78 AN ESSAY ON THE ROOTS OF INTEGERS, 



4. 15 p 4 p 4 -f 20 p 3 p 3 -f 15 p" p* -J- 6^? p ~f_ l is expounded by 6 230 s 

 -}- 15-230* 4- 20230 3 4- 15-230 2 + 6-230 -f 1, and since 4 substituted as 

 directed, produces by the sum 6-230 5 '4 -f- 15230 4 -4 3 -f- 20230 3 -4 s 

 4. 15230 2 -4 1 + 6-230-4 5 -f- 4 6 a number 16,134,619,913,216 smaller than 

 18,535,911,758,593 or R' p° 4- C, and since 4 is the greatest number which 

 will do so, so 4 expounds the c of the sum 6 p 5 p 5 c -j- 15 p 4 <p 4 c 2 4- 20 p 3 p 3 c 3 

 -f 15 p" p 2 c 4 -f 6 p p c 5 4- c s as by Par. 22.) 



('Z> // . ) Then 4, the third found figure of the Root, expounds c (Par. 23.) 

 and since 23 expounds a <p 4- b or p, and there is but one figure in c, so by 

 Lem. 6.) (a p -j- b) p -f c or p p -f c is expounded by 234. And since 

 p' by Par. 24 = pp 4- c so p' is expounded by 234. And hence 234 is 

 the highest approximate Root of | Ap 6 4- B | p 6 -j- C or P p 6 -f- C or P' } or 

 166,571,800,758,593 the three first periods of the given number. 



As is easily tried, for 234 6 zz 164,170,508,913,216, which is less, and 

 235* = 108,425,239,515,625, which is greater than 166,571,800,758,593. 



c". Then 16,134,619,913,216, the third Subtrahend, expounds 6p s p 5 c 4- 

 15p>V + 20jo 3 <p 3 c 3 4- 15p 2 <p*c* -j- Qppc s -f- c 6 by Par. 23.) 



<Z". Since by Par. 22) P f -f C — ^° ^ 4- R' p 6 -J- C so P p 6 4. C 

 — ■ p G p 6 zz W p 6 -J- C. Subtract from both sides of this equation the 

 third Subtrahend by Art. c"). And it becomes P f 4- C — p 6 p 6 — 6p s 

 p s c — 15 p* p" e — 20 p 3 p 3 c 3 — 15 p a p* c 4 — 6 j9 p c s — c 6 — R' p 6 -f- 

 C — (6 |J S p 5 c 4- 15 ^ 4 £ 4 c* -f 20 p 3 p 3 c 3 -\- lb p % tf c* -f 6 p <p c 5 4- c* 5 ) 

 - (P <p 6 + C) — (p p 4 cf - (by Par. 21.) P' — p' 6 - R". Then since 

 R' p 6 -|- C is expounded by 18,535,911,758,593, and 6 p s p 5 c 4- 15 p* p 4 c 

 4- 20 p 3 p 3 c 3 4- 15 j9* ?> c c 4 -)- 6 p pc 5 -f c 6 is expounded by 16,134,619 

 913,216, so R' p 6 -f C — (6p s £ s c 4- 15 jt> 4 <p 4 c 5 -|- 20 p 3 p 3 c 3 -J- 15 p* p* c 



4 „1 



* „4 



