Evolution by Subtraction* 191 



equation, we arrive at a value, actual or approximate, for n y 

 the required root. For the cube root, the corresponding 

 equation is considerably more involved; and beyond this point 

 our authors abandon the method. 



" Let us examine this traditional method of evolving square 

 and cube roots. The second and following steps are metho- 

 dical enough, being in fact solutions of a known equation, and 

 lead surely to correct results. But the first step depends 

 entirely on an exercise of memory ; it is not performed by 

 method, but by an arbitrary process of recalling previous 

 knowledge ; it is, in a word, guesswork. Take, for instance, 

 the square root of 6561 ; having given that the first figure of 

 the result is 8, the rest follows of course ; but how do we know 

 the first figure ? Or suppose the given square were 64 simply ; 

 our text-books furnish absolutely no method whatever for find- 

 ing the square root of a number of less than three digits. 



" Yet there is a method, not mentioned in our books, a me- 

 thod of singular, almost ridiculous, simplicity, that will evolve 

 with certainty the square root of any exact square whatever. 

 The rule is no more than this : — Subtract successively the even 

 numbers; the last remainder will be the square root. Here, 

 for, example, is the square root of 64 above mentioned, evolved 

 by an unerring rule. The process of subtraction has been 

 continued until it left us the required root, and could not then 

 be carried any further. 



n 2 = 64 



_2 



62 



58 

 _6 



52 

 J* 



44 

 10 



34 

 12 



22 

 14 



n = 8 



" Now let us turn to the cube root, and take this simple 

 rule: — For the nth subtrahend, multiply n by 6, and add the 

 previous subtrahend. Hie last remainder will be the cube root. 

 Thus the first subtrahend is 6 ; the next, 6x2 + 6 = 18; the 

 third, 18 + 18 = 36; then 24 + 36 = 60; and so on. The series 



