APPENDIX II.

ON UNLIMITED SPHERICAL TRIANGLES AND THEIR SOLUTION.*

OF THE VARIOUS TRIANGLES FORMED BY THE SAME THREE POINTS ON THE SPHERE.

1. If any two points, A and B, be taken upon the surface of the sphere, the arc of a great circle joining them may be considered to be either the arc A B (< 180°), or 360° -A B; or if we do not limit the arcs to values less than a circumference, we may consider it to have an indefinite number of values expressed generally by the formula 2 na, a denoting that value which is less than or a semicircumference, and n any whole number or zero.

2. If two arcs of great circles intersect in a point A, the angle which they form may be considered to be either the angle A (90°), or 180° — A, or 180° +Aa, or 360° -A; or, taking the most general view of angular magnitude, the angle will have an indefinite number of values expressed by the formula mA, A denoting the value which is less than, and m any whole number or zero.

3. If, therefore, any three points, A, B, C, be taken on the surface of the sphere, and great circles, made to pass through each pair, we shall have an infinite series of triangles whose sides will be generally expressed by

a, b, c, denoting the arcs less than 7 joining the pairs of points BC, AC, A B, respectively; A, B, C, the angles less than formed at those points by the intersection of these arcs; and n and m, any whole numbers, or zero.

4. It is evident, however, that we cannot assume that any three values of the sides from the series (1), combined with any three values of the angles from (2), will form a spherical triangle. Some general relations of the parts composing a triangle must first be established, from which corresponding values of n and m in (1) and (2)

* Introduced by Gauss. Notwithstanding the elegance and generality thus given, to the solutions of many astronomical problems, nothing is to be found on this subject in our trigonometrical works. The present paper is from Prof. Chauvenet of the U. S. Naval Acad. The explanatory notes are the author's.

may be deduced. Although these general relations are well known, it may not be out of place to add here a concise demonstration of them.

Let the point c,* one of the angular points of the spherical triangle ABC, be referred by rectangular co-ordinates to three planes, one of which, the plane of xy, is the plane of the great circle ▲ B; let the axis of x be the diameter of the sphere passing through B, and let the origin be the centre of the sphere. The formulas of transformation from these co-ordinates to polar co-ordinates, the origin being the same, the polar axis being the axis of x, and the fixed plane the plane of xy, are

where B denotes the angle which the plane passing through the polar axis and the point c makes with the fixed plane; R, the radius- vector in this plane, or distance of the point c from the origin; and a the angle which this radius-vector makes with the polar axis. B is an angle of the spherical triangle, and a is the side opposite the angle a; and, according to the principles of analytical geometry, в and a may be altogether unlimited, due regard being had to the signs of their trigonometric functions, and to those of x, y, and z.

Let us now transform from these rectangular co-ordinates to others also rectangular, the origin and the plane of x y remaining the same, but the axis of x in the new system passing through the point A, and therefore making with the first axis the angle c, c also expressing the side of the triangle opposite the angle c. The known formulas of transformation become

Ꮓ

B

* The rest of Art. 4 implies some knowledge of Analytical Geometry. It may be readily understood, however, by the mere student of trigonometry from the annexed diagram, with the following explanations. R in formulas (3) is equal to o c in the diagram. The projection of R or of oc, on o в which is called the axis of x, that is to say the distance between the foot of a perpendicular from c on o B and the point o is the value of x in the formulas, the projection of o c on a line called the axis of y, drawn from o in the plane A OB perpendicular to o B, is the value of y in the formulas, and the projection of R or oc on oz perpendicular to the plane A O B, is the value of z in the formulas. The first of formulas (3) is now obvious enough; in the second R sin a evidently expresses the value of a perpendicular from c to oв in the plane c o B, and this perpendicular multiplied by the cosine of the angle which it makes with its projection on the plane A o в equal to the angle of the two planes or the angle B, expresses the length of the projection on the plane A O B, which is evidently equal to the projection of R on the axis of y; or multiplied by the sin в expresses the height of c above the plane A O B, which is equal to the projection of R on o z the axis of z. N. B. The axes of x, y, and z are at right angles each to the plane of the other two. So also are those of x', y', and z'.

The student will readily deduce formulas (4) by the rules of Plane Trigonometry.

Finally, the formulas of transformation from this last system of rectangular again to polar co-ordinates, the origin being the same, the fixed plane the plane of x'y' and the polar axis the axis of x', are

Now, if the values of x, y, z, x', y', z', given by (3) and (5) be substituted in (4), we have at once the following system of equations:

cos a

cos c cos b+ sin c sin b cos a
sin a cos B sin c cos b-cos c sin b cos a
sin a sin B =
sin b sin A

(6)

which are the known fundamental formulas of spherical trigonometry, but established without imposing any restrictions upon the values of the parts of the triangle. From this investigation it appears that these formulas may be regarded as formulas of transformation from one system of polar co-ordinates to another, or rather from one system of spherical co-ordinates to another. For example, the co-ordinates of a star referred to the pole of the equator and the meridian of a place whose colatitude is c, are its polar distance a, and its hour angle в; the co-ordinates of the same star referred to the pole of the horizon and the meridian, are its zenith distance b, and its azimuth A; and the formulas (6) express the relations by means of which we can pass from one of these systems to the other.

(5.) Let us now inquire what are the corresponding values of the sides and angles in the series of triangles expressed by (1) and (2). Let a, b, c, A, B, C, denote the values of the parts of one of these triangles, which, if we please, we may suppose to be the triangle whose parts are less than . Then since

sin (2 n x+4)=sin p, cos (2 n x+4) = cos 4,

the equations (6) will be satisfied by the substitution of 2 n +a, 2 n a + b, &c., for a, b, and c; and therefore the triangle (a, b, c, A, B, C) is the first of an infinite series obtained from it by the successive addition of 2 to each or all of its parts, every triangle of the series being such, that the relations of its parts are expressed by (6), when a, b, c, A, B, C, are assumed to represent those parts.

It is evident, also, from the principle of "uniformity of direction" observed in the preceding demonstration in reckoning the sides and angles, that we must be able to satisfy the equations, by making either all the sides, or all the angles, or all the sides and angles, negative at the same time, and, considering each of the triangles

* The student will do well to conceive the position of the angular points of the triangles on the surface of the sphere with these variations.

N. B. That the sides are all negative together, or the angles together, or both together.

The same thing stated in the text may be made evident by referring to equations

thus obtained as the first of a series, as above, we have three more series. We have then the four series following:

In all the terms of these series, n may have the same or different values; and we thus have all the possible combinations of the values represented by (1) and (2), so long as m in (2) is even. But if we substitute 2 n + 1 for m we shall find that the following series will satisfy the equations (6) :

the 6th, 7th, and 8th of which series are derived from the 5th, as the 2d, 3d, and

4th were derived from the 1st, in the preceding paragraph.

By successively exchanging a for b and c, we find eight more series, namely,

(6), which involve all the relations of the six elements of a spherical triangle, and which will be satisfied by changing simultaneously a, b, and c into a,b,c, or A, B, C, into — A, B, -c or both; observing the general rule that sin (—ø) =— sin

& and cos (p) = cos p.

The student will try these elements given in the 5th series, in eqs. (6), observing that cos {(2n+1) = + 9 } = cos (180° + $) =—cos and sin {(2 n + 1) = + ♦ } = sin 4.