VII.] INDUCTION. 129 



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

 lau- 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. 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 pur- 

 pose is effected. By this process we should usually com- 

 pare two lists if we are allowed to mark them. For each 

 name in the first list we should strike oif 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. 



These two modes of proving an identity are so closely 

 allied that it is doubtful how far we can detect any differ- 

 ence in their powers and instances of application. The 

 first method is perhaps more convenient when the pheno- 

 mena to be compared are rare. Thus we prove that all 

 the musical concouls coincide with all the more simple 

 numerical ratios, by showing that each concord arises from 

 a simple ratio of undulations, and then showing that each 

 simple ratio gives rise to one of the concords. To examine 

 all the possible cases of discord or complex ratio of 

 undulation would be impossible. By a happy stroke of 

 induction Sir John Herschel discovered that all crystals 

 id' quartz which cause the plane of polarization of light 

 to rotate are precisely those crystals which have plagi- 

 hediul faces, that is, oblique faces on the corners of the 

 prism unsymmetrical with the ordinary faces. This 

 singular relation would bo proved by observing that all 

 [•lagihedral crystals possessed the power of rotation, and 

 vice versd all crystals possessing this power were plagi- 

 hedral. But it might at the same time be noticed that 

 all ordinary crystals were devoid of the power. There is 

 no reason why we should not detect any of the four pro- 

 f)ositions A = AB, B = AB, a = al, h = ah, all of which 

 follow from A = B (p. 1 15). 



Sometimes the terms ot the identity may be singular 

 objects ; thus we observe that diamond is a combustible gem, 

 and being unable to discover any other that is, we aflirui — - 



K 



