AN ESSAY ON THE ROOTS OF INTEGERS, 



Af+B £ 6 +C p 6 +D £ 6 4-EorP'V+EorP'" by 



166 <p 6 + 571,800 <p 6 -f 758,593 p 5 -f 887,308 <p 6 + 296,025, or 

 166, 571,800,758,593,887,308,296,025. 



And hence V" <p c -f F or P ,T by the same number as M (Par. 25.) 



(b.) Now the highest approximate Root of 166 or A is 2. Hence 

 2, the first found figure of the Root, expounds a (Par. 17). 



(c.) Then 2 6 = 64 == first Subtrahend, expounds a 6 . 



(d.) Then 166 — 64 = 102 expounds A — a 6 or R and is first 

 Remainder. 



(e.) Since B is expounded by 571,800 and consists of 6 figures, so 

 102,571,800 by Lem. 6 expounds R <p 6 -f B, and is first Resolvend. 



(f.) Since a is expounded by 2, so a p is expounded by 20, and a* <p* 

 or°a7j s by20% and a 3 q? or~a^| 3 by20 3 ,&c. hence6a s <f s -f 15 a 4 ^ 4 + 20 a 3 p 3 

 + 15 a* p* -f 6 a p ■+ 1 is expounded by 6- 20 5 + 1520 4 + 20'20 3 -f 1520 2 

 -J- 6-20 -j- 1, and since 3 substituted as directed produces by the sum 6'20 5 *3 

 + 15-20 4 -3 4 + 2020 3 -3 3 -f 15-20 3 3 4 if- 6-203 5 -f- 3 6 a number 84,035,889 

 smaller than 102,571,800 or R <p 6 -f- B, and since 3 is the greatest number 

 which will do so, so 3 expounds the b of the sum 6 a s <p 5 b -J- 15 a 4 p 4 ¥ 

 + 20 a 2 tf ¥ -j- 15 a % p* b* 4- 6 a <p h s 4. ¥ as by Par. 19). 



