Rules for determining whether the sides and angles of spheri- cal right triangles are obtuse or acute (495), (505), (517), The sines, &c. of angles less than 180° (496), (507), Theorems on spherical triangles which contain two or more Page. when the hypothenuse and a leg are known (566), (567), 19 when a leg and the opposite angle are known (572), (574), 20 when a leg and the adjacent angle are known (585), (586), 23 Theorems for the solution of spherical oblique triangles, The sines of the sides are proportional to the sines of the op- Solution of Spherical Oblique Triangles, when two sides and the included angle are known (615 – 620), when two angles and a side opposite to one of them are known 38 when the three sides are known (674), or (686-688), or 41 when the three angles are known (791), or (803-805), or 55 rules for determining when the sides and angles are obtuse or 29 36 The sum of the sides and angles (713), (822), Napier's Rules for oblique triangles (749), (758), (844), (852), 48 Theorems on isosceles and equilateral spherical triangles (731), (734), (836), (839), Theorems on the equality of triangles (561), (863), Sines, &c. of angles greater than 180° (705), 48. 46. SPHERICAL TRIGONOMETRY. 3 CHAPTER I. Definitions. 1. Spherical Trigonometry treats of the solution of spherical triangles. A Spherical Triangle is a portion of the surface of a sphere included between three arcs of great circles. In the present treatise those spherical triangles only are treated of, in which the sides and angles are less than 180°. 2. The angle, formed by two sides of a spherical triangle, is the same as the angle formed by their planes. 3. An isosceles spherical triangle is one which has two of its sides equal. An equilateral spherical triangle is one which has all its sides equal. 4. A spherical right triangle is one which has a right angle, all other spherical triangles are called oblique. We shall in spherical trigonometry, as we did in 1 plane trigonometry, attend first to the solution of right triangles. CHAPTER II. Spherical Right Triangles. SECTION. I. Napier's Rules for the Solution of Spherical Right Triangles. 5. Problem. To investigate some relations between the sides and angles of a spherical right triangle. Solution. The importance of this problem is obvious; for, unless some relations were known between the sides and the angles, they could not be determined from each other, and there could be no such thing as the solution of a spherical triangle. Fig.1 Let, then, ABC (fig. 1.) be a spherical right triangle, right angled at C. Call the hypothenuse AB, h; and call the legs BC and (427) AC, opposite the angles A and B, P respectively a and b. Let O be the centre of the sphere. Join OA, OB, OC. B The angle A is, by art. 2, equal to the angle of the planes BOA and COA. The angle B is equal to the angle of the planes BOC and BOA. The angle 257 1 of the planes BOC and AOC is equal to the angle (428) C, that is, to a right angle; these two planes are, therefore, perpendicular to each other. Moreover, the angle BOA, measured by BA, is equal to BA orh; BOC is equal to its measure BC (429) or a, and AOC is equal to its measure AC or b. Through any point A' of the line OA, suppose a plane to pass perpendicular to OA. Its intersections A'C' and A'B', with the planes COA and BOA must (430) be perpendicular to OA', because they are drawn through the foot of this perpendicular. As the plane B'A'C' is perpendicular to OA, it must be perpendicular to AOC; and its intersection B'C', with the plane BOC, which is also perpendicular to AOC, must likewise be perpendicular to AOC. Hence B'C' must be perpendicular to A'C' and OC which pass through its foot in the plane AOС. All the triangles A'OB', A'OC', B'OC', and (431) A'B'C' are then right-angled; and the comparison of them leads to the desired equations, as follows: First. We have from triangle A'OB' by (5) and (429), |