A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

146 Applications of Spherical Trigonometry.
132. Legendre's Theorem.If the sides of a spherical triangle be small compared with the
radius of the sphere, and if a plane triangle be constructed whose
sides are equal in length to those of the spherical triangle, then each
angle of the spherical triangle exceeds the corresponding angle of
the plane triangle by one-third of the spherical excess.
DEM.-Let a, b, c be the lengths of the sides of the spherical
triangle, r the radius of the sphere, then the circular measures
of the sides are
a b c
r' r? r'
respectively; hence
a b c
cos - - cos-. cos -
Cr r r
cos A =. b. c
sin-. snr r
a: 6
and, substituting for cos-, cos -, &c., their values given in
Pl. Trig., ~ 158, we get, neglecting powers higher than the
fourth of 1
r'
cos = b2t —
l-Ar2r2 24r- 2r2+ 24r4J 1 — 2r2C 24r4r/
r2 6r2 6r
(b2c-i-2-'a2 a4- 64-c - 6b22\ / b2 + c2\
2bc 24bcr2 ) 6r2
(b2+c2-a2 a4-b-c4_6^ C2) b2 + C2
= + 1 + 6r2
~ 2bc 24bcr2 6r2
b2 + c2 _ a2 a4 + 4 C4 + 2(a2b2+ b22+ a2)
2bc 246cr2