230 H. M^COLL ON THE GROWTH AND 



air into its component gases." In the special case of 

 f[x — 5) no ambiguity exists when the letter / is used in 

 no other sense throughout the investigation on which we 

 happen to be engaged; and when it is used in another 

 sense, all risk of confusion is obviated by simply em- 

 ploying instead of/, in the expressions /(a:), /(.r— 5), &c., 

 some other symbolic letter, such as ^. 



We may recapitulate the results so far arrived at 

 thus : — 



1. A new symbol, or an old symbol with a new meaning, 

 to be accepted as a permanent addition to the existing 

 stock, should represent an expression or relation that 

 tends to occur frequently, and that cannot be logically 

 expressed by any short combination of existing symbols. 



2. Provided we can avoid ambiguity, it is better to 

 employ an old symbol in a new sense than to invent a 

 totally new symbol, 



3. The symbol chosen should be of short and easy 

 formation and lead to symbolic expressions of simple and 

 symmetrical forms. 



The illustrations hitherto given have been borrowed 

 from mathematics, because these are more familiar to the 

 general reader than any that could have been taken from 

 symbolical logic; yet the latter is in every respect the 

 simpler science of the two, — simpler in the conceptions* 



* That the conceptions which underlie the very elements of symbolical 

 mathematics are by no means easy to gi'asp will be admitted by any one 

 who has attempted to explain to beginners the real logical meaning of the 

 "Kule of Signs" in multiplication and division of ordinary algebra. This 

 difficulty meets the tyro on the very threshold of the science. When he has 

 advanced a few steps further, he is confronted with a symbolical paradox 

 which it has taxed the ingenuity of the subtlest mathematical intellects to 

 explain, and of which no rational explanation whatever was given until 

 within very recent times ; I allude to the useful and important yet per- 

 plexing symbol V — i • 



