148 AN ESSAY ON THE ROOTS OF INTEGERS, 



from the Latus thus taken, is always less than the given number. And 

 this difference is considerable in every power except the Square. And for 

 finding the denominator of the fraction in the operation on the Cube, we 

 may multiply the figures of the external Row into itself, increased by unit, 

 and the product into three, and add to that unit." 



After the prolix detail in the former part of this paper, it would be 

 very useless to make many comments on the above extract, and I have 

 only therefore to add a few cursory observations. 



51). The directions given for erazing the added items, and merely 

 writing their sum in the same place, will account for the Ranks being in 

 the extract directed to be so much shorter than they appear in the full 

 Diagram given by me. 



52). The Rationale of the directions given for the treatment of 

 cypher, when it occurs as one of the found figures of the Root, is so easily 

 understood, that it would be needless to elucidate them by any expla- 

 nation. 



53). The last sentence respecting the denominator of the fraction 

 in the Cube is also easily comprehended. The figures of the external 

 Row are those of the approximate integral Root written above the Pul- 

 pit Diagram, and are consequently — in of Par. 28). Then the Rule of 



the Text evidently is [ m X in -f- 1 X 3 -J- 1 = 3 m" -f 3 m -f- 1 == 

 (in -\- l) 3 — m 3 . 



54). What I have here said of the increase of the error of deficiency 

 corresponding to the increase of the index of the power, would, if true, be 



