138 THE PRINCIPLES OF SCIENCE. [CHAP. 



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 = ABC 

 gives aB = aBC, Ab = A&C, AB = ABc, ab = abC, 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 = Abe 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 the apparently 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 = Abe, so we find that 

 db = abC is equivalent to ac = acB. 



Among 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 propositions which I could invent having been 

 treated, pairs of propositions were next investigated. Thus 



