2. When the angle is at the circumference of the primitive. 3. When the angle is in the plane of the primitive. Let AEN be the given angle. N A Find the poles P and X of the oblique circles AED, NEF, forming the given angle AEN. From P the angular point E reduce P, X, to U, W, on the primitive. Then UW TE is the measure of the angle AEN, or DEF. And the supplement of UW is the measure of the angle NED, or AEF. PROBLEM IX. To draw a small circle parallel to a given great circle, at any given distance. 1. When the small circle required is parallel to the primitive. Let the distance be 13° Set the given distance from B to O; reduce o to X on the B right circlé BD. Then D the circle XE, described with the centre C, and radius CX, will be the parallel circle required. 2. When the small circle required is parallel to a right circle. Let the distance be 45° from the right circle BD, कि F 1 Note. The point K, which is the centre of the circle FGH, may be found by drawing the tangent FK from the point F. For, the centre is where it intersects the diane. ter AY produced. 3. When VOL. II. N nn R 3. When the small circle required is parallel to an oblique circle. i Let the distance be X 30° from the oblique circle AED. B Find the pole P of the given oblique circle; from A reduce P to N; set from N, on each D side, the complement of Y the given distance of the E parallel, that is 60°, to R, and to Q; reduce R, Q, to X, Y. Then bisect XY in 0; with centre O, and radius OX, or OY, draw XBY for the cir: cle required. A Ad PROBLEM X. To measure any arc of a projected small circle. 1. When the given small circle is parallel to the primitive. Cor. Any arc of the primitive circle, as DC, is reducible, in this manner, to the similar arc AB of a given parallel circle. 2. When 2.. When the given small circle is parallel to a right circle. B Draw two oblique circles X through X, Y, the extrem Y E ities of the given arc, as AXD, AYD, cutting the right circle in O, B, and passing through the poles A, D. Then OB, reduced to the primitive, is equal to TU, and TU is the measure of the arc XY. Cor. By inverting the process any number of degrees may be set on a small circle, parallel to a right circle, from any given point. 3. When the given small circle is parallel to an oblique circle. Let XY be the given arc of XYZ, parallel to ABC. Through the poles P, p, and the points X, Y, the extremities of the given arc, draw the circles PXp, PYp, perpendicular to the given oblique circle ABC. Then OQ is equal to XY, and to LM. And LM is the measure of the arc XY. Cor. By inverting the process any number of degrees may be set on a small circle, parallel to an oblique circle. PROJECTIONS N PROJECTIONS OF THE SPHERE. A PROJECTION OF THE SPHERE is a perspective representation of the circles of the sphere. Each of the following figures, or general projections on the planes of great circles, contains a hemisphere. In the stereographic projection, the hemisphere, opposite to the eye, falls within the primitive, to which hemisphere this kind of projection is generally limited. If it be applied to any part of the hemisphere next to the eye, the representation extends beyond the primitive. PROBLEM I. To project the sphere stereographically on the plane of the equator. 1. Draw the primitive circle WNES, of a convenient bigness, to represent the plane of the equator ; and draw. the diameters WE, NS, at right angles to each other, which here represent the equinoctial and solstitial colures. 2. Divide each of the quadrants NE, ES, SW, WN, in, to nine equal parts ; by which means the equator will be divided into 36 parts, each =10. Then through each of these divisions draw lines to P, the pole. Each of these 10° may be divided more minutely. 3. Lay a rule over N and a, and it will cross PW at x; over Nh, and it will cross PW at y; and over Nc; and it will cross at z, &c. With radii Px, Py, Pz, &c. and centre P, through the points x, y, z, &c. draw concentric circles, which will be parallels of latitude 10° distant from each other. The tropic and polar circle may be projected by setting 23° 28', and 66° 32", from W toward S, reducing them to WP, and describing circles through the points of intersection in WP, concentric with the parallels of latitude. 4. If the projection be intended for the northern hemi. sphere, the tropic is that of cancer, and the northern half of the ecliptic is to be projected through W.E. In the projection of the southern hemisphere is the tropic of capricorn, and the southern half of the ecliptic touching it. Stereographic |