that the result (viz., 2,400,995,198,002,401) was equal to the square
of 48,999,951. He was again asked to multiply the product by 25, and
in naming the result (viz., 60,024,879,950,060,025) he said it was
equal to the square of 244,999,755.

"On being interrogated as to the manner in which he obtained these
results, the boy constantly said he did not know HOW the answers came
into his mind. In the act of multiplying two numbers together, and
in the raising of powers, it was evident (alike from the facts just
stated and from the motion of his lips) that SOME operation was going
forward in his mind; yet that operation could not (from the readiness
with which his answers were furnished) have been at all allied to the
usual modes of procedure, of which, indeed, he was entirely ignorant,
not being able to perform on paper a simple sum in multiplication or
division. But in the extraction of roots, and in the discovery of
the factors of large numbers, it did not appear that any operation
COULD take place, since he gave answers IMMEDIATELY, or in a very few
seconds, which, according to the ordinary methods, would have
required very difficult and laborious calculations, and prime numbers
cannot be recognised as such by any known rule."

I should hope that many of the above figures are wrong. I have
verified them carefully with Dr. Carpenter's quotation, but further
than this I cannot and will not go. Also I am happy to find that in
the end the boy overcame the mathematics, and turned out a useful but
by no means particularly calculating member of society.

The case, however, is typical of others in which persons have been
found able to do without apparent effort what in the great majority
of cases requires a long apprenticeship. It is needless to multiply