168 History of the Theory of Numbers. [Chap, vi 
*Th. Schroder^^ and J. Hartmann^^ treated periodic decimals. 
W. Shanks^^ gave Lambert's method (Bernoulli,^ end) for shortening 
the work of finding the length of the period for 1/A^. 
G. Salmon^^ remarked that the number 71 of digits in the period is known 
if we find two remainders which are powers of 2, since 10" = 2'' and 10'' = 2' 
imply 10"^"''^= 1; also if we find three remainders which are products of 
powers of 2 and 3. Muir''^ noted that it is here impUed that aq — bp equals 
n, whereas it is merely a multiple of n. 
J. W. L. Glaisher^" proved that, for any base r, 
1 
ir-iy 
.012...r-3r-l, 
a generalization of 1/81 = .012345679. 
W. Shanks^^ gave the length of the period for 1/p, when p is a prime 
< 30000, and a list of 69 errors or misprints in the table by Desmarest,^^ 
and 11 in that by Burckhardt.^° 
Shanks^- gave primes p for which the length n of the period for 1/p is a 
given number ^ 100, naturally incomplete. Shanks^^ gave additional 
entries p for n = 26, ?7 =99; noted corrections to his former table and stated 
that he had extended the table to 40000. Shanks^^ mentioned an extension 
in manuscript from 40000 to 60000. An extension to 120000 in manu- 
script was made by Shanks, 1875-1880. The manuscript, described by 
Cunningham, ^-^ who gave a list of errata, is in the Archives of the Royal 
Society of London. 
Shanks®^ stated that if a is the length of the period for 1/p, where p is a 
prime >5, that for 1/p'* is ap""'^ [^vithout the restriction by Thibault,^^ 
Muir^^]. 
G. de Coninck^® stated that, if the last digit (at the right) of A is 1 or 9, 
the last digit of the period for 1/A is 9 or 1 ; while, if A is a prime not ending 
in 1 or 9, its last digit is the same as the last in the period. 
Moret-Blanc^^ noted that the last property holds for any A not divisible 
by 2 or 5. For, if a is the integer defined by the period for 1/A, that for 
{A — l)/A is {A — l)a, whence a+ (A — 1)0 = 10" — 1, if n is the length of the 
periods. He noted corrections to the remaining nine laws stated by Coninck 
and implied that when corrected they become trivial or else known facts. 
"Progr. Ansbach, 1872, 
"Progr. Rinteln, 1872. 
"Messenger Math., 2, 1873, 41-43. 
"/bid., pp. 49-51, 80. 
^^Ibid., p. 188. 
«iProc. Roy. Soc. London, 22, 1873-4, 200-10, 384-8. Corrections by Workman.*" 
"/bwi., pp. 381^. Cf. Bertram", Loof." 
*Hbid., 23, 1874-5, 260-1. 
^Ibid., 24, 1875-6, 392. 
"Messenger Math., 3, 1874, 52-55. 
"Nouv. Ann. Math., (2), 13, 1874, 569-71; errata, 14, 1875, 191-2. 
*Ubid., (2), 14, 1875, 229-231. 
