1878.] 



Mr Glaisher, On circulating decimals. 



199 



where r is the least of those primitive roots of q for which 

 as base the index of 10 is least, the values of r, corresponding 

 to the different values of q, being given at the end of the tabled It 

 is stated by the editor, Dr Sobering, (p. 497) that the manuscript 

 book from which the table was taken ends with the words : 

 Explicitus October 11, 1795. This table is referred to in Art. 316 

 of the Disquisitiones arithmeticce which begins " Secundum htec 

 principia pro omnibus denominatoribus formoe p"- infra 1000 

 tabulam periodorum necessariarum construximus, quam integram 

 sive etiam ulterius continuatam occasione data publici juris 

 faciemus." Tabula III. at the end of the Disquisitiones is a 

 similar table extending only as far as 100 : but there are differences 

 in points of detail. It will be seen- that the table in vol. ii. of 

 Gauss's Werhe is not so complete, or for some purposes so con- 

 veniently arranged, as Mr Goodwyn's 'table of circles.' I have 

 had the two tables compared, and the discrepancies decided by 

 division. As was to be expected, considering the circumstances of 

 the publication of Gauss's table, the values in Goodwyn were 

 found to be the more correct : but the results of this comparison 

 are not at present in a fit state for publication. 



§ 10. With regard to the arrangement of a table for the conver- 

 sion of vulgar fractions into decimals, the most elegant theoretical 

 method would be, I think, to give as it were the full divisions 

 in the form in which Mr Goodwyn actually performed them. 

 Thus the actual divisions for the number 39 are 



and this might be printed : 



■^ Gauss's table is explained in some detail by Professor Cayley, British Associa- 

 tion Report, 1875 (Bristol), pp. 317—318. 



YoL. III. Pt. v. 15 



