148 THE PRINCIPLES OF SCIENCE. 



phenomena to be compared are rare. Thus we prove 

 that all the musical concords coincide with all the more 

 simple numerical ratios, by showing that each concord 

 arises from a simple ratio of undulations, and then show- 

 ing that each simple ratio gives rise to one of the con- 

 cords. 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 of quartz which rotate the plane of polar- 

 ization of light are precisely those crystals which have 

 plagihedral faces, that is, oblique faces on the corners 

 of the prism unsymmetrical with the ordinary faces. 

 This singular relation would be proved by observing that 

 all plagihedral crystals possessed the power of rotation, 

 and vice versd all crystals possessing this power were 

 plagihedral. But it might at the same tune be noticed 

 that all ordinary crystals were devoid of the power. 

 There is no reason why we should not observe any of the 

 four propositions A = AB, B = AB, a = ab, b = ab, all of 

 which follow from A = B (see p. 133). 



Sometimes the terms of 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 

 affirm 



Diamond = combustible gem, 



In a similar manner we ascertain that 



Mercury = metal liquid at ordinary temperatures, 

 Substance of least density = substance of least atomic 



weight. 



Two or three objects may occasionally enter into the 

 induction, as when we learn that 



Sodium I potassium metal of less density than 



water, 



Venus I Mercury I Mars = major planet devoid of 

 satellites. 



