SPHERICAL TRIGONOMETRY. INTRODUCTION. THE following definitions and properties of spherical triangles belong to spherical geometry, and are premised here as principles on which the demonstrations of the propositions in spherical trigonometry depend.* Definitions. 1. A sphere or globe is a solid contained under one uniform round surface, which is every where equally distant from a point within it called the centre. A sphere may be conceived to be formed by the revolution of a semicircle about its diameter, which remains unmoved, and is called the axis of the sphere. 2. A diameter of a sphere is a straight line passing through the centre, and terminated both ways by the convex surface. 3. The circles of the sphere are of two denominations, great circles, and small circles. A great circle of the sphere is that which divides its surface into two equal parts. A small circle of the sphere is that which divides its surface into two unequal parts. Thus, the equator of a common globe is a great circle, and any parallel of latitude is a small circle. 4. Hence the plane of any great circle passes through the centre of the sphere, and divides the sphere into two equal parts. * These definitions and elementary propositions, with their demonstrations, will be annexed to the second edition of Playfair's Geometry, under the title of Elements of Spherical Geometry. They constitute no part of spherical trigonometry, according to the genuine meaning of the terms which express the title of that science; and therefore cannot, with propriety, be prefixed to a regular and systematic treatise of spherical trigonometry. 5. The poles of any circle are the two extremities of that diameter of the sphere which is perpendicular to the plane of the circle. 6. Hence either pole of any circle is equidistant from every part of its circumference; and each pole of a great circle is 90° from the circumference. 7. A spherical angle is an angle on the surface of a sphere, contained between the arcs of two great circles which intersect each other. 8. The measure of a spherical angle is the arc of a great circle intercepted between the two arcs which form the angle, and drawn at the distance of 90° from the angular point. 9. A spherical triangle is a portion of the surface of a sphere contained by the arcs of three great circles which intersect one another. Note. The three arcs are called the sides of the triangle, and the three angles which every two arcs form by their intersection are called its angles. Also, both the sides and the angles of spherical triangles are computed in degrees, minutes, and seconds, in the same manner as the angles of plane triangles. 10. A right-angled spherical triangle is that which has one right angle, or an angle of 90°. 11. A quadrantal spherical triangle is that which has one of its sides a quadrant, or 90°. 12. An oblique-angled spherical triangle is that which has each of its sides, or angles, greater or less than 90°. 13. Any two sides, or angles, of a spherical triangle are said to be like, or of the same kind, or of the same affection, when they are both greater or less than 90°. 14. If one side or angle of a spherical triangle be equal to, or greater than 90°, and the other side or angle less, they are said to be unlike, or of different kinds, or of different affections. Properties of the Sphere. 15. Every section of a sphere by a plane passing through it is a circle. 16. The centre of a sphere is the centre of all its great circles, and its axis is the common section of all the great circles which pass through its two extremities. 17. A great circle can be drawn through any two points on the surface of a sphere, and a small circle can be drawn through any three points on its surface. 18. All parallel circles of the sphere have the same pole; and no two great circles can have a common pole. 19. Any two great circles of the sphere cut each other twice at the distance of 180°, and make the angles at the intersections equal. 20. A great circle of the sphere is perpendicular to any other circle, when its plane is perpendicular to the plane of the other; and conversely. 21. A great circle passing through the poles of any other great circle cuts the other circle at right angles; and if a great circle cut any other circle at right angles it will pass through its poles. Note. Most of these principles will be evident by inspecting the nature and position of the circles drawn on an artificial globe. The sixth article becomes obvious by observing that all the meridians pass through the north and south poles of the globe, and are perpendicular to the equator, and to all the parallels of latitude. 22. If two arcs of great circles intersect each other, the vertical or opposite angles will be equal. 23. An angle made by the intersection of any two great circles of the sphere is equal to the angle of inclination of the planes of those circles. 24. The distance of the poles of any two great circles of the sphere is equal to the angle of inclination of the planes of those circles. General Properties of Spherical Triangles. 25. Any side, or any angle, of a spherical triangle is less than 180°, or two quadrants. 26. The greater side is opposite to the greater angle, and the less side to the less angle. 27. The sum of any two sides is greater than the third side, and their difference is less than the third side. 28. The difference of any two sides is less than 180°, or a semicircle; and the sum of the three sides is less than 360°, or two semicircles. 29. The sum of the three angles is greater than 180°, or two right angles; and less than 540°, or six right angles. 30. The sum of any two angles is greater than the supplement of the third angle. 31. A spherical triangle is equilateral, isosceles, or scalene, according as its angles are all equal, or only two of them equal, or all unequal. 32. If each of the three angles be acute, or right, or obtuse, then each of the three sides will be less than 90°, or equal to 90°, or greater than 90°; and conversely. 33. Half the sum of any two sides is of the same kind as half the sum of their opposite angles. Or, the sum of any two sides is of the same kind, in respect of 180°, as the sum of their opposite angles. 34. If three arcs of great circles be described from the angular points of any spherical triangle, as poles, the sides and angles of the new triangle, so formed, will be the supplements of the opposite angles and sides of the former triangle; and conversely. Thus, DE = 180° C, EF = 180° - A, FD = 180° - B, and D = 180° - BC, E = 180° - AC, F = 180°AB. Again, AB = 180° - F, BC = 180° -D, AC = 180° - E, and A = 180° - EF, B = 180° - FD, C = 180°DE. FC D Affections of Right-angled Spherical Triangles. BE 35. The sides are of the same kind as their opposite angles; and conversely. 36. The hypothenuse is less or greater than 90°, according as a side and its adjacent angle, or the two sides, or the two angles, are like or unlike. 37. A side is less or greater than 90°, according as its adjacent angle and the hypothenuse, or the other side and the hypothenuse, are like or unlike. 38. An angle is acute or obtuse according as its adjacent side and the hypothenuse, or the other angle and the hypothenuse, are like or unlike. Other Properties of Right-angled Spherical Triangles. 39. If the hypothenuse be 90°, one of the sides and its opposite angle will be 90° each; and the other side and angle will be of the same number of degrees. |