AN ESSAY ON THE ROOTS OF INTEGERS, 



-f 15 a- <p° b* + 6 a<p b 5 -f b 5, having that Digit substituted for b. And 

 since 6 a s <p s b -{- 15 a* <p* b" 4. 20 a 3 <p 3 b 3 -f- 15 a? f b* -f 6 a <p b s -f- 6 6 , must 

 also be not greater than R <p 6 -J- B, and since this b must be a Digit so it 

 is evident that the present operation from Art. 1 to 10) is equivalent to 

 seeking the b of Par. 20). and since 3 by Art. g et seq.) is the found Digit, 

 so 3 also expounds the b of Par. 20) and then — 



g. Since by Art. t) 12 = 6 a, and since 3 contains one figure, so 

 123 —Gap -J-Z»by Lem. 6,) and is the upper number in the Rank of 

 the Latus. 



c. Then 123 X 3 = 369 = (6 a <p -f b) X h — 6 a p b -J- b\ and is 

 the product written in the Rank of the Square. 



r. Then since by Art. v) 60 = 15 a", with its units put under the 

 3d place of 369, so by Lem. 7,) their sum in this situation = 369 -f- 60 X 10 2 

 = 369 -J- 6,000 — 6,369 = (6 a <p b 4. b") f 1 5 a" f = 15 or f -f 6 a p b -j- b", 

 and is the upper number in the Rank of the Square. 



v. Then 6,369 X 3 = 19,107 = (15 a* f + 6 a <p + b°~) x b — 15 a~ tf- b 

 -J- 6 a<p b" -j- 6 3 , and is the product written in the Rank of the Cube. 



(p. Then since by Art. /) 160 — 20 «*', with its units put under the 

 4th place of 19,107, so by Lem. 7,) their sum in this situation — 19,107 

 + 160 X 10 3 = 19,107 -f 160,000 - 179,107 = (15 a 2 tfb + 6 a <p b Q -f V) 

 -f- 20 a 3 X <p 3 = 20 a 3 <p 3 4 15 a" <p" b 4. 6 a <p b" -j- b 3 , and is the upper 

 number in the Rank of the Cube. 



X- Then 179,107 X 3 = 537,321 = (20 a 3 f 4- 15 a 2 p 2 6 -f 6 a <p ¥ 4. 6 5 ) 

 X 6 — 20 a 3 <p 3 b 4- 15 a 2 f b 2 4- 6 a <p b 3 + b\ and is the product written 

 in the Rank of the Biquadrate. 



