190 Mr Glaisher, On circulating decimals. [Oct. 28, 



57 _ 57 ^ 1 57 X 5* ^ 1 ( 35625 

 ^'^^■' 656" 2^x41" 10* 41 "10*1 41 



= ~ {868|f } = -08681^ = -0868 90243. 



It is scarcely necessary to remark that whatever the de- 

 nominator q may be, ^ (q) represents the number of remainders 

 that occur in the divisions, and also the number of proper fractions 

 which in their lowest terms have q for denominator. 



§ 4. The full titles of Mr Goodwyn's works of 1823 are 

 " A tabular series of decimal quotients for all the proper vulgar 

 fractions of which, when in their lowest terms, neither the nu- 

 merator nor the denominator is greater than 1000. London . . . 

 1823," and "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." The former contains 5 pages of 

 introduction, &c. and 153 pages of tables, and the latter contains 

 5 pages of introduction, &c. and 118 pages of tables; the paging 

 however ends at p. 111. The two form one volume in a copy 

 before me ; but they were also published separately as appears 

 from the words "the Table of Circles, which is subjoined to the 

 work, but sold, also, as a separate publication ..." which occur ia 

 the introduction to the 'Tabular series.' Mr Goodwyn's name 

 does not appear in connexion with either work, but there is no 

 doubt about the authorship, as reference is made to " a short 

 specimen" which the author had published in 1818 under the title 

 " The First Centenary of a Series of Concise and Useful Tables..."; 

 and this specimen bears the name of Henry Goodvvyn. 



Mr Goodwyn's volume of 1823 contains three tables, the most 

 important being the ' table of circles ' which occupies 107 pages. 

 It contains all the periods (or circles) of every denominator q, prime 

 tO' 10, from q = i to q— 1024. The following are specimens. 



41 



'02439 -04878 -67317 -14634 

 •97560 -95121 -92682 -85360 



127 



'007874015748031496062 

 99212598425196850393t 



•023622047244094488188 -070866141732283464566 



976377052755905511811 920133858267716535433 



