192 3£r Glaisher, On circulating decimals. [Oct. 28, 



we can add tlie remaining 88 digits of the period, viz. 4897... 5306 

 as in § 3. 



Similarly, entering the' tabular series' with ^g- we find '08689024, 

 entering the small table with 656 we find 41; and looking among 

 the periods of 41 in the 'table of circles' [the periods are quoted 

 near the beginning of this section] for one containing the digits 

 9024 we find the complete period to be 90243, so that the re- 

 quired decimal is -086890243. 



§ 5. It will thus be seen that Mr Goodwyn's tables give with 

 great ease the complete decimal value of a vulgar fraction, when 

 this vulgar fraction has been found in the 'tabular series'; but as 

 the fractions in the 'tabular series' are arranged in order of magni- 

 tude it is not very easy to find a given vulgar fraction among the 

 series of arguments. This Mr Goodwyn himself acknowledges in 

 his introduction to the 'Tabular series' (]823) for he writes (p. iv.) : 

 " But though it is easy to find, from this Tabular Series, a fraction 

 either exactly or nearly equivalent to a given decimal, the table does 

 not show, with equal readiuess, the decimal corresponding to a vulgar 

 fraction. This latter object is best effected by such an arrangement 

 as that adopted in the first of the tables contained in the ' Cente- 

 nary ' above referred to, under the title of ' Tables of complete 

 Decimal Quotients.' Tlie Table of complete Decimal Quotients was 

 the source, indeed, from which the Tabular Series was derived ; 

 and, again, the principal source from which the Table of complete 

 Decimal Quotients originated was 'The Table of Circles,' which was 

 necessarily completed before either the Table of Decimal Quotients 

 or the Tabular Series were began. But whether that table shall be 

 printed, or the present work completed, will depend on the recep- 

 tion which this result of the Computer's labour may meet with 

 from the public." 



The full title of the "First centenary ".is "The 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." The work is of quarto size. The ' First 

 centenary' contains pp. xiv. + 18: then follows the 'Tabular series' 

 (with a separate title-page) which contains pp. vii. + 30 (pp. 17 — 30 

 forming an appendix). The ' first centenary ' consists of a series 

 of a hundred small tables, corresponding to the first hundred 

 numbers. The following is a specimen : — 



