: cos D :: tan B: tan E, that is cos (C+B): cos (C--B) : : tan 1⁄2 BC tan (C+B); which is the second part of the proposition. Therefore, &c. Q. E. D.

COR. 1. By applying this proposition to the triangle supplemental to ABC (11.), and by considering, that the sine of half the sum or half the difference of the supplements of two arches, is the same with the sine of half the sum or half the difference of the arches themselves; and that the same is true of the cosines, and of the tangents of half the sum or half the difference of the supplements of two arches; but that the tangent of half the supplement of an arch is the same with the cotangent of half the arch itself; it will follow, that the sine of half the sum of any two sides of a spherical triangle, is to the sine of half their difference as the cotangent of half the angle contained between them, to the tangent of half the difference of the angles opposite to them: and also that the cosine of half the sum of these sides, is to the cosine of half their difference, as the cotangent of half the angle contained between them, to the tangent of half the sum of the angles opposite to them.

COR. 2. If therefore A, B, C be the three angles of a spherical triangle, a, b, c the sides opposite to them,

I. sin (A+B) sin (A-B) : : tan c: tan (a-b).

II. cos

:

(A+B): cos

(A-B) :: tanc tan (a+b).
(a+b) : sin (a—b) : : tan C : tan (A—B).

III. sin
IV. cos (a+b): cos † (α-b) :: tan †C : tan ¦ (A+B).

PROBLEM I.

In a right angled spherical triangle, of the three sides and three angles any two being given, besides the right angle, to find the other three.

This problem has sixteen cases, the solutions cf which are contained in the following table, where ABC is any spherical triangle right angled at A.

TABLE for determining the affections of the Sides and Angles found by the preceding rules.

AC and B of the same affection, (14).

If BC 90°, AB and B of the same affection, otherwise difdifferent,

(Cor. 15.)

If BC 90° C and B of the same affection, otherwise different,

AB and C are of the same affection,

1

2

(15.)

3

(14.)

4

(Cor. 15.)
(14.)

If AC and C are of same affection, BC90°; otherwise
BC790°,

B and AC are of the same affection,

Ambiguous.

5 6

When BC 90°, AB and AC of the same; otherwise of

different affection,

AC and B of the same affection,

When BC90°, AC and C of the same; otherwise of dif

ferent affection,

BC 90°, when AB and AC are of the same affection,

B and AC of the same affection,

C and AB of the same affection,

AB and C of the same affection,

AC and B of the same affection,
When B and C are of the same affection, BC 90°, other-
wise, BC 790°,

The cases marked ambiguous are those in which the thing sought has two values, and may either be equal to a certain angle, or to the supplement of that angle. Of these there are three, in all of which the things given are a side, and the angle opposite to it; and accordingly, it is easy to shew, that two right angled spherical triangles may always be found, that have a side and the angle opposite to it the same. in both, but of which the remaining sides, and the remaining angle of the one, are the supplements of the remaining sides and the remaining angle of the other, each of each.

Though the affection of the arch or angle found may in all the other cases be determined by the rules in the second of the preceding tables, it is of use to remark, that all these rules except two, may be reduc

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ed to one, viz. that when the thing found by the rules in the first table is either a tangent or a cosine; and when, of the tangents or cosines employed in the computation of it, one only belongs to an obtuse angle, the angle required is also obtuse.

Thus, in the 15th case, when cos AB is found, if C be an obtuse angle, because of cos C, AB must be obtuse; and in case 16. if either B or C be obtuse, BC is greater than 90°, but if B and C are either both acute, or both obtuse, BC is less than 90°.

It is evident, that this rule does not apply when that which is found is the sine of an arch; and this, besides the three ambiguous cases, happens also in other two viz. the 1st and 11th. The ambiguity is obviated, in these two cases, by this rule, that the sides of a spherical right angled triangle are of the same affection with the opposite angles. Two rules are therefore sufficient to remove the ambiguity in all the cases of the right angled triangle, in which it can possibly be removed.

It may be useful to express the same solutions as in the annexed table. Let A be at the right angle as in the figure, and let the side opposite to it be a; let b be the side opposite to B, and c the side opposite to C.