332 Mr. J. W. L. Glaisher on a 



London : . . . . 1823 " (introduction pp. iii-v, and tabular series 

 1-153); (2) " 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 " (Introduction pp. iii-v, 

 and 118 pp. of tables). Both these works were anonymous ; 

 but there is no doubt about the authorship, as the Introduction 

 to the " Tabular series " commences, " A short specimen of 

 the following Tabular Series of Decimal Quotients was pub- 

 lished, by the Computer, in 1818, in ' The First Centenary of 

 a Series of Concise and Useful Tables,' " and the tract of 1818 

 bears Mr. Goodwyn's name. The ' Table of Circles ' was 

 subjoined to the l Tabular Series ' of 1823, but sold also as a 

 separate publication*. 



§ 10. The ' Tabular series ' of 1823 contains the first eight 

 (and occasionally nine or even ten) digits of the decimal values 

 of all fractions having both numerator and denominator not 

 exceeding 1000 arranged in order of magnitude, from y^oo 

 to^i. e. of those whose decimal values begin with *0). At 

 the conclusion is "End of Part I,; " and it was the author's in- 

 tention that Part II. should contain the fractions whose decimal 

 valuesbegin with-1, Part III. those beginning with '2, Part IV. 

 with "3, and Part Y. with '4. Parts I-Y. would thus contain 

 the decimal values of the fractions up to J; and it would be 

 unnecessary to print the other half of the table, as the argu- 

 ments and results would be complementary to those in the first 

 half. Part I. is all that was published. 



The i Table of Circles ' (1823) contains all the periods or 

 " circles " of the fractions having denominators prime to 10, 

 from 1 up to 1024. 



By means of the two tables the complete decimal value of 

 every vulgar fraction less than ^ having in its lowest terms 

 a denominator not exceeding 1000 may be obtained at once. 

 For example, from the 'Tabular Series' we find^\ 5 2 = '07918552 ; 

 and entering the i Table of Circles ' with 221 (i. e. with the 

 residual factor of the denominator when powers of 2 and 5 

 have been thrown out) we can complete the period from among 

 the "circles " of 221, by means of the "circle''' which contains 

 the digits 7918552, which is '9909502262443438914027149 

 3212669683257918552036i, so that the remaining digits of 

 the period are 036199 . . . 8325. 



The ' Tabular series ' of 1818 is similar to that of 1823, but 

 only includes fractions having both numerator and denominator 



* Introduction to Tabular Series (1823), p. iv. 



