An introduction to the theory of numbers

music, sound, comets, carriage wheels and decimal coinage.As a geologist, Farey is apparently forgotten, and, if that were all there were to say about him, I doubt that he would find his way into the Dictionary of National Biography today.It is really very astonishing that Farey's official biographer should be so completely unaware of his subject's one real title to fame.For, in spite of the Dictionary of National Biography, Farey is immortal; his name stands prominently in Dickson's History and in the German encyclopaedia of mathematics, and there is no number-theorist who has not heard of "Farey's series".Just once in his life Mr. Farey rose above mediocrity and made an original observation.He did not understand very well what he was doing, and he was too weak a mathematician to prove the quite simple theorem he had discovered.It is evident also that he did not consider his discovery, which is stated in a letter of about half a page, at all important; the editor of the Philosophical Magazine printed a very stupid criticism in the next volume, and Farey, usually a rather acrid controversialist, ignored it completely.He had obviously no idea that this casual letter was the one event of real importance in his life.We may be tempted to think that Farey was very lucky ; but a man who has made an observation that has escaped Fermât and Euler deserves any luck that comes his way.*Farey's observation was this.The Farey series of order n is the series, in order of magnitude, of the irreducible rational fractions between 0 and 1 whose denominators do not exceed n.Thus 0 11112 12 3 1 ryi'T'T'T' 3 ; s ' 7 ; 2 ' 432534561 T' T' T' T' T' T' 6~' T' T* It should be added that Farey's discovery had been anticipated 14 years before by C. Haros: see Dickson's History, vol. 1, p. 156.Cauchy happened to see Farey's note and attributed the theorem to him, and everyone else has followed Cauchy's example.