Evolution by Subtraction. 199 



to look even for an approximation to the required root, as a 

 general rule. But our investigation of the series has shown 

 us that n m — n is always the sum of n terms ; and consequently, 

 in approximating to the mth root of n m ±d, we can discover n 

 at once by counting the number of terms subtracted. It must 

 be remembered that the first term is in every case zero, and 

 therefore the number of actual subtractions we have performed 

 will be n — 1. 



Having thus found w, which is so far an approximation to 

 the required root, we may now obtain a nearer approximation 

 by the help of the last remainder, which we have seen to be 

 n±d. For we can now determine the value and sign of d by 

 subtracting n from the last remainder, and we know that our 

 given number is n m ±d. There are, of course, known methods 

 by which the mth root of n m ±_d may be pretty accurately 

 found ; but, omitting these, let me point out that this method 

 itself will furnish a means of approximating tolerably nearly 

 to the root, by adding to n (or, if the sign of d prove to be 

 negative, subtracting from n) the quotient of d divided by the 

 next term of the subtrahend series following the last one sub- 

 tracted. To write it algebraically, 



''%/{n m ±d) = n±~ nearly. 



For example, let it be required to find the 11th root of 

 8590035271. After 7 subtractions we arrive at a last re- 

 mainder 100687. Therefore 7 + 1 = 8 is an approximate root. 

 The next subtrahend would have been 22791,125016 ; accord- 

 ingly we may take as a still closer approximation to the re- 

 quired root 



«, 100687-8 Bnnnnii - 



8 + 22791125016 =z8 ' QQ0Q44neal ' 1 y- 



Our investigation, then, will lead us to a restatement of 

 Cole's method in some such form as this: — 



To find the ?nth root of a given number, find a sufficient 

 number of terms of the series whose sum to n terms is n m — n ; 

 subtract these terms successively from the given number as 

 far as such subtraction is possible. The number of such sub- 

 tractions + 1, if it be equal to the last remainder, will be an 

 exact mth root. If not, it will be an approximate mth root ; 

 and for a nearer approximation, find the difference between 

 this first approximate root and the last remainder, and add (or 

 if the remainder was the less, subtract) the quotient of that 

 difference divided by that term of the series which follows 

 next after the last term subtracted. 



