Solutions of Sir H.C. Englefieli’s Mathematical Question. 114 
membering the cubes of the nine digits. For all numbers ending 
in the same digit, having their cubes ending also in a constant 
digit, it follows that the terminating figure of the cube will im- 
mediately indicate the unit’s place of the root ; that is, If the ter- 
Mminating figure of the cube be Be pe rs epi Be. bg iv 
that of the Toot Ce ale 8, 7s 4, 5,6, 35 2 9. 
And with regard to the other - of the root, or that in the 
tens place, it is found with equal ease from the leading figures of 
the cube, viz. If the three rght-hand figures be supposed to be 
pointed off, the nearest cube root to the remaining figures will 
be the figure in the tens place of the root: or, which is the same, 
If after pointing off the three right-hand figures, those remaining 
exceed 1, 8, 27, 64, Lass 216, &c. the figure in the tens place 
will be accordingly 1,2 2,3, 4, 5, 6, &ec. and which, therefore, is 
known as readily from the thrée leading figures of the cube (or 
from the two leading ones, if the number of them be also known) 
@s that of the unit’s place is by the termination of the cube. 
Suppose, for example, the cube root of 658503 were required. 
The nearest integral cube root of 658 is 8, and the termi- 
nating figure being 3, that of the root is 7; therefore the root 
is 87. And it would have been just the same if we had only 
known the two first figures 65, provided we had at the same time 
known that there were ¢hree figures in the period, or, which is the 
same, Six figures in the pr -oposed cube; and in the same manner 
may the cube root of any other cube number, within the pro- 
posed limits, be immediately ascertained. 
It is undoubtedly on these principles that Z. Colbourn performs 
his extractions ‘of the cube root, and which he can apply with 
great ease to 12 figures, or to four, i in the root. But in order to 
effect this, it is necessary for him to know the two terminating 
figures of the first 100 numbers, which he is perfectly master of; 
but whether absolutely from memory, or from a very rapid hia 
tiplication it is difficult to say. At all events, he is able to name 
them with great rapidity; and I have by mea table of this kind, 
which was set down from his dictation, and is correct in every 
figure. 
He undoubtedly owes much to an extraordinary memory, as 
well as to a remarkably ready perception of the qualities and pro- 
perties of numbers ; of which I could mention many instances, 
relating to the summation of arithmetical and geometrical pro- 
gressions, the naming of prime numbers, &c. In one instance, 
he repeated to me every prime number in its order from 1 to 600, 
with only one mistake, ‘which he corrected immediately, and was 
very anxious to he allowed to proceed to 1000. But here it was 
obvious that he did not depend upon his memory only, but on a 
sort of computation. 1 know it has been stated that what he can 
: perform 
