284 CALCULATING PRODIGIES [CH. XIII 



We shall underrate his rapidity if we allow as much as 

 a second for each of these steps, but even if we take this low 

 standard of his speed of working, he would have given the 

 answer in 7 seconds. By this method he never had at one time 

 more than two numbers to add together, and the factors are 

 arranged so that each of them has only one significant digit : 

 this is the common practice of mental calculators. It will also 

 be observed that here, as always, Bidder worked from left to 

 right: this, though not usually taught in our schools, is the 

 natural and most convenient way. In effect he formed the 

 product of (100 + 70 + 3) and (300 + 90 + 7), or (a + b + c) and 

 (d + e +/) in the form ad + ae ... + ef. 



The result of a multiplication like that given above was 

 attained so rapidly as to seem instantaneous, and practically gave 

 him the use of a multiplication table up to 1000 by 1000. On 

 this basis, when dealing with much larger numbers, for instance, 

 when multiplying 965,446,371 by 843,409,133, he worked by 

 numbers forming groups of 3 digits, proceeding as if 965, 446, 

 &c, were digits in a scale whose radix was 1000 : in middle 

 life he would solve a problem like this in about 6 minutes. 

 Such difficulty as he experienced in these multiplications seems 

 to have been rather in recalling the result of the previous step 

 than in making the actual multiplications. 



Inaudi also multiplies in this way, but he is content if one of 

 the factors has only one significant digit: he also sometimes makes 

 use of negative quantities : for instance he thinks of 27 x 729 as 

 27 (730 - 1) ; so, too, he thinks of 25 x 841 in the form 84100/4: 

 and in squaring numbers he is accustomed to think of the 

 number in the form a + b, choosing a and b of convenient forms, 

 and then to calculate the result in the form a 2 + 2ab + 6 s . 



In multiplying concrete data by a number Bidder worked 

 on similar lines to those explained above in the multiplication 

 of two numbers. Thus to multiply £14. 15s. 6|d by 787 he 

 proceeded thus : 



We have £(787) (14) = £11018. 0*. Od. 



to which we add (787) (15) shillings =£590. 5s. Od. making £11608. 5s. Od. 

 to which we add (787) (27) farthings =£22. 2*. 8Jd making £11630. 7s.8%d. 



