162 History of the Theory of Numbers. [Chap, vi 
H. Goodw-jTi-^ gave for each integer d^ 100 a table of the periods for 
n/d, for the various integers n<d and prime to d. Also, a table giving 
the first eight digits of the decimal equivalent to everj^ irreducible vulgar 
fraction < 1/2, whose numerator and denominator are both ^ 100, arranged 
in order of magnitude, up to 1/2. 
GoodwATi"' -^ was without doubt the author of two tables, which refer 
to the preceding ''short specimen" by the same author. The first gives 
the first eight digits of the decimal equivalent to every irreducible \'ulgar 
fraction, whose numerator and denominator are both ^ 1000, from 1/1000 
to 99/991 arranged in order of magnitude. In the second volume, the 
"table of circles" occupies 107 pages and contains all the periods (circles) 
of ever}' denominator prime to 10 up to 1024; there is added a two-page 
table showing the quotient of each number ^ 1024 by its largest factor 2°5''. 
For example, the entry in the "tabular series" under -^^ is .08689024. 
The entry in the two-page table under 656 is 41. Of the various entries 
under 41 in the "table of circles," the one containing the digits 9024 gives 
the complete period 90243. Hence /^V = -086890243. 
Glaisher"^ gave a detailed account of Goodwyn's tables and checks on 
them. They are described in the British Assoc. Report, 1873, pp. 31-34, 
along -wdth tables showing seven figures of the reciprocals of numbers 
< 100000. 
F. T. Poselger'^ considered the quotients 0, a, h,. . . and the remainders 
1, a, j8, . . . obtained by di\'iding 1, ^, A~, ... by the prime p; thus 
A a A' .,.B 
—=a-i — , — = aA+b-\ — ,.... 
P P P P 
Adding, we see that the sum lH-a-}-/3-|- ... of the remainders of the period 
is a multiple TTzp of p; also, w(A — 1) =a+6-f- . . .. Set 
M = k+...+hA'-'--\-aA*-\ 
where A belongs to the exponent t modulo p. Then 
— = -+MS, S = l+A'+ . . . +A^"-'\ 
P P 
"The first centenarj' of a series of concise and useful tables of all the complete decimal quotients 
which can arise from dividing a unit, or any whole number less than each divisor, by all 
integers from 1 to 1024. To which is now added a tabular series of complete decimal 
quotients for all the proper vulgar fractions of which, when in their lowest terms, neither 
the numerator nor the denominator is greater than 100; with the equivalent vulgar 
fractions prefixed. By Henry Goodwyn, London, 1818, pp. xiv + lS; vii+30. The first 
part was printed in 1816 for private circulation and cited by J. Farey in Philos. Mag. and 
Journal, London, 47, 1816, 385. 
"A tabular series of decimal quotients for all the proper vulgar fractions of which, when in their 
lowest terms, neither the numerator nor the denominator is greater than 1000, London, 
1823, pp. v + 153. 
"A table of the circles arising from the division of a unit, or any other whole number, by all the 
integers from 1 to 1024; being all the pure decimal quotients that can arise from this 
source, London, 1823, pp. v + 118. 
"Abhand. Ak. Wiss. BerUn (Math.), 1827, 21-36. 
