e quadrature are aware that the several increases of
numbers of decimals to which [pi] has been carried have been indications of
a general increase in the power to calculate, and in courage to face the
labor. Here is a comparison of two different times. In the day of
Cocker,[137] the pupil was directed to perform a common subtraction with a
voice-accompaniment of this kind: '7 from 4 I cannot, but add 10, 7 from 14
remains 7, set down 7 and carry 1; 8 and 1 which I carry is 9, 9 from 2 I
cannot, etc.' We have before us the announcement of the following _table_,
undated, as open to inspection at the Crystal Palace, Sydenham, in two
diagrams of 7 ft. 2 in, by 6 ft. 6 in.: 'The figure 9 involved into the
912th power, and antecedent powers or involutions, containing upwards of
73,000 figures. Also, the proofs of the above, containing upwards of
146,000 figures. By Samuel Fancourt, of Mincing Lane, London, and completed
by him in the year 1837, at the age of sixteen. N.B. The whole operation
performed by simple arithmetic.' The young operator calculated by
successive squaring the 2d, 4th, 8th, etc., powers up to the 512th, with
proof by division. But 511 multiplications by 9, in the short (or 10-1)
way, would have been much easier. The 2d, 32d, 64th, 128th, 256th, and
512th powers are given at the back of the announcement. The powers of 2
have been calculated for many purposes. In Vol. II of his _Magia
Universalis Naturae et Artis_, Herbipoli, 1658, 4to, the Jesuit Gaspar
Schott[138] having discovered, on some grounds of theological {65} magic,
that the degrees of grace of the Virgin Mary were in number the 256th power
of 2, calculated that number. Whether or no his number correctly
represented the result he announced, he certainly calculated it rightly, as
we find by comparison with Mr. Shanks."

There is a point about Mr. Shanks's 608 figures of the value of [pi] which
attracts attention, perhaps without deserving it. It might be expected
that, in so many figures, the nine digits and the cipher would occur each
about the same number of times; that is, each about 61 times. But the fact
stands thus: 3 occurs 68 times; 9 and 2 occur 67 times each; 4 occurs 64
times; 1 and 6 occur 62 times each; 0 occurs 60 times; 8 occurs 58 times; 5
occurs 56 times; and 7 occurs only 44 times. Now, if all the digits were
equally likely, and 608 drawings were made, it is 45 to 1 against the
number of sevens being as distant from the probabl