

 twenty-four more of the list of 256 series. I proceed in 

 this way to work out the results of every form of law 

 which I can find or invent. If in the course of this work 

 I obtain any series of combinations which had been pre- 

 viously marked off, I learn at once that the law giving 

 these combinations is logically equivalent to some law 

 previously treated. It may be safely inferred that every 

 variety of tlie a[>parently new law will coincide in meaning 

 with some variety of the former expression of the same 

 law. I have sufficiently verified this assumption in some 

 cases, and have never found it lead to error. Thus as 

 AB = ABC is equivalent to Ac = Ahc, so we find that 

 ah = ahC is equivalent to ac = acB. 



Am-onsc the laws treated were the two A = AB and 

 A = B which involve only two terms, because it may of 

 course happen that among three things two only are in 

 special logical relation, and the third independent ; and 

 the series of combinations representing such cases of re- 

 lation are sure to occur in the complete enumeration. All 

 single pro])ositions which I could invent having been 

 treated, pairs of propositions were next investigated. Thus 



