[ 107° J v1 
XXIL. Solutions of Sir H. C. ENcirrienp’s Mathematica 
Question given in the last Number of this Journal. 
Liverpool, Feb. 4, 1814. 
Sirs,—I BEG leave to present you with the following solution 
to the question proposed by Sir H. C. Englefield, Bart. F.R.S. 
F.A.S. &c. in the last number of your most excellent Magazine ; 
and at the same time take the liberty to remark, that in the stating 
of the question I think a small typographical error has crept; 
for the two last, read the two first (11th line, page-64). 
All numbers under a million must consist of six figures or less, 
and consequently the cube reots of all such numbers must con- 
tain two figures or less. In numbers whose roots have only one 
figure, no difficulty can be found, as their roots can be obtained 
from simple inspection: the remaining case is then that of such 
cubes that their roots may contain two digits. In order to ob- 
tain the first digit, from the given number of figures subtract 
three, and of the remaining figures we by the supposition know 
the two first, and thus we obtain a number consisting of either 
one, two, or three digits: the nearest root of this number may 
be immediately found by simple inspection; and that root will 
be the first digit of the number required. 
The second digit of the root may be obtained by considering 
that all numbers must end either in 0; or in some of the nine 
digits: now the cubes of these digits are respectively 0, 1, 8, 27, 
64, 125, 216, 343, 512 and 729; therefore the cubes of all num- 
bers ending in 0, 1, 2,3, 4,5, 6,7, 8, or 9, must terminate in 0, 
1, 8, 7,4, 5, 6,3, 2,0r9. From this consideration it is evident, 
that knowing the terminal digit of any cube, we can immediately 
by inspection determine that of the root, and thus the first digit 
being before found, we obtain the root itself, 
Thus, in the example the learned Baronet has given: from the 
given number of figures (6) subtract three, and three will re- 
main; we therefore, affix a cypher to the two given figures 43, 
and obtain 430, whose nearest root is 7. The terminal figure 
being 6, we find that the last figure of the root must also be 6; 
and therefore the root required must he 76, whose cube is 438996. 
I take the liberty of proposing, for the amusement of your scien- 
tific readers, the two following questions: I think they have 
both been before proposed, but imagine that they admit of more 
interesting solutions than have hitherto appeared. 
1. Required a method of determining at sight, or by simple 
inspection, whether any given number be a prime number or not. 
2. By what methods could the Romans solve simple arith- 
metical questions with their mode of notation ? 
, I remain, gentlemen, 
4 Your obedient humble servant, 
To Messrs. Nicholson and Tilloch, Ecrrron 8, Eyrgs, 
