Farey sequence

A Farey sequence is a sequence of numbers named after the English geologist John Farey (1766–1826) who wrote about such sequences in an article called "On a curious property of vulgar fractions" in the Philosophical Magazine in 1816. Farey says that he noted the "curious property" while examining the tables of Complete decimal quotients produced by Henry Goodwin. To obtain the Farey sequence for a fixed number n, consider all rational numbers between 0 and 1 which, when expressed in their lowest terms, have denominator (the number on the bottom of a fraction) not exceeding n. Write the sequence in ascending order of magnitude beginning with the smallest. The "curious property" is that each member of the sequence is equal to the rational whose numerator (the number on top of a fraction) is the sum of the numerators of the fractions on either side, and whose denominator is the sum of the denominators of the fractions on either side. For example, the Farey sequence for n = 5 is (0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1), from which it can be seen that 2/5 = (1 + 1)/(3 + 2), 1/3 = (1 + 2)/(4 + 5), 1/2 = (2 + 3)/(5 + 5), 2/3 = (3 + 3)/(5 + 4), and so forth. Farey wrote:


I am not acquainted whether this curious property of vulgar fractions has been before pointed out?; or whether it may admit of some easy or general demonstration?; which are points on which I should be glad to learn the sentiments of some of your mathematical readers ...


One "mathematical reader" was Augustin Cauchy, who gave the necessary proof in his Exercises de mathématique, published in the same year as Farey's article. Farey was not the first to notice the property. C. Haros, in 1802, wrote a paper on the approximation of decimal fractions by common fractions. He explains how to construct what is in fact the Farey sequence for n = 99 and Farey's "curious property" is built into his construction.