is not adapted to logarithmic computation, and would, if employed practically, lead to a very tedious and complicated calculation. We can, however, put this expression under a form adapted to logarithmic calculation, by having recourse to an algebraical artifice, and introducing a subsidiary angle. Adding and subtracting 2 ab on the right hand side. = a2 + b2 2 ab + 2 ab 2 ab cos. C log. c = log. (a — b) + log. sec. — log. R The angle is known from the equation. log. tan. = log. 2 + (log. a + log. b) + log. sin. - log. (a - b) 2 being thus determined, log. sec. can be found from the tables, and the value of c becomes known. The angle, which is introduced into the above calculation, in order to render the expression convenient for logarithmic computation, is called a subsidiary angle. The above transformation may be effected in a manner somewhat different, as before. Assume c2 = a3 + b* 2 ab cos. C = a2 + b2 + 2 ab 2 ab 2 ab cos. C = (a + b) — 2 ab (1+cos. C) log. c = log. (a+b)+ log. cos. · C 2 C 2 log. R As before the angle must be determined from the equation. that 2ab is always less than (a + b), this is easily done. But since (a—b)' is necessarily a positive quantity, it must always be greater than 0 (except in the particular case a = b, where it is = 0), and therefore 2 ✓ab cos. (a + b) C 2 is always less than unity, and consequently an angle may always be found whose sine is equal to it. In solving the same case of oblique-angled triangles, we determined the difference of the angles A, B from the equation. In the solution of certain astronomical problems, the logarithms of the sides a, b are given, but not the sides themselves, and these logarithms being given, Whence B The angle 2 log. tan. = log. R+ log. blog. a thus becomes known from the logs. of a and b, without calculating a and b. In the same way we may have, A GREAT variety of geometrical problems may be solved with much elegance by the introduction of trigonometrical formulæ. We shall give a few examples. PROB. I. To express the area of a plane triangle in terms of the sides of the triangle. Let CD be a perpendicular from C upon AB. bc 2 B = 2 ⋅ b c • √s (sa) (s — b) (s — c) ... Chap. ш bc To express the radius of a circle inscribed in a given triangle, in terms of the sides of the triangle. To express the radius of a circle circumscribed about a given triangle, in terms of the sides of the triangle. Given the three angles of a plane triangle, and the radius of the inscribed circle, to find the sides of the triangle. Let A, B, C, be the three given angles, r the radius C P Given the three angles of a plane triangle, and the radius of the circumscribing circle, to find the sides of the triangle. SPHERICAL TRIGONOMETRY. HAVING demonstrated in the treatise on Spherical Geometry, several important properties of the circle of the sphere, and of spherical triangles, we shall now proceed to deduce various relations which exist between the several parts of a spherical triangle. These constitute what is called Spherical Trigonometry; and enable us, when a certain number of the parts are given, to determine the rest. The first formula which we shall establish, serves as a key to all the rest, and is to spherical trigonometry what the expression for the sine of the sum of two angles is to plane trigonometry. CHAPTER I. 1. To express the cosine of an angle of a spherical triangle in terms of the sines and cosines of the sides. Let ABC be a spherical triangle, O the centre of the sphere. Let the angles of the triangle be denoted by the large letters A, B, C, and the sides opposite to them by the corresponding small letters, a, b, c. At the point A, draw A T a tangent to the arc A B, and A t a tangent to the arc A C. Then the spherical angle A is equal to the angle T At between the tangents (Spher. Geom. prop. IV.). Join O B, and produce it to meet A T in T. Join T, t; B A T |