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

86 Spherical Excess.
the perpendiculars AF, BG, C-H. Then evidently the three
pairs of triangles AB'F, CB'ff; A'BG, A',CH; ADF, BEG
are, two by two, equal in every respect. Hence it is easy to
see that the angle BAF = (A + B + C); that is, = 90~ + E;
therefore DAF = 90~ - E; also DF is the complement of A'B',
and DA of half AB; that is, DA = 90~ -.
2'
80. Cagnoli's Theorem.-From the values of cos f a, sin a b,
sin - c (~~ 32, 33), we get
sin b sin - sinA
sm E= --;
cos a
but sin A =
sin b sin '
Hence sin E= - (351)
2 ccos a b c os b cos
This is CAGNOLI'S Theorem.
81. By the transformation of ~ 78, we get
sin (A - E) = 2 cosan (352)
cos 1 a sin - b si n
Hence, by interchange of letters,
sin (B - E) = 2sin cosb sin c (353)
2 sinL a cos b b sin c
sin (7 - E) = --— ^- --- (354)
sin(C- sin - a s in b cos o ( 54
82. To find the value of cos E.
From the triangle DAF, ~ 79, we get
sin DAF = sin iFD sin AD,
cos A'B'
or cos E=; (355)
Cos -c