Property of Vulgar Fractions. 329 



the fraction next greater than — ■ —z. in magnitude, and 



& m + r + 1 & J 



the series of fractions greater in magnitude than — 



° m+r+1 



remains unaffected. Thus if the property is true for denomi- 

 nators not exceeding m + r, it is true for denominators not 

 exceeding m + r + 1. But it is true for denominators not ex- 

 ceeding m or 7n+l, and is therefore true generally. 



If, therefore, in any such series as those considered in § § 1-5 

 all the fractions having numerators exceeding any given num- 

 ber be removed, the properties will still hold good for the 

 series of fractions that remain. For instance, if in the ex- 

 ample at the end of § 5 the fractions having numerators 

 exceeding 45 be removed, there remains the series 

 20 27 34 41 7 

 303' 409' 515' 62l' 106' 

 for which the properties are true. 



§ 7. I come now to the history of the properties stated in § 1. 

 The first property was published by Mr. John Farey, in a 

 letter to the Philosophical Magazine (vol. xlvii. 1816, pp. 385- 

 386), entitled " On a curious Property of Vulgar Fractions." 

 This letter commences : — " On examining lately, some verv 

 curious and elaborate Tables of ' Complete decimal Quotients,' 

 calculated by Henry Goodwyn, Esq. of Blackheath, of which 

 he has printed a copious specimen, for private circulation 

 among curious and practical calculators, preparatory to the 

 printing of the whole of these useful Tables, if sufficient en- 

 couragement, either public or individual, should appear to 

 warrant such a step : I was fortunate while so doing to deduce 

 from them the following general property." Mr. Farey then 



states the first property, viz. that if -±, ^ — be three conse- 

 cutive fractions, then -^ = * , 3 , and illustrates it in the case 



where the greatest denominator is 5. He concludes with the 

 words, " I am not acquainted, whether this curious property of 

 vulgar fraction's has been pointed out ? ; or whether it may 

 admit of any easy or general demonstration ?; which are points 

 on which I shall be glad to learn the sentiments of some of 

 your mathemetical readers." 



An account of the property appeared under the title " Pro- 

 priete curieuse des fractions ordinaires," in the Bulletin des 

 Sciences par la Societe Philomatique de Paris for 1816 

 p. 112*. An example is given in which the greatest denomi- 



* By an error of paging, the page-numbers 105-112 occur twice, and 

 p. 121 follows the second p. 112. It is the first p. 112 that is here re- 

 ferred to. 



