and that object will, evidently, have been attained, when equations have been found, which exhibit the relations between four of the proposed quantities, or their functions, taken in all the possible combinations of four quantities out of six. Now, the whole number of such combinations

is (5) fifteen; but yet, when they are compared

with one another, there are found, amongst them, only four combinations, that are essentially different: and the four theorems, marked (I), (II), (IX), (X), furnish the necessary equations. So that the combinations, and the theorems which embrace them, admit of the following classification:

Combination 1. Three sides and an angle (I.)

2. Two sides and two angles opposite to

them (IX.)

3. Two sides and two angles, not both of them opposite (X.)

4. One side and three angles (II.)

The problems, which those several combinations supply, and which are really distinct problems, may be reduced to six cases: and these cases have been separately discussed, in Art. 263; because, although the four theorems that have been cited, are sufficient for the solution of any proposed spherical triangle, of which any three parts are given, they do not always afford the best prac

tical methods of solution.

When a summary is wanted, of the modes of proceeding, generally to be followed, in the solution of oblique-angled spherical triangles, the following table may be consulted.

Three rules have been already referred to, for removing the ambiguity, incident to the solution of some of the cases of oblique-angled spherical triangles: viz.

1. The greater side is opposite to the greater angle. 2. The measure of any side, or angle, is less than 180o, or, the measure of the half of any side, or angle, is less than 90°.

3. No side, nor any angle, can have a negative value.

It is evident, however, that these rules can only be applied, after the calculation of the unknown parts of the triangle: but if, in the fifth case, S' be the side opposite to the given angle; and if, in the sixth case, A' be the angle opposite to the given side; it has been proved (Art. 144.) that whenever

S' is greater than S, and less than 180° – S, or, whenever

A' is greater than A, and less than 180°-A, then, the unknown part is of the same species as the given part opposite to it: and this rule may be applied to determine the species of the parts of the proposed triangle, previously to any calculation having been entered into.

Thus, in either of the two ambiguous cases, after two sides and the two angles opposite to them have been determined, if the solution be carrried on, by drawing an arch from one of the angles, at right angles to the opposite side, it will be known, from Art. 134, whether that arch fall within, or without, the triangle. In the former case, the unknown quantity will be the sum, and, in the latter case, it will be the difference, of the resulting segments.

Further, when an astronomical problem is reduced to the solution of a spherical triangle, the circumstances of the problem will, of themselves, sometimes, exclude one of the two values found for an unknown part. In computing, for example, the zenith distance of a star, from data obtained by actual observation, if two values be found, one less, and the other greater than 90°, the latter value is, manifestly, excluded; because, if the zenith distance had really exceeded 90°, the star would have been invisible, and the observation upon it could not have been made.

PART II.

THE ELEMENTS OF

Spherical Trigonometry.

SECTION IV.

ON THE COMPUTATION OF SPHERICAL SURFACES.

PROP. I.

(265.) Problem. To exp

express

the surface of a spherical triangle, in terms of its three angles, and of the radius of the sphere.

Put for the sphere's radius; π for the circumference of a circle, of which the diameter is unity: then (Archim. or Legendre Geom. and E. 2. 12.) the surface of the hemisphere is expressed by 2r2; and, if T be put for the surface of the spherical triangle, of which the angles are A, A', A",

T: 2r2π :: (A+A′+A′′)−180° : 360° (Art. 222.)