CH. XIIl] CALCULATING PRODIGIES 291 



about the same. In Dase's case, if the time occupied is pro- 

 portional to n", we must have x less, than 3. From this, some 

 have inferred that probably Dase's methods were different in 

 character to those used by Bidder, and it is suggested that the 

 results tend to imply that Dase visualized recorded numerals, 

 working in much the same way as with pencil and paper, while 

 Bidder made no use of symbols, and recorded successive results 

 verbally in a sort of cinematograph way; but it would seem 

 that we shall need more detailed observations before we can 

 frame a theory on this subject. 



The cases of calculating prodigies here mentioned, and as 

 far as I know the few others of which records exist, do not 

 differ in kind. In most of them the calculators were unedu- 

 cated and self-taught. Blessed with excellent memories for 

 numbers, self-confident, stimulated by the astonishment their 

 performances excited, the odd coppers thus put in their pockets 

 and the praise of their neighbours, they pondered incessantly 

 on numbers and their properties ; discovered (or in a few cases 

 were taught) the fundamental arithmetical processes, applied 

 them to problems of ever increasing difficulty, and soon acquired 

 a stock of information which shortened theit work. Probably 

 constant practice and undivided devotion to mental calculation 

 are essential to the maintenance of the power, and this may 

 explain why a general education has so often proved destructive 

 to it. The performances of these calculators are remarkable, 

 but, in the light of Bidder's analysis, are not more than might 

 be expected occasionally from lads of exceptional abilities. 



19—2 



