194 Mr Olaisher, On circulating decimals. [Oct. 28, 



commences with the words " Since the ' First Centenary, &c.,' and 

 its Introduction were printed, which was in March, 1816..." and 

 in a paper which appeared in the Philosophical Magazine for May, 

 1816 (vol. xlvii. p. 385) Mr J. Farey speaks of "some 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 encouragement, either public or 

 individual, should appear to warrant such a step." 



Mr Farey V paper evidently relates to the ' Tabular series,' and 

 as it seems clear from the contents of the work of 1818 that this 

 appeared for the first time, with its introduction and appendix, as 

 an addition to the 'First Centenary' of 1818, it is to be presumed 

 that Mr Goodwyn showed Mr Farey some portion of the ' tabular 

 series' in manuscript in 1816. 



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 mathe- 

 matical attainments are highly and justly appreciated; and, it is hoped, that, 

 amongst them, some will have leism-e and inclination to honour him with their 

 sentiments on the Specimen, which is thus submitted to- their consideration j 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 any one, who may favour him with his opinion, shall not be divulged with- 

 out his express consent. Ht. Goodwyn, Blackheath, Kent, March 5th, 1816."" 



1 The object of Mr Farcy's paper is to draw attention to the following property 

 of vulgar fractions. If all the proper fractions in their lowest terms having both 

 numerator and denominator not greater than a given number n, be written down in 

 order of magnitude, then each fraction is equal to a fraction whose numerator and 

 denominator are respectively equal to the sum of the numerators and denominators 

 of the two fractions on each side of it, e.g. for n — 1 the fractions are 



111121213253456 oni\ 1 1 "*" Xrn 



Tj ¥j 5' ¥> 7' 'S' 6"' 2' bf 3> T> ?) "5J F> T' "'"^^ "^~7 + 5 ' ^^ & + ~i 



There are two theorems : (i) the difference of any two consecutive fractions is 

 equal to the reciprocal of the product of thek denominators; (ii) any three 

 consecutive fractions are connected by the relation mentioned above. Mr Farey 

 observed that the second theorem was true generally for any value of n, and 

 published it in the paper in the PhilosoiMcal Magazine cited above. The first 

 theorem (from which the second is at once deducible) is also true generally, but 

 Mr Farey does not allude to it. An account of Mr Farey's paper appeared in the Bulletin 

 de la societe philomatique de Paris, t. xii. (1816) p. 112 [by some error the page- 

 numbers 105 — 112 occur twice]. Cauchy proved both theorems on pp, 133 — 135 of 

 the same volume, and the proof is reprinted in his Exercices de Mathematiques, 

 t. II. (1826) pp. 114 — 116. Mr Goodwyn mentions both theorems on p. v of the 

 introduction to the 'tabular series' of 1818, but in the introduction to the 'tabular 

 series ' of 1823 (p. iv) he only refers to the latter. It thus appears that the second 

 theorem was first published by Mr Farey, and the first by Cauchy. In the British 

 Association report, 1873, (p. 33) I have erroneously ascribed both theorems to Cauchy. 

 [Since this paper was communicated to the Society I have written and sent to 

 the PMlosopJdeal Magazine a detailed historical account of the two theorems, with 

 demonstrations of them. J. W. L. G., February 20, 1879.] 



