444 Method of naming Roots of Cubes under Ten Figures. 



3. The remainder, after dividing by 11, is 7 j whose correr 

 sponding nmiiber 6 being subtracted from 2 + 1 4 11, or 14, 

 leaves 8 for the middle figure of the root, which is therefore 182. 



The principles on which arc founded the rules for obtaining 

 thejirst and last figures of the root, are perhaps sufficiently 

 evident. The rule, however, for the middle figure, on which 

 alone I rest my claim for originality in the present instance, de- 

 pends upon the following obviou'5 property. If a given immber 

 divided by 1 1 leave a certain remainder r, and the cube of r 

 be divided by 1 1, the new remainder / left after this second di- 

 vision will be the same as what would remain, if the cube of 

 the original number were divided by 1 1 ; and further, if r be 

 0, 1, 2, 3, 4, 5, (), 7j S, 9, or 10, it will be found upon trial 

 that r' will be 0, 1, 8, 5, 9, 4, 7, 2, 6, 3, or 10: or inverselv, 

 if / be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, r will be 0, 1, 7, 9, 

 5, 3, 8, 6, 2, 4, or 10. Consequently since / is obtained by 

 dividing the proposed cube by 11, r may be found by the series ; 

 7. e. the remainder that should result from the division of the 

 roof by 1 1. But it is well known that if the remainder arising 

 from casting the elevens out of the ei>en digits of any given num- 

 ber, be subtracted from the remainder arising from casting the 

 elevens out of the odd digits, the resulting remainder is what 

 would remain, if the given number were divided by 11. IlencQ 

 if r be subtracted from the ximi of the first and last figures al- 

 ready found, borrowing 11, if necessary, it is obvious that the 

 difference will be the middle figure of the root. 



If, sir, the method I have just described, which is, to the best 

 of mv knowledge, entirely new as respects its extension to cubes 

 above six figures, and which certainly far surpasses the common 

 method in conciseness and facility of application, be deemed by 

 vou worthy of a i)lace in your Magazine, your insertion of it 

 wll oblige yours, &c. 



Islington, June Ifi, 1818. JoHN EVANS Jun. 



P. S. — Before I conclude this commuriication, allow me just 

 to hint, that in the rule I have suggested, the method of casting 

 out elevens niay be sometimes conveniently substituted for the 

 division by 1 1 . 



It may likewise not be improper to remark that the series 

 0, 1, 7, 9, 5, 3, S, G, 2, 4, 10, may be easily and firmly fixed 

 in the memory, by observing that if we omit the 0, the sum of 

 any two terms equally distant from the extremes is always 11 j 

 thus 1 -j- 10 = 7 + 4 &c.= 11: also that the first five consist 

 of all the odd numbers, the last five of all the even numbers un- 

 der 1 1 : and lastly, that the differences of the series are in the 

 following regular order, 6, 2, —4, —2, 5,-2, —4, 2, 6. 



LXXIX. His^ 



