The Farey series Fn consists of all the proper fractions, in their lowest terms and in order of magnitude from 0/1 to 1/1, whose denominators do not exceed n. The series Fn may be formed from Fn–1 by inserting the mediant (a + c)/(b + d) between two adjacent terms a/b and c/d of Fn–1 provided b + d ≤ n. The process requires inspection of terms and rewriting of earlier series. Professor Neville showed (Proc. London Math. Soc., Series 2, Vol. 51), how the order n may be increased approximately in the ratio 3 to 2 at one time, but inspection and repetition will still be necessary.