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



laws, which have been mentioned, will remain unaffected, in 

 the former case, neither of them will hold throughout the 

 remaining terms." 



§ 12. In his letter quoted in § 7, Mr. Farey speaks of 

 " some very curious and elaborate Tables of c Complete decimal 

 Quotients calculated by Henry Goodwyn .... of which he 

 has printed a copious specimen for private circulation . . . ." 

 This appeared in the Philosophical Magazine for May 1816, 

 and would seem at first sight to point to the publication of a 

 specimen of the i Tabular Series ' previous to this date. A 

 copy of the specimen alluded to is in the library of the Royal 

 Society. It contains the i First Centenary ' only; and the 

 following address is prefixed : — " The Calculator of about a 

 Chiliad of Tables, from the application of which, in various 

 ways, he has himself derived considerable benefit, has been 

 induced to print the annexed Centenary as a Specimen. En- 

 couraged likewise by Friends — not, perhaps, quite impartial, 

 — to give them some publicity, yet still doubtful in himself 

 whether they deserve general notice, he adopts this method, 

 which, he trusts, will not be deemed obtrusive or impertinent, 

 of presenting this portion of his labours to a few Individuals. 

 To these Gentlemen, indeed, he has not, in all instances, the 

 good fortune of being personally known, but their scientific 

 knowledge and mathematical attainments are highly and 

 justly appreciated ; and, it is hoped, that amongst them, some 

 will have leisure, and inclination, to honour him with their 

 sentiments on the Specimen, which is thus submitted to their 

 consideration ; since he is anxious to confide to their decision, 

 whether the Tables themselves are worthy of publication, — or 

 may sink into oblivion with their Author. 



" As the above is a private Address, it seems needless for 

 him to add, that the name of anyone, who may favour him 

 with his opinion, shall not be divulged without his express 

 consent. Hy. Goodwyn, Blackheath, Kent, March 5th, 1816." 



The i First Centenary ' (pp. xiv + 18) is exactly similar to 

 the i First Centenary ' of 1818 ; and as the fractions are not 

 arranged in order of magnitude, it contains nothing that in 

 any way suggests either of the properties that form the subject 

 of this paper. It seems pretty clear that no part of the 

 • Tabular Series ' was published previous to 1818; for the title- 

 page to the tract of 1818 runs " The First Centenary ... to 

 which is now added a Tabular Series . . . ;" and the introduction 

 to the ( Tabular Series' (1818), commences, " Since the ' First 

 Centenary, &c.' and its Introduction were printed, which was 

 in March, 1816, it has appeared to the Calculator . . . ." 



It would thus appear that Mr. Goodwyn published no Table 



