156 History of the Theory of Numbers. [Chap, v 
C. Haros-"*^ proved the results rediscovered by Farey-^° and Caiichy.^^- 
J. Farey-^" 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 frac- 
tion 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 
1112 13 2 3 4 
Z' T' -JT' T' IT' T' 7' T' T' 
and 
1_1 + 1 2_1 + 1. 
4 5+3' 5 3+2* 
Henry Goodwyn mentioned this property on page 5 of the introduction 
to his "tabular series of decmial quotients" of 1818, published in 1816 for 
private circulation (see Goodwyn,^^' ^- Ch. VI), and is apparently to be 
credited with the theorem. It was ascribed to Goodwyn by C. W. Merri- 
field.2" 
A. L. Cauchy^^^ proved that, if a/b, a'/b', a"/b" are any three consecu- 
tive fractions of a Farey series, b and b' are relatively prime and a'b—ab' = 1 
(so that a'/b'-a/b = l/bb'). Similarly, a"b'-a'b" = l, so that a+a": b+b" 
= a': b', as stated by Farey. 
StouveneP^^ 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-\-b)/b', 
M. A. Stern-^^ studied the sets m, n, and m, m-\-n, n, and m, 2m-\-n, 
m-\-n, m-\-2n, n, etc., obtained by interpolating the sum of consecutive 
terms. G. Eisenstein^^" briefly considered such sets. 
*A. Brocot^^'' considered the sets obtained by mediation [Farey] from 
U/1, 1/0: oil. 01121. 
T' T' ITJ T' Y' 1' T' TJ">- • •• 
Herzer^^^ and Hrabak^^^ gave tables with the limits 57 and 50. 
G. H. Halphen^^^ considered a series of irreducible fractions, arranged 
in 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 
2<9Jour. de I'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. 
"'Math. Quest. Educat. Times, 9, 1868, 92-5. 
"*Bufl. 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. 
"»Jour. de mathdmatiques, 5, 1840, 265-275. 
»«Jour. fur Math., 55, 1858, 193-220. "laBericht Ak. Wiss. Berlin, 1850, 41^2. 
"'Calcul des rouages par approximation, Paris, 1862. Lucas.'" 
2^«Tabellen, Basle, 1864. ""Tabellen-Werk, Leipzig, 1876. 
"'Bull. Soc. Math. France, 5, 1876-7, 170-5. 
