USE OF A SYMBOLICAL LAxNGUAGE. 227 



discover any general principles or rules which should 

 guide us in this important matter of admission or rejec- 

 tion? Let us examine a few of the symbols which we 

 now possess, and see whether any such rules can be 

 discovered. 



The ratio which the circumference of a circle has to its 

 diameter, namely, 3-14159 &c., is one that occurs fre- 

 quently, and for this reason mathematicians express it by 

 a single arbitrary symbol tr. The ratio w hich the diagonal 

 of a square has to its side, namely, i'4i42i &c., is another 

 ratio w^hich also occurs frequently, and yet mathemati- 

 cians do not express this by any single arbitrary symbol, 

 nor would any mathematician think the introduction of 

 such a symbol desirable. Why is this? The answer is 

 obvious : the latter ratio may be expressed, ivithout any 

 fresh definition or explanation, by a very brief and simple 

 combination of existing symbols, namely by the combi- 

 nation V2 ; while we know of no brief and easily formed 

 combination of existing symbols, requiring no fresh defi- 

 nition, which would accurately and unambiguously express 

 the former ratio. 



From these and other analogous examples we may safely 

 assume as one guiding principle, that some conventional 

 symbol of abbreviation should be used as a substitute for 

 any expression that has a tendency to recur frequently, 

 provided that no suitable combination of existing symbols 

 [i. e. a combination short, simple, and requiring no fresh 

 definition or explanation) can be found to replace it. 



The next point is, as a rule, more important and also 

 less easily decided. It is this : — Granting the necessity for 

 some new symbol of abbreviation, what kind of symbol 

 should be selected ? 



In the case of the symbol %, to which we have already 

 alluded, this question of suitable selection is, it is true, of 



q2 



