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



nator is 7. The property is stated to be taken from the Phi- 

 losophical Magazine for May 1816; but Mr. Farey's name is 

 not mentioned. 



A proof was given by Canchy in the same volume of the 

 Bulletin (pp. 133-135), under the title " Demonstration d'un 

 theoreme curieux sur les nombres ;" this proof was reprinted 

 in t. i. (1826) pp. 114-116 of his Exercices de Mathematiques. 

 The first three paragraphs of the paper in the Bulletin 

 are : — 



" On trouve dans le dernier ISTumero de ce Bulletin l'enonce 

 d'une propriete remarquable des fractions ordinaires observee 

 par M. J. Farey. 



" Cette propriete n'est qu'un simple corollaire d'un theoreme 

 curieux que je vais commencer par etablir. 



" Theoreme. — Si, apres avoir range dans leur ordre de gran- 

 deur les fractions irreductibles dont le denominateur n'excede 

 pas un nombre entier donne, on prend a volonte, dans la suite 

 ainsi formee, deux fractions consecutives, leurs denominateurs 

 seront premiers entre eux, et elles auront pour difference une 

 nouvelle fraction dont la numerateur sera l'unite." 



Cauchy thus discovers for himself and proves the second 



property, viz. that ~ — ~ — ^-r-, and deduces the first, or Mr. 



Farey 's, property from it. It is clear, from the first paragraph 

 of his paper, that he must have referred to Mr. Farey's ori- 

 ginal letter in the Philosophical Magazine, since, as has been 

 mentioned, Mr. Farey's name does not occur in the account 

 in the Bulletin. 



m § 8. In the ' Educational Times'* for 1868 Mr. C. W. Mer- 

 rifield proposed the question, " Mr. Henry Goodwyn published 

 in 1818 a table, in which all proper fractions (reduced to their 

 lowest terms) in which the denominator did not exceed 100, nor 

 the numerator 50, were arranged in order of magnitude. He' 

 observed the following property, a general proof of which is 



required. Let any three consecutive fractions be ^; ^p ^, 



Ni+N" N r 1 2 3 



then — — ppr = tT»" anc ^ indicated a mode of solution depend- 



ing on the property D^ — D 2 Ni = 1 : a solution was also given 

 by Mr. Morgan Jenkins. 



Probably all the proofs of the second property depend ulti- 

 mately on the same principles. In the proof in § § 2 and 3 an 

 attempt has been made to render the analysis as elementary 



* Mathematical Questions with their Solutions. From the ' Educational 

 Times/ vol. ix. pp. 92-95. 



