1878,] Mr Glaisher, On circulating decimals. 203 



primes, which was published anonymously in the first series of the 

 Messenger of Mathematics, vol. II. pp. 1 — Q, is a short but valuable 

 contribution to the subject. When engaged during the early part 

 of the present year in forming as complete a list as possible of the 

 rules relating to circulating decimals, I found this paper more 

 useful than any other that I met with. On p. 39 of the same 

 volume of the Messenger the full decimal value of yw6J (which has 

 a period of 18G0 digits) is given. This was intended to illustrate 

 Mr Hudson's paper, and it is pointed out that the complemen- 

 tary portion begins at the 931st digit, the double at the 1202nd, 

 the half at the 660th, &c. The number of the Messenger contain- 

 ing Mr Hudson's paper, and the period of 1861, was published in 

 March, 1863, and as the folding shee't of Messrs Suffield and Lunu 

 containing the period of 7,699 (see § 8) is dated April 29, 1863, 

 the latter was probably a consequence of Mr Hudson's writings. 

 Mr Shanks subsequently calculated the period of the prime 17,389, 

 which contains 17,388 digits, but it has not been printed \ Mr 

 Suffield published in 1863^ a tract, Synthetic division in arithmetic, 

 with some introductory remarks on the period of circidating decimals 

 (pp. iv. + 19), and also, on a separate leaf, A specimen of ' Synthetic 

 Division' (without date, but probably issued immediately after the 

 tract). The 'introductory remarks on the period of circulating 

 decimals,' which occupy three pages, contain the more important 

 rules on the subject, and the whole method of synthetic division 

 is closely connected with the theory of circulating decimals, so 

 that the work is an important one. As the book is scarce it 

 may be convenient to mention that The rationale of circulating 

 numbers by Henry Clarke (London, 1777 and 1794), merely 

 contains the ordinary arithmetical processes for the addition of 

 circulating decind.als, conversions of vulgar fractions into decimals, 

 &c., and does not relate to the theoretical principles. I do not 

 know of any account of the properties of circulating decimals 

 which at all approaches completeness ; and the list of rules in 

 § 2, although purposely restricted, is more complete than any 

 account that I have met with. On the other hand, however, 

 it would be very difficult to discover any property that had not 

 been previously published in some form or another. 



1 See Proc. Roy. Soc. vol. xv. p. 429 (1867). 



2 The preface is dated January, 1863. 



