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

40 Connecting Sides and Angles of a Spherical Triangle.
Cor. —If we change cos to sin in the expression for tan 2 (a - b)
we get that for tan f (a + b).
The theorems contained in the equations (157)-(160) may be
expressed as proportions, and are called NAPIER'S ANALOGIES,
after their discoverer. It may be remarked that the last pair
can be got from the first by means of the polar triangle; also
that the second and fourth may be inferred from the first and
third by multiplying them respectively by the equation
tan (A - B) tan 2 (a - b)
tan (ZA + B) tan (a + )'
Several proofs of these important theorems are known, but the
foregoing are probably the simplest.
EXERCISES.-VIII.
1. Show that cos a sin b = sin a cos b cos C + cos A sin c. (161)
Multiply equation (61) by sin a, and replace sin a cot sin C by cos A
sin c.
2. Prove that sin C cos a = cos A sin B + sin A cos B cos C. (162)
SECTION IV.-THIRD CLASS.
Formulae containing Six Elements.
42. DELAMBRE'S AN&LOGIES.-1~. To prove
sin (A + B) cos(a - b)
cos 6 C cos c
DEM.-sin (A + B) = sin A cos s B + cos A- sin } B;
and substituting for sin A, cos }A, &c., their values from
~~ 26, 27, we get
sin (+ B) - sin(s-b) +sin(s -a)sins.sin(s-)
sin c sin a. sin b
Hence
sin(A+B) sin(s-b)+sin(s-a) coso(a-b) (16
cos C sin = (163)
cos 4 C sin c cos -<