Napier's Analogies. Proof. This theorem is at once obtained by applying § 79 to the polar triangle. 91. Theorem. The cosine of half the sum of two sides of a triangle is to the cosine of half their dif ference, as the cotangent of half the included angle is to the tangent of half the sum of the other two angles, or in (figs. 32 and 33), cos.(a-c): cos. (a-c)=cotan. B: tang. (A+C). (370) Proof. This theorem is at once obtained by applying $80 to the polar triangle. 92. Corollary. These two theorems, similar to § 79 and § 80, were given by Napier for the solution of the case, in which two sides and the included angle are given. By means of them the other two angles can be found without the necessity of calculating the third side. In using them, regard must be had to the signs of the terms by means of Pl. Trig. § 61. 93. EXAMPLES. 1. Given in a spherical triangle two sides 149°, and = 49°, and the included angle = 88°; to find the other angles. 13°, and=9°, 2. Given in a spherical triangle two sides and the included angle = 176; to find the other angles. Ans. 2° 13'12", and 1° 51′ 14′′. SPHERICAL ASTRONOMY. CHAPTER I. THE CELESTIAL SPHERE AND ITS CIRCLES. 1. Astronomy is the science which treats of the heavenly bodies. 2. Mathematical Astronomy is the science which treats of the positions and motions of the heavenly bodies. The elements of position of a heavenly body are (Geo. §8) distance and direction. 3. Spherical Astronomy regards only one of the elements of position, namely, direction, and usually refers all directions to the centre of the earth. 4. In spherical astronomy, all the stars may, then, be regarded as at the same distance from the earth's centre, upon the surface of a sphere, which is called the celestial sphere. Upon this imaginary sphere are supposed to be drawn various circles, which are divided into the well known classes of great and small circles. [B. p. 47.] |