154 AN ESSAY ON THE ROOTS OF INTEGERS, 



61. But this observation will by no means apply to powers higher 

 than the Square, as will appear from the following examples in Cubes. 



Let A be a surd Cube, of which a is the approximate Root, and r the 

 remainder as before. Then on the principles of Par. 28), the assumed 



Root of A is a -f- — — — a -f . Then 



(a + I) 3 — a 3 3 a 2 + 3 a + 1 



Let A = 2 .— I 3 -j- 1. Then a = 1, r — 1, and 3 « 2 f 3 a -j- 1 = 7 

 and assumed Root — If. 



4 3 ' 



Then l^^^l-h^H--^--}- — i— — l-§ff , and deficiency - -H 

 Let A — 3 — l 3 -f 2. Then a — 1, r — 2, and assumed Root — If.. 



Then 2T) 3 = 1 + f + if + -2t = 2 W^ and deficiency - .§£§.. 

 Let A = 4 — I 3 -j- 3. Then a — 1, r = 3, and assumed Root tz 1 



Then Tf~[ 3 - 1 + f -f -ff -f ff^ - 2fff, and deficiency - l^s 



Let A=5= l 3 -f- 4. Then a — 1, /• — 4, and assumed Root — If. 



ThenTf | 3 — 1 -(- V 2 + H -I- wr = 3 lHHr > and deficiency — l-^- 

 Let A z 6 - l 3 | 5. Then a — 1, r — 5, and assumed Root — If. 



Then If] 3 - 1 + y + jf + -Mi = 44", and deficiency = fff. 

 Let A - 7 = 1 J f 6. Then a — 1, r = 6, and assumed Root ~ If. 



