GIVEN. SOUGHT. SOLUTION. 5. Sin BC: sin AC :: sin A :sin B (7). The affection of B The angle B is ambiguous, unless it can opposite to be determined by this rule, the other that according as AC + BC given side is greater or less than 180°, AC. A + B is greater or less than 180° (Šp. Ge. 10). From ACB, the angle sought, 6. draw CD perpendicular to Two sides AB; then AC and BC, | The angle R: cos AC :: tan A : cot and an angle ACB con ACD (c. 3); and tan BC: A tained by the tan AČ:: cos ACD : cos opposite to given sides | BCD (11). ACD + BCD | one of them, AC and BC. = ACB, and ACB is amBC. biguous, because of the am biguous sign + or Let fall the perpendicular CD from the angle C contained by the given sides upon the The third | side AB. side R:cos A ::tan AC:tan AD AB. (c. 5); cos AC: cos BC :: cos AD: cos BD (9). AB = AD + BD; wherefore AB is ambiguous. GIVEN. SOUGHT. SOLUTION. By proposition 13, any angle The three 11. may be found when the sides, three sides are given. To AB, AC, One of the find A, sin AB sin AC: and angles R = sin ( S AB) · sin BC. A. (S - AC): sino į A. This is a convenient rule when expressed logarithmically. Suppose the supplements of 12. the three given angles A, B, C, to be a, b, c, and to be the sides of a spherical triangle. Find, by the last The three One of the case, the angle of this triangles sides angle opposite to the side A, B, C. BC. a, and it will be the sup-| plement of the side of the given triangle opposite to the angle A, that is, of BC (Sp. Ge. ll); and there fore BC is found. In the foregoing table, the rules are given for ascertaining the affection of the arc or angle found, whenever it can be done. Most of these rules are contained in this one rule, which is of general application :-That when the part found is either a tangent or a cosine, and of the tangents or cosines employed in the computation of it, either one or three belong to obtuse angles, the angle found is also obtuse. This rule is particularly to be attended to in cases 5 and 7, where it removes part of the ambiguity. . Schol.--The preceding rules are sufficient for the solution of all the cases of Spherical Trigonometry. There are various other rules, however, which may be used in some cases with advantage; but the investigation of them, and also the explanation of the preceding rule for determining the affection of the part sought, belong properly to Analytical Trigonometry. PROJECTIONS. GENERAL DEFINITIONS. 1. The representation on a plane, of the important points and lines of an object, as they appear to the eye when situated in a particular position, is called the projection of the object. 2. The plane on which the delineation is made, is called the plane of projection, or primitive. 3. The point where the eye is situated, is called the point of sight, or the projecting point. 4. The point on the plane of projection, where a perpendicular to it from the point of sight meets the plane, is called its centre. . 5. The line joining the point of sight and the centre, is called the axis of the primitive. 6. Any point, line, or other object to be projected, is called the original in reference to its projection. • 7. A straight line drawn from the point of sight to any original point, is called a projecting line. 8. The surface, which contains the projecting lines of all the points of any original line, is called a projecting surface. When the original line is straight, the projecting surface will be a projecting plane. Cor. The projection of any point is the intersection of its projecting line with the primitive. FIRST BOOK. STEREOGRAPHIC PROJECTION OF THE SPHERE. DEFINITIONS. 1. The stereographic projection of the sphere is that in which a great circle is assumed as the plane of projection, and one of its poles as the projecting point. 2. The great circle, upon whose plane the projection is made, is called the primitive. 3. By the semi-tangent of an arc, is meant the tangent of half that arc. 4. By the line of measures of any circle of the sphere, is meant that diameter of the primitive, produced indefinitely, which is perpendicular to the line of common section of the circle and the primitive. PROPOSITION I. Every great circle which passes through the projecting point is projected into a straight line, passing through the centre of the primitive; and every arc of it, reckoned from the other pole of the primitive, is projected into its semitangent. Let ABCD be a great circle, passing through A, C, the poles of the primitive, and intersecting it in the line of common section BED, E being the centre of the sphere. From A, the projecting point, let there be drawn straight lines AP, AM, AN, AQ, to any number of points P, M, N, Q, in the circle ABCD. These lines will intersect BED, which is in the same IE ID plane with them; let them meet it in the points p, m, n, 9; then p, m, n, %, are the projections of P, M, N, Q. And thus the whole circle ABCD is projected into the straight line BED, passing through the centre of the primitive. Again, because the points C and M are projected into E and m, the whole arc MC will be projected into the straight line mE, which, to the radius AÈ, is the tan mAE = tan # MC. Thus, the arc MC is projected into its semi-tangent mE; PC into its semi-tangent pE, &c. All arcs, therefore, on the circle ABCD, reckoned from the pole C, are projected into their semi-tangents. |