INDUCTION. 153 



Our premises will be 



A - BIG -I- DIE, &c. 

 B = 



and so on for the rest of the metals. Now evidently 

 Kbc = (DIE IF! ...... )lc, 



ar.d by substitution as before we shall obtain 



Abe = A&cX, 



or in words, ' All metals not gold nor silver are opaque ; ' 

 at the same time we have 



A(B I C) - AB I AC = ABaH- ACx = A(B t C)x, 

 or ' Metals which are either gold or silver are not opaque/ 

 In some cases the problem of induction assumes a much 

 higher degee of complexity. If we examine the properties 

 of crystallized substances we may find some properties 

 which are common to all, as cleavage or fracture in definite 

 planes ; but it would soon become requisite to break up 

 the class into several minor ones. We should divide 

 crystals according to the seven accepted systems and 

 we should then find that crystals of each system possess 

 many common properties. Thus crystals of the Eegular 

 or Cubical system expand equally by heat, conduct heat 

 and electricity with uniform rapidity, and are of like 

 elasticity in all directions ; they have but one index of 

 refraction for light ; and every facet is repeated in like 

 relation to each of the three axes. Crystals of the system 

 which possess one principal axis will be found to possess 

 the various physical powers of conduction, refraction, 

 elasticity, &c., uniformly in directions perpendicular to 

 the principal axis, but in other directions their properties 

 vary according to complicated laws. The remaining systems 

 in which the crystals possess three unequal axes, or have 

 inclined axes, exhibit still more complicated results, the 



