108 Solutions of Sir H.C. Englefield’s Mathematical Question. 
Chichester, Feh. 5, 1814. 
Sir H. ENGLEFIELD’s mathematical question published in your 
last number may, I believe, be solved in the following manner : 
The cube root of every number under a million must of course 
consist of not more than two places of figures. Of these the 
Jast may be immediately determined, by observing that with re- 
gard to six of the ten integers, viz. 1, 4, 5,6, 9, and 0, the 
unit’s place is always the same in the cube and its root ; while 
2 and 8, 3 and 7, are reciprocally changed one into the other in 
the processes of involution or evolution. 
The other figure of the root may be obtained as in the com- 
mon rule for extracting the third power, viz. by deducting 3 from 
the given number of places, and taking the root of the cube next 
less than the amount indicated by the one, two, or three figures 
remaining on the left hand. 
Thus, in the example proposed, where we have given the unit’s 
place 6, the two first figures 43, the number of places 6; . 
The unit’s place in the root will of course be 6, as in the cube, 
Deducting 3 from the number of places, there remain 3; 
therefore, affixing a cypher to the two first figures, we obtain 430, 
The cube next less than this is 8343, of which the root is 7. The 
root required is therefore 76. 
In this way, I find I can obtain the root of any cubic number 
almost instantaneously ; but I can hardly suppose so simple and 
obvious a principle to have escaped the notice of mathematicians: 
B: 
Paddington, Feb. 9, 1814. 
Sirs,—Havine been induced by the letter of Sir H. C. Engle- 
field, inserted in your last, to turn my attention to the subject 
of his inquiry; I am enabled by the following simple process to 
name immediately the root of any cube under a million, whose 
two first figures, last figure, and number of figures are given. 
All that appears necessary for the performance of this problem 
is a knowledge of the cubes of the digits. As the root consists 
of two figures, its tens will be indicated by, and must be sought 
for in, its two first figures, and its units in the last figure given. 
By obsérving the number of figures in the cube we previously 
discover the value of the two first figures given, whether they 
contain the units only, or the tens and units, or the hundreds and 
tens of the first figure of the root when cubed: having ascer- 
tained this, we compare these figures with the cubes of the digits, 
and call that digit with whose cube they correspond, or to which 
they are next in superiority, the first figure of the root. 
The last figure given, which is of course the unit of the cube, 
will produce the second figure, or unit of the root, by this simple 
rule; 
