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

Second Class. 39
By interchanging the letters a, b, and at the same time the
letters, B in the formulae (151)-(154), we get four others,
which, however, may be regarded as not differing essentially
from those given.
SECTION III.-SECOND CLASS.
Formulae containing Five Elements.
40. NAPIER'S ANALOGIES.-If we multiply the identity
tan A. + tan B
tan - (A + B) = tn t- tan i by tan C,
1 - tan -A. tan ^and substitute for tan 4 A. tan 1 C, &c., their values, from
~ 28, Cor., we get
tan (A + B) tan sin(s - b) + sin(s - c) cos (a - b)
tan } (, + B) tanC —.. G = --
sin s - sin (s - a) cos (a +
Hence tan.i(.+B)= ' cotos C. (157)
cos ~.(a + b)
Similarly, tan (A-B) sin ( cot C. (158)
sin n (a + b)
Cor. 1.-If in the expression for tan (A + B) we change
cos to sin, we get the expression for tan ( (A - B).
41. Again, we have,a,. l tan a a cot c + tan } b cot ce
tan - (a + b) cot - e = -
tan(a+b)eot1 - tan a a tan b
and substituting for tan aa cot 2c, &c., their values from ~ 34,
Cors. 1, 2, we get
sin(A -E) + sin(B - E) cos (A -B)
tan(+b) cot = =
sin (C - E) - sin E cos (. + B)'
Hence tan (a+ b) = cs ( tan c. (159)
cos - (A + B)
sin (A - B)
Similarly, tan j (a - b) = sitan c. (160)
sin 4 (A + B)