Property of Vulgar Fractions. 331 



as possible. All that is assumed is that if — be the last con- 

 vergent to j- y then aq— bp = ±1. It is a well-known theorem, 



and one given in elementary treatises on algebra, that any 

 convergent to a fraction is nearer to it than any other 

 fraction having a less denominator than that of the con- 

 vergent ; and the proof* of this includes a proposition proved 



in § 2, viz. that - is nearer to j than any fraction on the 



same side of it having a less denominator than b. But it was 

 more convenient to establish the result in § 2 independently, for 

 the sake of completeness, as in any case it must have been 



a — 



shown that ^ — — was nearer to T than any fraction on the same 

 b — q b 



side of it having a denominator less than b. The lemma in § 2, 



which really forms an interesting theorem, cannot of course be 



new, but it is certainly little known. I may mention that, as 



far as I am aware, the only complete investigation of the theory 



v 

 of the successive minima of the expression - — 0, 6 being a 



3fi 



given fraction, is given by Professor H. J. S. Smith in the 

 addition to his " Note on Continued Fractions" (' Messenger of 

 Mathematics/ vol. vi. (May 1876) pp. 7-14). 



§ 9. Mr. Henry Goodwyn, who is referred to by Mr. Farey 

 in his letter in the Philosophical Magazine (see § 7), pub- 

 lished in 1818 a quarto tract entitled a 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, ror the deno- 

 minator, is greater than 100 : with the equivalent vulgar 

 fractions prefixed. By Henry Goodwyn. London: 1818." 

 There is an introduction (pp. v-xiv), followed by the first 

 centenary itself, which occupies pp. 1-18. Then there is a 

 fresh title-page for the " Tabular Series," which consists of pp. 

 iii-vii (introduction), pp. 1-15 (the tabular series itself), and 

 pp. 17-30 (appendix). 



In 1823 Mr. Goodwyn published two octavo works, Adz.: — 

 (1) " A tabular Series of decimal Quotients for all the proper 

 vulgar fractions of which, when in their lowest terms, neither 

 the numerator nor the denominator is greater than 1000. 



* Todhunter's < Algebra/ art. 610; Gross's ' Algebra/ art. 149. 



