ig 
Solutions of Sir H.C. Englefield’s Mathematical Question. 109 
rule: 1,4,5,6, 9,0, remain unaltered, while 3 and 7, 2 and 8, 
are respectively convertible into each other; or each when oc~ 
curring in the cube may be represented in the root by what it 
wants of 10. : 
The instance adduced by Sir H. C. Englefield is a cube of 
6 figures, the two first of which are 43, and the last is 6. Now 
as in a cube of 6 figures, the eubed tens of the root exist 
in the three first figures, these two first, here given, must con- 
tain the hundreds and tens of that cube, and 43 represent 430 
(the amount of the unit’s place being immaterial). The regular 
cube next below 430 is 343, whose root (7) is consequently the 
fist figure required. For the second figure, 6 in the cube gives, 
as before explained, 6 in the root, and the whole root is there- 
fore 76. | 
I know not if this be the method used by Sir H. C. Engle- 
field, and shall be happy to see any others suggested by your 
correspondents ; but must observe that this appears amply suffi- 
cient for the purpose, as by it any one, after a practice of less 
than half an hour, may tell the root of any number, within the 
bounds of the question, as soon as proposed to them. 
I remain, sirs, 
Your very obedient servant, 
Joun Ditton. 
To Messrs. Necholson and Tilloch. 
—_—— 
Lansdown Crescent, Bath, Feb. 10, 1814. 
Sirs,—Tue solution of the following question proposed is 
your last Magazine is not difficult. 
«Of any cube number under a million, give the figure of the 
unit, the two first figures, and the number of places, instantly 
and without any aid of writing to name its cube root.” 
In every cube number, two figures, the first and last, may be 
known by simple inspection; the first being the nearest cube 
root under the first or left hand period, whether consisting of 
one, twe, or three figures, and the last being invariably indicated 
by the final digit. Thus, if the final digit be 2, the last figure 
of the root will be 8; if it be 3, the root indicated will be 7: a 
table of the cubes of all the single digits at once exhibits the 
unvarying correspondence between the final digit of the cube and 
its root. Now, according to the conditions of the question, the 
cube whose root is sought, can never exceed six places of figures : 
consequently the root must consist merely of a first and a last 
figure, which we have seen may be instantly discovered by sim- 
inspection. It is true, the intermediate figures between the 
final digit and the two first figures are not given; but the number 
of places being given, it is (for an obvious reason) only necessary 
mentally 
