35. The section of a sphere cut by a plane must be a circle 36. The circumferences of two great circles bisect each other ib. A spherical angle is measured by the plane angle formed by tan- 37. The great circle distance between the poles of two intersecting ib. Every great circle passing through the poles of another is at 38. Any side of a spherical triangle is less than the sum of the other 39. The three sides are together less than a whole circumference 41. Characteristic property of the supplemental triangle 42. The angles of every spherical triangle are together greater than 44. The fundamental equations of Spherical Trigonometry 45. Deduction of geometrical properties from these equations similarly involving angles and sides 81 54. Solution of quadrantal triangles 55. Solution of oblique-angled triangles 56. When the three sides are given 57. When the three angles are given 58. When two sides and the included angle are given 59. When two angles and the interjacent side are given CHAP. II. Application of Spherical Trigonometry to Astronomical PART I, ELEMENTS OF PLANE TRIGONOMETRY. CHAPTER I. EXPLANATION OF THE TRIGONOMETRICAL LINES. mathe (Article 1.) PLANE TRIGONOMETRY is that branch of pure matics of which the primary object is to determine the several parts of a plane triangle from having certain other dependent parts given. By the parts of a plane triangle we mean these six things, viz. the three sides and the three angles, and if any three of these six be given, provided only that a side be among them, the other three may always be determined either by geometrical construction, as shown in the Elements of Geometry, or by numerical computation, as will be seen hereafter. From the foregoing definition it appears that quantities of two kinds, perfectly distinct from each other and admitting of no comparison, are concerned in Trigonometry, viz. straight lines and angles. By means of certain happy contrivances, however, the whole business of trigonometry, and, indeed, the general theory of angular magnitude is conducted by help of linear quantities only; the angles themselves not entering into the computations, but certain straight lines dependent upon them and serving as indexes to them. (2.) Before we explain the nature of these trigonometrical lines, it will be necessary first to show how angular magnitude is measured. B |