CH. XIII] CALCULATING PRODIGIES 287 



given number lies between 500 2 or 250,000 and 600 2 or 360,000, 

 the left-hand digit of the root must be a 5. Reflection had 

 shown him that the only numbers of two digits, whose squares 

 end in 61 are 19, 31, 69, 81, and he was familiar with this fact. 

 Hence the answer was 519, or 531, or 569, or 581. But he 

 argued that as 581 was nearly in the same ratio to 500 and 

 600 as 337,561 was to 250,000 and 360,000, the answer must 

 be 581, a result which he verified by direct multiplication in a 

 couple of seconds. Similarly in extracting the square root of 

 442,225, he saw at once that the left-hand digit of the answer 

 was 6, and since the number ended in 225 the last two digits 

 of the answer were 15 or 35, or 65 or 85. The position of 

 442,225 between (600) 2 and (700) 2 indicates that 65 should be 

 taken. Thus the answer is 665, which he verified, before 

 announcing it. Other calculators have worked out similar 

 rules for the extraction of roots. 



For exact cube roots the process is more rapid. For 

 example, asked to extract the cube root of 188,132,517, Bidder 

 saw at once that the answer was a number of three digits, and 

 since 5 s = 125 and 6 8 = 216, the left-hand digit was 5. The only 

 number of two digits whose cube ends in 17 is 73. Hence the 

 answer is 573. Similarly the cube root of 180,362,125 must be 

 a number of three digits, of which the left-hand digit is a 5, and 

 the two right-hand digits were either 65 or 85. To see which 

 of these was required he mentally cubed 560, and seeing it was 

 near the given number, assumed that 565 was the required 

 answer, which he verified by cubing it. In general a cube root 

 that ends in a 5 is a trifle more difficult to detect at sight by 

 this method than one that ends in some other digit, but since 

 5 8 must be a factor of such numbers we can divide by that and 

 apply the process to the resulting number. Thus the above 

 number 180,362,125 is equal to 5 8 x 1,442,897 of which the 

 cube root is at once found to be 5 (113), that is, 565. 



For still higher exact roots the process is even simpler, and 

 for fifth roots it is almost absurdly easy, since the last digit of 

 the number is always the same as the last digit of the root, 

 Thus if the number proposed is less than 10 10 the answer 



