132 AN ESSAY ON THE ROOTS OF INTEGERS, 



$ . Then since by Art. y) 4,197,615 =. 15 p*, with its units put 

 under the 5th place of 986,144,576 so by Lem. 7) their sum in this situa- 

 tion = 986,144,576 + 4,197,615 X 10 4 — 986,144,576 -f 41,976,150,000 = 

 42,962,294,576 — (20 p 3 p 3 c -f 15 f p 3 c 2 -f- 6 p p c 3 -J- c 4 ) -f 15 p 4 x p* = 

 15 p 4 <p 4 -f- 20 p 3 p 3 c -f- 15 jt> 2 <p 2 c 2 -f- 6 j9 <p c 3 ~{- c 4 , and is the upper number 

 in the Rank of the Biquadrate. 



a/. Then 42,962,294,576 X 4 = 171,849,178,304 - (15j9* p 1 -f 20 j» 3 p 3 c 



-f- 15 jo 2 <p 2 c" -f 6 jo <p c 3 -J- c 4 ) X cr 15 p* p* c -f 20 jp 3 <p 3 c 2 -{- 15 p" <p" c 3 -\- 

 6 2> p c* -f- c 5 , and is the product written in the Rank of the Quadratus 

 Cubi. 



V. Then since by Art. t') 38,618,058 = 6 p s , with its units put under 

 the 6th place of 171,849,178,304 so by Lem. 7) their sum in this situation = 

 171,849,178,304 -f 38,618,058 X 10 5 — 171,849,178,304 + 3,861,805,800,000 

 — 4,033,654,978,304 — (15jo 4 <p 4 c + 20 jw 3 p 3 c* + 15 p~ p z c 3 + 6 p p c* 4. c 5 ) 

 -f Gp 5 X p 5 = 6 p 5 p 5 -f-15 p l p* c -f 20 jy 3 <p 3 c 2 + 15 jtr <p 2 c 3 -f- 6 p p c* -f- c 5 , 

 and is the upper number in the Rank of the Quadratus Cubi. 



<_/. Then 4,033,654,978,304 x 4 = 16,134,619,913,216 = (6 p 5 p 5 -f 



15 p 4 <p, 4 c -f 20 ;/ p 3 c 2 4- 15 j» 2 <p 2 c 3 -|- 6 jo p c 4 + c 5 ) X c — 6 £> s <p s c -f 

 15 // <p 4 c z -f 20 p 3 p 3 c 3 -J- 15 jw* <p* c 4 -f 6 jo <p c s -j- c 6 ) and is less than 

 18,535,911,758,593, or R' p 6 -f C by Art /. 



<—»'. Then since 4 is the greatest number which answers this condi- 

 tion, so 4 is the third figure of the Root, and agrees with the third figure 

 of the Root found by the European method in Par. 34, Art. ¥.) 



ey„ And 16,134,619,913,216 expounds the third Subtrahend, which 

 agrees with the third Subtrahend found by the European method in 

 Par. 34, Art. c ff ). 



