INDUCTION. 147 



himself that the letters in Astronomers make No more 

 stars, that Serpens in akuleo is an anagram of Joannes 

 Keplerus, or Great gun do us a sum an anagram of Au- 

 gustus de Morgan, it will certainly be necessary to break 

 up the act of comparison into several successive acts. The 

 process will acquire a double character, and will consist in 

 ascertaining that each letter of the first group is among 

 the letters of the second group, and vice versd, that each 

 letter of the second is among those of the first group. 

 In the same way we can only prove that two long lists of 

 names are identical, by showing that each name in one 

 list occurs in the other, and vice versd. 



This process of comparison really consists in establish- 

 ing two partial identities, which are, as already shown 

 (P- T 33) equivalent in conjunction to one simple iden- 

 tity. We first ascertain the truth of the two propositions 

 A = AB, B = AB, and we then rise by substitution to the 

 single law A = B. 



There is another process, it is true, by which we may 

 get to exactly the same result, for the two propositions 

 A = AB, a = ab are also equivalent to the simple identity 

 A = B (p. 133). If then we can show that all objects 

 included under A are included under B, and also that all 

 objects not included under A are not included under B, 

 our purpose is effected. By this process we should 

 usually compare two lists if we are allowed to mark them. 

 For each name in the first list we should strike off one in 

 the second, and if, when the first list is exhausted the 

 second list is also exhausted, it follows that all names 

 absent from the first must be absent from the second, 

 and the coincidence must be complete. 



The two modes of proving a simple identity are so 

 closely allied that it is doubtful how far we can detect 

 any difference in their powers and instances of application. 

 The first method is perhaps more convenient where the 



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