162 HlSTOBY OF THE THEOBY OF NlJMBEBS. [CHAP. VI H. Goodwyn21 gave for each integer dg 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 every irreducible vulgar fraction < 1/2, whose numerator and denominator are both ^ 100, arranged in order of magnitude, up to 1/2. Goodwyn22' 23 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 vulgar 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 every denominator prime to 10 up to 1024; there is added a two-page table showing the quotient of each number g 1024 by its largest factor 2°56. For example, the entry in the "tabular series" under -/gV 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 T5?V= .086890243. Glaisher78 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 with tables showing seven figures of the reciprocals of numbers < 100000. F. T. Poselger24 considered the quotients 0, a, 6, ... and the remainders 1, a, /3, . . . obtained by dividing 1, A, A2, ... by the prime p] thus .. P P P P Adding, we see that the sum l+a+/3+ ... of the remainders of the period is a multiple mp of p; also, m(A — l) =a+&+ .... Set where A belongs to the exponent t modulo p. Then wThe first centenary 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+18; 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. 22 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-H53. MA 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. MAbhand. Ak. Wiss. Berlin (Math.), 1827, 21-36.