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LXXVTII On a new and expedil'iom Method of naming at 

 sight the Roots ojcowplele Cules under Ten Figures. By 

 John Evans Jiiu. A.M. 

 Sir, _ In a former volume of your valuable Miscellany, I re- 

 collect having met with some communications, in which is given 

 a method of naniing at sight the root of any complete cube num- 

 ber under seven figures. The principle was bneflv this :-The 

 root will evidently not exceed two figu.es ;-of which the first or 

 digit of tens, will be the root of the greatest cube contained m 

 the 4th, 5th and 6th figures of the given number, counting from 

 the right hand:-and the latter, or terminaling figure of the 

 root, will be found by considering that if the cube terminate m 

 1, 4, 5, 6, or 9, the root will terminate in the^f^mejigure ; it 

 in 2, 3, 7, or 8, the termination of the root will be b, /, J, or ^, 

 the respective deficiencies from 10. 



Now, sir, it has occurred to me, that by uniting another p in- 

 ciple with the method just alluded to, we may name at sight, it 

 required, the root of any cube number not exceeding nine figures. 

 This is effected by the following rules : 



1. It is obvious that the required root will never exceed tnree 

 figures ■.-i\^e first of which will be the root of the greatest cube 

 contained in the 7th, 8th, and 9th figures counting from t. 



the 



"^2!*The terminating figure of the root may be found as before 

 from the termination of the given cube. 



3 To obtain the middle figure of the root, divide the guen 

 cube by 11, and according as the remainder is, 

 0,1, 2, 3, 4, 5, 6, 7, 8,9, or 10, 

 take 0, 1,7.9, 5,3, 8, 6, 2,4, or 10; 

 which being subtracted from the sum of the>5^ and /as^ figures 

 Tfound as alcove) borrowing 11, if necessary, will give the miu- 

 J/e figure of the root. -niQrcooR 



Take as an example the complete cube f '>43jS336. 



1 The greatest iube in 504, the 7th, Sth, and 9th figures, 

 Is 343 or (7)': therefore the fust figure of the root is 7- 



2 Since the cube ends in 6, the icrmination o\ the root is 6 



3 Dividing the given cube l,y 1 1, the reimnnder is 9, to which 

 4 corresponds in tiie series given above. Then the two former 

 Lures 7 and 6 added together give 13, from winch 4 being de- 

 ducted, leaves 9 for the middle figure of the root. Hence the 



root recpiircd is 796. «no«i;«8 



Again, let the proposed cube be 602856^. 



1. The root of the greatest cube m 6 is I. 



2. The tewnination will be 10 — 8 = 2. 



3. The 



