GIVEN.

SOUGHT.

SOLUTION.

By proposition 13, any angle may be found when the three sides are given. To find A, sin AB sin AC: R2 sin (SAB) · sin (SAC): sin2A. This is a convenient rule when expressed logarithmically.

Suppose the supplements of 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 case, the angle of this triangle opposite to the side a, and it will be the supplement of the side of the given triangle opposite to the angle A, that is, of BC (Sp. Ge. 11); and therefore 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.

P

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 plane with them; let them meet it in the points p, m, n, q; then p, m, n, q, are the projections of P, M, N, Q.

B

n

E

M

C

உ

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 AE, 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.

F

COR. 1.-Each of the quadrants contiguous to the projecting point is projected into an indefinite straight line, and each of those that are remote, into a radius of the primitive.

COR. 2. Every small circle which passes through the projecting point is projected into that straight line which is its common section with the primitive.

COR. 3.-Every straight line, in the plane of the primitive, and produced indefinitely, is the projection of some circle on the sphere passing through the projecting point.

COR. 4. The stereographic projection of any point in the surface of the sphere is distant from the centre of the primitive, by the semi-tangent of that point's distance from the pole opposite to the projecting point.

PROPOSITION II.

Every circle on the sphere which does not pass through the projecting point, is projected into a circle.

If the circle be parallel to the primitive, the proposition

is evident.

For, a straight line, drawn from the projecting point to any point in the circumference, and made to revolve about the circle, describes the surface of a cone, which is cut by a plane (namely, the primitive), parallel to the base; and therefore the section (the figure into which the circle is projected) is a circle.

If the circle MN be not parallel to the primitive BD; let the great circle ABCD, passing through the projecting

point, cut it at right angles, in the diameter MN, and the primitive in the diameter BD. Through M, in the plane

=

of that great circle, let MF be drawn parallel to BD; let AM, AN, be joined, and meet BD in m, n. Then, because AB, AD, are quadrants, and BD, MF, parallel, the arc AM AF, for BMDF; since, if B and F were joined, the alternate angles would be equal; hence the angle AMF or Amn = ANM. Thus, the conic surface, described by the revolution of AM, about the circle MN, is cut by the primitive in a sub-contrary position (Conic Sections); therefore the section mn is, in this case, likewise a circle.

COR. 1.-The centres, and poles of all circles, parallel to the primitive, have their projections in its centre. COR. 2.-The centre, and poles of every circle, inclined to the primitive, have their projections in the line of

measures.

COR. 3.-All projected great circles cut the primitive in two points diametrically opposite; and every circle in the plane of projection, which passes through the extremities of a diameter of the primitive, or through the projections of two points that are diametrically opposite on the sphere, is the projection of some great circle. For the original great circles cut the primitive in two points diametrically opposite.

COR. 4.-A tangent to any circle of the sphere, which does not pass through the projecting point, is projected into a tangent to that circle's projection; also, the circular projections of tangent circles touch one another.

COR. 5.-The extremities of the diameter, on the line of measures of any projected circle, are distant from the centre of the primitive, by the semi-tangents of the circle on the sphere's least and greatest distances from the pole opposite to the projecting point.

COR. 6.-The extremities of the diameter, on the line of measures of any projected great circle, are distant from the centre of the primitive, by the tangent and cotangent of half the complement of the great circle's inclination to the primitive.

For BM (second figure) measures the inclination of the circle MN to the primitive BD, and MC is its complement, and angle MAC half its complement. Also, since MAN is a