QTJAi^TITATIVE EVOLUTION OF KNOWLEDGE. 159 



which the perceptions of sensible equality have arrived. A 

 savage with a piece of stone in his hand, and another piece 

 lying before him of greater bulk but of the same kind (a 

 fact which he infers from the equality of the two in colour 

 and texture) knows about what effort he must put forth to 

 raise this other piece; and he judges accurately in propor- 

 tion to the accuracy with which he perceives that the one 

 is twice, three times, four times, <fec. as large as the other ; 

 that is — in proportion to the precision of his ideas of equali- 

 ty and number. And here let us not omit to notice that 

 even in these vaguest of quantitative previsions, the concep- 

 tion oi equality of relations is also involved. For it is only 

 in virtue of an undefined perception that the relation be- 

 tween bulk and weight in the one stone is equal to the re- 

 lation between bulk and weight in the other, that even the 

 roughest approximation can be made. 



But how came the transition from those uncertain per- 

 ceptions of equality which the unaided senses give, to the 

 certain ones with which science deals ? It came by placing 

 the things compared in juxtaposition. Equality being pre- 

 dicated of things which give us indistinguishable impres- 

 sions, and no accurate comparison of impressions being 

 possible unless they occur in immediate succession, it re- 

 sults that exactness of equality is ascertainable in propor- 

 tion to the closeness of the compared things. Hence the 

 fact that when we wish to judge of two shades of colour 

 whether they are alike or not, we place them side by side ; 

 hence the fact that we cannot, with any precision, say which 

 of two allied sounds is the louder, or the higher in pitch, 

 unless we hear the one immediately after the other ; hence 

 the fact tliat to estimate the ratio of weights, we take one 

 m each hand, that we may compare their pressures by rap- 

 idly alternating in thought from the one to the other ; hence 

 the fact, that in a piece of music, we can continue to make 

 equal beats when the first beat has been given, but cannot 



