156 HISTORY OP THE THEOKY OF NUMBEKS. [CHAP, v
C. Haros249 proved the results rediscovered by Farey250 and Cauchy.252 J. Farey250 stated that if all the proper vulgar fractions in their lowest terms, having both numerator and denominator not exceeding a given number n, be arranged in order of magnitude, each fraction equals a fraction whose numerator and denominator equal respectively the sum of the numerators and sum of the denominators of the two fractions adjacent to it in the series. Thus, for n=5, the series is
and
*> i> f > f >
4 5+3' 5 3+2*
Henry Goodwyn mentioned this property on page 5 of the introduction to his "tabular series of decimal quotients" of 1818, published in 1816 for private circulation (see Goodwyn,21'22 Ch. VI), and is apparently to be credited with the theorem. It was ascribed to Goodwyn by C. W. Merri-field.261
A. L. Cauchy252 proved that, if a/6, a'/b', a"/b" are any three consecutive fractions of a Farey series, b and &' are relatively prime and a'b—ab' = 1 (so that a'/b'-a/b = l/W). Similarly, a"b''-a'V-1, so that a+a": b+b" = a': 6', as stated by Farey.
Stouvenel263 proved that, in a Farey series of order n, if two fractions a/b and c/b are complementary (i. e., have the sum unity), the same is true of the fraction preceding a/b and that following c/b. The two fractions adjacent to 1/2 are complementary and their common denominator is the greatest odd integer ^n. Hence 1/2 is the middle term of the series and two fractions equidistant from 1/2 are complementary. To find the third of three consecutive fractions a/b, a'/b', x/y, we have a-\-x = a'z, b+y = b'z (Farey), and we easily see that z is the greatest integer ^ (n+&)/&'.
M. A. Stern254 studied the sets m, n, and mf m+n, n, and m, 2m+n, m+n, m+2n, n, etc., obtained by interpolating the sum of consecutive terms. G. Eisenstein254a briefly considered such sets.
*A. Brocot255 considered the sets obtained by mediation [Farey] from 0/1,1/0: 011. 01121.
T> T' TT> T' T} T' T> in---*
Herzer266 and Hrabak257 gave tables with the limits 57 and 50.
G. H. Halphen258 considered a series of irreducible fractions, arranged hi order of magnitude, chosen according to a law such that if any fraction/ is excluded then also every fraction is excluded if its two terms are at least
M9Jour. de 1'dcole polyt, cah. 11, t. 4, 1802, 364-8.
"oPhilos. Mag. and Journal, London, 47,1816,385-6; [48,1816,204]; Bull. Sc. Soc. Philomatique
de Paris, (3), 3,1816,112. '"Matn. Quest. Educat. Times, 9, 1868, 92-5. "'Bull. Sc. Soc. Philomatique de Paris, (3), 3,1816,133-5. Reproduced in Exercices de Math.,
1, 1826, 114-6; Oeuvres, (2), 6, 1887, 146-8. M8Jour. de mathe"matiques, 5, 1840, 265-275.
mJoiir. fur Math., 55,1858, 193-220. 25*aBericht Ak. Wiss. Berlin, 1850, 41-42.
^Calcul des rouages par approximation, Paris, 1862. Lucas.378
^Tabellen, Basle, 1864. »7Tabellen-Werk, Leipzig, 1876.
M8Bull. Soc. Math. France, 5, 1876-7, 170-5.