10 AN ESSAY ON THE ROOTS OF INTEGERS, 



<x ! \ The upper number transferred in the Rank of the Latus = 140,736= 



<— > 17 . The sought number or sixth figure of the Root = 7. 



e^\ The sixth Subtrahend =29,823,008,824,922,999,565,181,681,169. 



^J\ The sixth and last Remainder = 987,654,321. 



Hence then the integral approximate Root of the given number is 

 234,567, and the last Remainder 987,654,321 is the Numerator of a Frac- 

 tion, which is to be added to the integral Root, so as to afford a nearer 

 approximate to truth. And the Denominator of this Fraction is found 

 by the following process — 



£\ Add the last figure of the Root to the upper number in the Rank 

 of the Latus, and write the sum opposite to and immediately above it. 

 This sum is now the upper number in the Rank of the Latus. 



q. Multiply the last figure of the Root into this sum, and write the 

 product in the Rank of the Square opposite to and immediately above the 

 upper number in that Rank, 



£. Add together in the Rank of the Square this product and that 

 upper number, and write the sum opposite to and immediately above them, 

 This sum is now the upper number in the Rank of the Square. 



£. Multiply the last figure of the Root into this sum, and write the 

 product in the Rank of the Cube opposite to and immediately above the 

 upper number in that Rank. 



