The above result may be enunciated thus-- The sum of two sides is to their difference as the tangent of half the sum of their opposite angles is to the tangent of half their difference. By combining the last two formulæ the formula for the tangent of half an angle in terms of the sides is easily deduced, as follows {log. (s — b) + log. (s — c) + 20 — (log. s + log. (s — SPHERICAL TRIGONOMETRY The fundamental formula in spherical trigonometry as deduced from the spherical triangle having its solid angle at the centre of the sphere is that which connects the cosine of an angle with sines and cosines of the three sides of the triangle. In the spherical triangle A B C, Fig. 10, given the three sides a b c to find angle A. Cos. a = cos. b. cos. c + sin. b. sin. c. cos. A. A Cos. 2 = = · =s-a √sin. s. sin. (s — a) . cosec. b. cosec. c {L. sin. s + L. sin. (s — a) + L. cosec. b. L. cosec. c — 20 20 } TO FIND LOGARITHMIC FORMULA FOR THE SINE OF AN ANGLE IN L. sin. (sc) + L. sin. (s· b) + L. cosec. b + L. cosec. c · - 20 By combining these two formulæ the tangent formula is easily deduced as follows |