Or the side ab may be found by equation 18. This is the direct solution of the celebrated problem of clearing the observed distance between the moon and the sun, or a star, from the effect of parallax and refraction. A B Ex. 2. In the spherical triangle ABC, given AC 46° 18', AB 100° 26', and the angle A 39° 50', to find the rest. Result: ACB 136° 0' 54"; ABC 30° 41' 54"; BC 65° 6' 34". Ex. 3. In the triangle ABC, given AB 112° 56", BAC 40° 16', ABC 54° 20', to find the rest. Result: AC 79° 44′ 58′′; BC 51° 31' 30"; ACB 130° 30' 20". Ex. 4. Given, the side AB 96° 12', AC 57° 16', BC 49° 8', to find the angles. Result: BAC 31° 32′ 42"; ABC 35° 35′ 15"; ACB 136° 32′ 48". Ex. 5. Given, the angle BAC 50°, ABC 60°, ACB 85°, to find the sides. Result: AB 51° 59' 16"; AC 43° 13' 48"; BC 37° 17 26". SECTION IV. CONIC SECTIONS. ARTICLE 78. Definition 1. If, from a point in the circumference of a circle, a right line be drawn to pass through a fixed point which is not in the plane of that circle, and then caused to revolve round that fixed point so as to describe the whole circumference of the circle; the curve surface, described by this revolving line, is called a conical surface; and the solid included between this curve surface and the generating circle, is called a cone. Def. 2. The circle described by the revolving line is called the base, and the fixed point the vertex, of the cone. Def. 3. The straight line drawn from the vertex to the centre of the base, is called the axis of the cone. Def. 4. When the axis is at right angles to the plane of the base, the cone is called a right cone; but when the axis is oblique to that plane, the solid is termed a scalene cone. As the line which, by its revolution, describes the conical surface, may be indefinitely extended, two cones having a common vertex, and equal solid angles at the vertex, may be generated by the same revolution. ART. 79. Let the cone ABCD be cut by a plane which passes through its vertex A, and cuts the base in the right line BC; the common section of this plane with the surface B D of the cone, will be a triangle. The common section of the base and cutting plane is a right line (3.2 sup.); and the right lines drawn from B and C to the vertex, are in the cutting plane (2.2 sup.); and those lines correspond to the position of the revolving line when it passes through B and C; they are there fore in the conical surface. Q. E. D. ART. 80. Let the cone ABC be cut by a plane which is parallel to the plane of the base; then the section of this cutting plane with the conical surface, is a circle whose centre is in the axis of the cone. A H G D L E Let AF be the axis of the cone; DLE the cutting plane. In the circumference of the base take any point K; join FK; and through AF, FK, suppose a plane to pass, cutting the conical surface in AK, and the cutting plane in GH; then (Art. 79, and 3.2 sup.) AK and GH are right lines. Let also another plane ABC pass through the axis; its section with the base will be a diameter, because F is the centre of the circle; and the section of this plane with the conical surface is a triangle (Art. 79). Take BC and DE, the sections of this plane with the parallel planes BCK and DLE; then (14.2 sup.) DE and GH are respectively parallel to BC, FK; consequently, B F C As AF AG :: BF: DG :: FC : GE :: FK : GH. : But BF, FC and FK are all equal; therefore, DG, GE and GH are also equal; consequently (9.3), DLEH is a circle whose centre is G. Q. E. D. |