vii.] INDUCTION. 133 



planetary system have a progressive motion from west to 

 east, except the satellites of Uranus and Neptune. Even 

 some of the profoundest laws of matter are not quite 

 universal ; all solids expand by heat except india-rubber, 

 and possibly a few other substances ; all liquids which have 

 been tested expand by heat except water below 4 C. and 

 fused bismuth; all gases have a coefficient of expansion 

 increasing with the temperature, except hydrogen. In 

 a later chapter I shall consider how such anomalous 

 cases may be regarded and classified ; here we have only to 

 express them in a consistent manner by our notation. 



Let us take the case of the transparency of metals, and 

 assign the terms thus : 



A = metal D = iron 



B = gold E, F, &c. = copper, lead, &c. 



C = silver X = opaque. 



Our premises will be 



A = B -I- C -I- D -|. E, &c. 



B = Ba; 



C = Gx 



D=DX 



E = EX, 

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



Abe = (D -I- E -|-F -I- )Z>c, 



and by substitution as before we shall obtain 



Ale = A&cX, 



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

 at the same time we have 



A(B -|. C) = AB .|. AC = ABx .|- AGx = A(B -|- G)x, 

 or " Metals which are either gold or silver are not opaque." 

 In some cases the problem of induction assumes a much 

 higher degree 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 Regular 

 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 



