PROBLEM VIII.

To project the sphere stereographically on the plane of a circle

oblique to the horizon.

EXAMPLE.

Required a stereographic projection of the sphere on the plane of a great circle, declining 24° 30′ westward from the south at Cambridge, in lat. 42° 23′ 28′′ N, and reclining 36 30' from the zenith northward.

1. Project the great oblique circle HDOL, and quarter it by the diameters HO, DL.

2. Set the reclination northward from O to A, considering the upper as the northern part; reduce A to Z; then Z is the zenith of Cambrige, and Q is that of the place, where the oblique circle is a horizontal plane. Also reduce a, 90° distant from A, to z; aud through H, z, O, project HWZSO, the horizon of Cambridge, and continue it through O to E.

3. Set the western declination from H to y, from D to c, and from O to d. From the pole Z reduce y, c, d, to W, S, E, on the horizon, which will be the west, south, and east, points of the horizon of Cambridge.

4. From Q to F set the tangent of 53° 30', the complement of the reclination 36° 30'; from F raise the perpendicular FG, and continue the diameter EW till it intersect FG. Then with centre G, and radius GS, describe PZSK, the meridian of Cambridge.

5. A rule on W, Z, cuts the primitive at C. Set the complement of the latitude from C to B; and a rule over W, B, cuts the meridian at P, which is the north pole. Set a quadrant faom B to I, and also from I to M. Then a rule on W, I, cuts the meridian at Æ, the point where the equinoctial intersects it; and on W, M, it cuts the meridian produced in K, the south pole.

6. Draw the equinoctial RWETE through the three points W, E, E; and VQK, the axis, and meridian of the place Q.

7. Through the three points R, P, T, describe the circle PRKT, and draw the diameter UX perpendicular to PQK. Then describe the meridians, and parallels of latitude, as in the stereographic projection on the plane of the horizon; finding the intersections of the parallels with the axis by reducing from the point R.

Stereographic Projection on an oblique Circle.

L

NOTE. Every great circle of the sphere is the horizon of a certain place. And when the sphere is projected on an oblique circle, it is easy to determine the latitude of the place, where the primitive would be horizontal, and also its difference in longitude from the place, for which the projection is made.

Thus, in this projection, VP = Qe the latitude; and eÆ=kh the difference, in longitude from Cambridge, of the place, where this oblique circle would be the horizon.

SPHERIC TRIGONOMETRY.

DEFINITIONS.

1. SPHERIC TRIGONOMETRY teaches the rela

tions and calculation of the side and angles of spheric triangles.

2. A spheric triangle is a figure on the surface of a sphere, bounded by three arcs of great circles.

3. A spheric triangle, like a plane one, is equilateral, isoscelar, or scalenous, according as the three sides, two of them only, or no two of them, are equal.

4. A spheric triangle is right-angled, or rectangular, if it have one right angle, or more; quadrantal, or rectilateral, if one of the sides be a quadrant; and oblique, if it have neither a right angle, nor a quadrantal side.

5. In a right-angled spheric triangle, as in a plane one, the side, opposite to the right angle, is called the hypotenuse ; and the other two the legs, or sides.

6. Two sides of a spheric triangle are said to be alike, or of the same affection or kind, when they are each greater or less than a quadrant; and unlike, or of different affection or kind, when one is greater and the other less than a quadrant. Also two angles are alike, or of the same affection or kind, when they are both acute or obtuse; and unlike, or of different affection or kind, when one is acute and the other obtuse. Vol. II. Ddd