62 AN ESSAY ON THE ROOTS OF INTEGERS, 



(23.) In this case c when found must be a Digit. If not c n or > <p 



First, let c — <p then 6 p 5 <p 5 c -j- 15 p* <p* c- -f 20/> 5 <p 3 c 3 -f 15/> 2 p 2 c* 

 -\-&p<pc 5 -f c 6 = 6jo 5 p 6 -f- 15 jo 4 <p 6 -|-20;j 3 £° -f 15/> 2 p 6 -j- jt> <^ G -f£ 6 . 

 Now R' by Par. 21) /L 6p 5 -f 15 jw 4 -f 20 jo 3 -f 15 jo 2 -f 6p +1, and since' 

 R' is an integer so Maximum of R' is 6p 5 -f 15 p* -f-20^ 3 -f 15jo 2 -f- Qp 

 and Maximum ofR'p 6 is 6 p 5 <p 6 -f 15 p* p 6 -f 20 j» 3 <p 6 -f 15;j 2 £ 6 -j-6j0p 6 . 

 Again by Par. 17). C contains 6 figures and hence by Lem 4. Maximum 

 of C is <p 6 — 1. Hence then Maximum of R' <p 6 -f Cis 6 p 5 <p 6 -f- 15p* <p 6 -f 

 20 p 3 <p 6 -f I5p°- <p 6 -f 6p <p 6 -f p 6 — 1. But 6p 5 p 6 -f 15 p 4 p 6 -f 20 p 3 p 6 

 -f 15/? 2 <p 6 -f 6 p <p 6 -J-<p 6 exceeds 6 p 5 tp 6 -f 15 p 4 p 6 -f- 20^ 3 <p 6 -f 15 p 2 <p 6 

 + 6p <p 6 -J- <p 6 — 1 by Unit. That is, if c =z <p, then 6 p 5 <p 5 c -j- 15 p 4 p 4 c 2 

 -f- 20 p 3 <p 3 c 3 -f 15 jo 2 <p 2 c 4 -f 6p <p c 5 -f c 6 must exceed R' <p 6 -f- C at 

 least by 1, and yet is also Z_ R' <p 6 -J- C, which is absurd. A fortiori c 

 cannot be > <p. 



(24.) Then let P <p 6 -f- C be put == P' and p <p -f c = p' then since jt>' 

 is the greatest approximate Root of P' so p' 5 ^- P' and {p' -f l) 6 > P'. Let 

 p> — p/6 — R„" Then by similar reasoning to that by which R' was proved 

 Z. 6 p 5 -J- 15 p* -f 20 j? 3 -f- 15 p* -f 6 p -f- 1, may R" be proved Z_ 

 Qp' 5 -f 15p' 4 -f 20 j/ 3 -f- 15 ju' 2 -f-6jt/-f 1. And by continuing with P' and p' 

 the same reasoning that was applied to P and p, there will be found 

 p' <p -f- d or p" the highest approximate Root of P' <p 6 -f D or P." That 



is \p p -j- c | <p -f- d or | a <p -}- 6 | <p -J- c <p -J- «? is the greatest approxi- 



mate Root of [ P <p 6 -f c | <p$ -f_ D or | A p 6 -j- B | <p 6 + c | ^ 6 -f D and 

 P"— p'' will be equal to R"'. 



(25.) And by a continuation of the same reasoning, there will suc- 

 cessively be found— 



