Property of Vulgar Fractions. 335 



for the conversion of vulgar fractions into decimals, in which 

 the fractions were arranged in order of magnitude, prior to 

 the ' Tabular Series ' of 1818 ; and in this work both the pro- 

 perties are referred to. In the ' Tabular Series ' of 1823 only 

 the first is stated. The wording of Mr. Farey's letter implies 

 that he had seen not only the printed specimen of 1816, but 

 also Mr. Goodwyn's manuscript Tables. It is not clear, how- 

 ever, whether Mr. Farey discovered the property he enunciated 

 without any assistance from Mr. Goodwyn ; or whether, Mr. 

 Goodwyn having remarked the property as holding good in 

 the ' Tabular Series,' i. e. when the denominator is 100, Mr. 

 Farey merely extended it to the general case of any denomi- 

 nator. Whoever first began to arrange the fractions in order 

 of magnitude could scarcely fail to notice both properties ; and 

 the second, which relates to the difference of two consecutive 

 fractions, would probably present itself first. On the whole, 

 therefore, it seems most probable that only the extension to 

 the general case was due to Mr. Farey. In none of Mr. 

 Goodwyn's works is any allusion made to Mr. Farey or to 

 Cauchy. 



It seems curious that so elementary and remarkable a pro- 

 perty of fractions should not have been discovered until 1816. 

 It may of course be found that it had been published previously; 

 but supposing the discovery to be due to Mr. Goodwyn and 

 Mr. Farey, an explanation might be afforded by the fact that 

 the ' Tabular Series 'is probably the earliest Table of the kind, 

 and that the property would not be likely to present itself to 

 any one who had not arranged a complete series of proper 

 fractions having denominators less than a given number in 

 order of magnitude. 



§ 13. Mr. Goodwyn's works are almost unknown; and those 

 of 1823, which are the most important, are, as mentioned in 

 § 9, anonymous. The only references I have seen to them 

 are contained in Mr. Merrifield's question quoted in § 8, and 

 in De Morgan's articles on Tables in the Penny and English 

 Cyclopaedias. In the latter the works of 1823 only are de- 

 scribed, and by inadvertence the ' Tabular Series' is stated to 

 contain all fractions " which in their lowest terms have a 

 numerator not exceeding 99, and a denominator not exceeding 

 1000, in order of magnitude." In the English Cyclopaedia 

 (1861) De Morgan continues: — "Mr. Woolgar is our authority 

 for saying that there was a previous work by Goodwyn, ' First 

 Centenary of concise and useful Tables of Decimal Quotients ' 

 (1818) 4to. Mr. Goodwyn (of Blackheath) was an indefati- 

 gable calculator; and the preceding Tables are the only ones of 

 the kind which are published. His manuscripts, an enormous 



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