CH. XIIl] CALCULATING PRODIGIES 285 



Division was performed by Bidder much as taught in school- 

 books, except that his power of multiplying large numbers at 

 sight enabled him to guess intelligently and so save unnecessary 

 work. This also is Inaudi's method. A division sum with a 

 remainder presents more difficulty. Bidder was better skilled 

 in dealing with such questions than most of these prodigies, 

 but even in his prime he never solved such problems with the 

 same rapidity as those with no remainder. In public perform- 

 ances difficult questions on division are generally precluded by 

 the rules of the game. 



If, in a division sum, Bidder knew that there was no 

 remainder he often proceeded by a system of two-digit 

 terminals. Thus, for example, in dividing (say) 25,696 by 

 176, he first argued that the answer must be a number of 

 three digits, and obviously the left-hand digit must be 1. 

 Next he had noticed that there are only 4 numbers of two 

 digits (namely, 21, 46, 71, 96) which when multiplied by 76 

 give a number which ends in 96. Hence the answer must 

 be 121, or 146, or 171, or 196; and experience enabled him 

 to say without calculation that 121 was too small and 171 

 too large. Hence the answer must be 146. If he felt 

 any hesitation he mentally multiplied 146 by 176 (which he 

 said he could do "instantaneously") and thus checked the 

 result. It is noticeable that when Bidder, Colbum, and some 

 other calculating prodigies knew the last two digits of a product 

 of two numbers they also knew, perhaps subconsciously, that 

 the last two digits of the separate numbers were necessarily of 

 certain forms. The theory of these two-digit arrangements has 

 been discussed by Mitchell. 



Frequently also in division, Bidder used what I will call a 

 digital process, which a priori would seem far more laborious 

 than the normal method, though in his hands the method was 

 extraordinarily rapid : this method was, I think, peculiar to him. 

 I define the digital of a number as the digit obtained by find- 

 ing the sum of the digits of the original number, the sum 

 of the digits of this number, and so on, until the sum is less 

 than 10. The digital of a number is the same as the digital of 



