140 AN ESSAY ON THE ROOTS OF INTEGERS, 



70,073,957,724,640,906 



Hence the deficiency in this case is 728 — 726 ■ — and is 



86,482,825,840,140,625 

 16,408,868,115,499,719 



again diminished to 1 — • 



86,482,825,840,140,625 



45.) It is obvious that the operation and exposition may easily be 

 extended to any other power, by the method of Par. 30), and by having as 

 many Ranks as there are units in the index of the power and analogically 

 adapting the circle of operations to these Ranks. It would be both curi- 

 ous and entertaining to investigate those properties of figurate numbers 

 by which the upper transferred number in each Rank becomes the found 

 figures of the Root involved to the index of that Rank and multiplied by 

 the proper co-efficient of the Binomial Theorem, and the succeeding ope- 

 rations finally produce for each period, the last found figures of the Root 

 multiplied by ten, and having then added the next figure of the Root, 

 and the sum being involved to the index of the given power, and then 

 having subtracted the last found figures of the Root multiplied by ten, 

 and involved to the index of the given power. But such an inquiry would 

 swell the present paper beyond all bounds of moderation, and must there- 

 fore be omitted. 



4(3.) From all this ample detail, it appears that the advantages pro- 

 posed by the Arabian Arithmeticians in the complicated apparatus of 

 calculation required for the Pulpit Diagram, is first, that the Root may be 

 extracted, as it were mechanically, without previous knowledge of the 

 co-efficients of the Binomial Theorem, which are here produced by the mere 

 arrangement of the Ranks ; and next, that throughout all the intricacies 

 of this operation it should never be necessary to multiply by a number 

 higher than a Digit. I shall not undertake to decide, whether these objects 

 were sufficiently important to justify the employment of means so 



