226 H. M^COLL ON THE GROWTH ANT) 



quently. When any complex expression or relation is 

 perceived to have a tendency to recur again and again, we 

 economize thought, time, and space if we denote this 

 expression or relation by some simple, suggestive, and 

 easily formed symbol which we may always recognize as 

 doing duty for its more complex equivalent. 



The representive symbols thus invented combine after- 

 wards among themselves into new expressions and rela- 

 tions of more or less complexity, and give birth, in their 

 turn, when the necessity or convenience arises, to fresh 

 representative symbols, whose abbreviating power bears, on 

 an average, the same ratio to that of the symbols they dis- 

 place, as the abbreviating power of the latter bears to that 

 of their immediate progenitors. In strict conformity with 

 this law of symbolical growth the science of mathematics 

 has gradually attained its present wonderful power within 

 the limits of its application ; and in strict conformity with 

 the same law, the science of logic, which is now evidently 

 entering on quite a new phase of existence, will probably 

 before long, and within much wider limits of application, 

 surpass the achievements of mathematical science itself. 



Now it is clear that the power and progress of any 

 symbolical language must depend very largely upon the 

 judgment exercised, first, as to whether, in any proposed 

 case, a new symbol is really required, or would on the 

 whole be useful, and, secondly (supposing the need of a 

 new symbol to be admitted) , as to the kind of symbol that 

 should be selected. With regard to the first point, we 

 must remember that the introduction of a fresh symbol is 

 always accompanied by the disadvantage that it adds a 

 fresh item to the load which the memory has to carry, 

 and it is only when its advantages more than outweigh 

 this very serious drawback that it should be admitted as a 

 permanent addition to the existing vocabulary. Can we 



