THE COMPUTING BOY. '293 



bers ending with 5 will likewise terminate with 25, as I li3ve Observations 



shown in your Philosophical Journal, No. 99, where may also ^ M J J^of'ihe 



be seen some other curious properties relating to square num- talents, and 



bers. It is manifest, therefore, that, if the boy adopted this o"si«'i^y °£ 



, , , ' . • . the methods of 



method, he would not only make " many more errors in the computing by 



extraction of the square than in that of the cube root j" but Zerali Col- 

 that he. would, in most cases, fail three times out of four } and, 

 in some cases, 7iine times out of ten. 



Any of your readers may satisfy themselves respecting this 

 ambiguity, by referring to a table of square numbers, where 

 they will rind that the frst 25 square numbers contain all the 

 varieties of the two terminating figures of such numbers j and 

 that the squares of all numbers equally above and below 25 ; 

 as of 24 and 26 ; or of 23 and 27, &c. will have their two last 

 figures the same : this property may not have been noticed by 

 your readers in general, but those of them who are but slightly 

 acquainted with mathematics may satisfy themselves of its 

 truth and universality -, for since the difference of the squares 

 of the sum, and difference of any two numbers is equal to four 

 times the product of those numbers, it is manifest that the dif- 

 ference of the squares of two numbers of the form 25 + a, and . 

 25 — a, would be of the form 100a ; that is, this difference 

 would be some exact multiple'of 1 00 j and therefore two such- 

 squares could not differ in their units and tens places of figures 5 

 viz. in their two last digits ; hence, then (syice the two last 

 figures only are used in M. Ralliers method) would arise the 

 ambiguity which I have stated. It will be easily seen, that what 

 I have shown of numbers of the form 25 + a, and 25-— a, is 

 equally true of the general formulas 25w + tf and 25w— a. 



Having proved, that M. Ralliers rule is only of partial 

 utility in the extraction of the cube root, and of little or no 

 use in the square root, " I think it would be extremely unfair 

 to conclude, that either this method, or one very similar to 

 it is adopted by the boy. 



Suppose, however, Sir, that it were possible for the boy to 

 have answered such questions as related merely to the square 

 and cube roots of numbers by the help of the above rule, still 

 this will not explain the method by which he multiplies four 

 figures by four, or by which he ascertains the factors of any 

 number, however large, with a rapidity that has astonished some 



"f 



