32. Solution of isosceles triangles and quadrantal triangles
33. Solution of oblique spherical triangles (six cases)
34. Graphical solution of (oblique and right) spherical triangles

CHAPTER III.

PAGE

33

34

35

36

39

39

42

RELATIONS BETWEEN THE SIDES AND ANGLES OF SPHERICAL TRIANGLES.

36. Derivation of the Law of Sines and the Law of Cosines . 37. Formulas for the half-angles and the half-sides

38. Napier's Analogies

39. Delambre's Analogies or Gauss's Formulas

40. Other relations between the parts of a spherical triangle.

CHAPTER IV.

SOLUTION OF TRIANGLES.

44. Case III.

45. Case IV.

46. Case V.

Given two sides and their included angle
Given one side and its two adjacent angles

Given two sides and the angle opposite one of them

47. Case VI. Given two angles and the side opposite one of them 48. Subsidiary angles

53. To find the area of a sphere. Area of a zone.

58. Formulas for the spherical excess of a triangle

59. The number of spherical degrees in any figure on a sphere. The spherical excess of a spherical polygon

:

60. Given the area of a figure to find its spherical excess 61. The measure of a solid angle

63. Definitions. Spherical pyramid, segment, and sector 64. Volume of a spherical pyramid; of a spherical sector 65. Volume of a spherical segment

CHAPTER VII.
VII.

PRACTICAL APPLICATIONS.

PAGE

69

69

70

71

73

66. Geographical problem

APPLICATIONS TO ASTRONOMY.

82

382

68. The celestial sphere.

69. Points and lines of reference on the celestial sphere

70. The horizon system: Positions described by altitude and azimuth. 71. The equator system: Positions described by declination and hour angle

72. The altitude of the pole is equal to the latitude of the place of observation

76. The equator system: Positions described by declination and right

77. The ecliptic system: Positions described by latitude and longitude 93

APPENDIX.

NOTE A. On the fundamental formulas of spherical trigonometry

NOTE B. Derivation of formulas for Spherical Excess

QUESTIONS AND EXERCISES FOR PRACTICE AND REVIEW
ANSWERS TO THE EXAMPLES.

SPHERICAL TRIGONOMETRY.

CHAPTER I.

REVIEW OF SOLID AND SPHERICAL GEOMETRY.

On beginning the study of spherical trigonometry it is advisable to recall to mind or learn some of the definitions and propositions of solid geometry. A clear and vivid conception of the principal properties of the sphere is especially necessary. The definitions and theorems which will be used frequently in the following pages, are quoted in this chapter.*

Planes and Lines in Space. Diedral Angles. Solid Angles.

1. a. Two planes which are not parallel intersect in a straight line. (Euc. XI. 3.)

b. The angle which one of two planes makes with the other is called a diedral angle. Thus, in Fig. 1, the two planes BD and

* As far as possible, references are made to the text of Euclid; since, of the numerous geometrical text-books in English-speaking countries, his work is the one which is most largely used. Those who use a text-book other than Euclid's can substitute the appropriate references.

AE intersect in the straight line AB, and form the diedral angle FABC.

c. The planes AE and AC are called the faces, and the line AB is called the edge, of the diedral angle. The faces are unlimited in extent. The magnitude of the diedral angle depends, not upon the extent of its faces, but only upon their relative position. (Just as the magnitude of a plane angle depends, not upon the lengths of its boundary lines, but upon their relative position.)

d. If PR be drawn perpendicular to AB in the plane AE, and PS be drawn perpendicular to AB in the plane AC, the angle RPS is called the plane angle of the diedral angle.

e. If a plane is drawn perpendicular to the edge of a diedral angle, the intersections of this plane with the faces of the diedral angle form the plane angle of the diedral angle. (See Euc. XI. 4.) Thus, if the plane M be passed through p perpendicular to AB, the intersections, pr, ps, of the plane M and the planes AE, AC, form the angle rps which is the plane angle of FABC.

f. All plane angles of the same diedral angle are equal. (See Euc. XI. 10.) Hence, the plane angle can be taken as the measure of the diedral angle.

2. a. If a straight line be at right angles to a plane, every plane which passes through the line is at right angles to that plane. (Euc. XI. 18.)

b. If two planes which cut one another be each of them perpendicular to a third plane, their common section is perpendicular to the same plane. (Euc. XI. 19.)

3. a. When three or more planes meet in a common point, they are said to form a solid angle, or a polyedral angle, at that point.

The point in which the planes meet is called the vertex of the solid angle; the intersections of the planes are called its edges; the portions of the planes between the edges are called its faces; the plane angles formed by the edges are called its face angles; and the diedral angles formed at the edges by the planes are called the diedral angles (or the edge angles) of the solid angle.