sion remains unaltered; so that if we had set out with cos. B, as given by the second of (A), instead of with cos. A, we should have had the

that is, in any spherical triangle the sines of the sides are to each other as the sines of the opposite angles; so that when two of the three given quantities are a side and its opposite angle, the unknown, which is opposite to the third given quantity, may be determined by a simple proportion, or by an easy logarithmic process.

(47.) The equation (2) above might serve to find an angle, from knowing the three sides; it is, however, much less simple than the original expression (1), but neither of them are adapted to logarithms. In order to obtain one that is adapted, add 1 to each member of (1), and there results (form 24, p. 57),

but a and b + c are respectively the difference and sum of the two arcs

(a+b+c), and (b+c—a) ;

hence (form 4, p. 46),

cos.acos. (b+c) = 2 sin. § (a + b + c) sin. 1 (b + c − a); therefore, putting S for the sum of the three sides, we have

sin. S sin. (S— α)

cos. A =

(1).

sin. b sin. c

If instead of adding 1 to each side of (1) p. 76, we subtract each side from 1, and proceed as above, we shall obtain for sin. A the value,

and, by dividing this equation by the former, we have

and all these expressions are adapted to logarithms.

It is unnecessary to put down the corresponding expressions for the other angles, as they may be obtained from these by simply changing the letters: thus for sin. B, we have, by changing A for B and b for

from which it appears that if a > b, sin. § A > sin. § B, and therefore
A> B; also if b > a, sin. B > sin. † A; and therefore B > A.
Consequently the greater side is always opposite to the greater angle.
If b = c, the equation (2) becomes

(48.) We have thus got convenient formulas for the determination of the unknown parts, when two sides and an opposite angle are given, when two angles and an opposite side are given, and when all the three sides are given. We shall now seek the solution to the case in which two sides and the included angle are given, or two angles and the interjacent side; that is to say, we shall proceed to deduce an equation involving only the four quantities a, b, A and C.

For cos. c in the first of equations (A) substitute its value, as given by the third, and there results, after putting 1 sin.2b, for its equal cos.2b,

cos.acos.acos. a sin.2b+ sin. a sin. b cos. b cos. C+

sin. b sin. c cos. A;

or, cancelling cos. a on each side, dividing by sin.b, and transposing,

For sin. c in this equation substitute its value given by (3, p. 77), viz.

which is the equation we proposed to deduce, and from which we at once get an expression for cot. A, when the two sides a, b, and their included angle C, are given, or for cot. a when the two angles A, C, and interjacent side b are given. The remaining parts of the triangle may, obviously, be found by the relation (p. 77) between the sides and opposite angles; but if the third side, in terms of the other two, and the included angle, is required in a single formula, we must then recur to the fundamental equations (A), which obviously furnish that formula. But neither this nor that which we have just deduced are calculable by a single logarithmic operation; by the introduction, however, of a subsidiary arc the solution may be conducted by logarithms, although two operations will be necessary. But we shall explain this artifice in the next chapter, which will contain the practical application of the formulas deduced in this.

(49.) It now only remains for us to furnish a formula for the side of a spherical triangle in terms of the three angles, and this we may easily do by help of the formulas already given for an angle in terms of the sides, availing ourselves of the property of the supplemental triangle, viz. that the angles and sides of this are supplements of the sides and angles of the former (41). For let the formulas (47) refer to the supplemental triangle of that in question, then, by marking the letters of the former with an accent for distinction sake, we have

A' 180°-a, a' 180° - A, 180° - B, c'= 180

C, S'=

540°-S;

S' being the sum of the sides of the triangle in (47), and S the sum of the angles of the triangle with which we are now occupied.

Consequently,

cos. § A'=cos. (90° — § a) — sin. a, sin. 6′ = sin. (180° — B)=sin. B

sin. c' sin. (180° — C) = sin. C, sin. § S′ = sin. (270° — } S)=

- cos. S

sin. († S′ — a′) = sin. [90° — (} S — A)] = cos. (§ S — A);

therefore, by substituting these values the formula (2) becomes

As S exceeds 90° but falls short of 270° art. (42), cos. S is always negative, and, therefore, the numerators, of the first and third of these expressions, although appearing with a negative sign, are in reality positive.

(50.) By means of the polar triangle it is obvious that we may, in all cases as well as in this, convert any formula involving the sides and angles of a triangle into another, similarly involving the angles and sides; the sides in the one formula being replaced by the angles opposite to them in the other, and the angles being replaced by the opposite sides. To effect this change we need only write, instead of sin. and cos. in the original formula, sin. and — cos. of the opposite arc, whether side or angle.

Thus the fundamental equations (A) become in this manner changed into the following,

which plainly show that if the three angles of one triangle are equal to the three angles of another, the sides of the former must also be equal to those of the latter; and also that if two angles B, C, and interjacent side, a, of one triangle are respectively equal to two angles, and the interjacent side of another, the remaining angle A of the one must be equal to the remaining angle of the other; and thus all the parts of the one triangle are equal severally to those of the other.

(51.) The theory now delivered is sufficient for the solution of every case of spherical triangles; but we shall add two more theorems applicable to the case in which the two sides and included angle are given to find the other angles, and to that in which two angles and the interjacent side are given to find the other sides. These theorems have the advantage of being very simple, and are of a form easily retained in the memory. They were first given by Lord Napier, and are known by the name of Napier's Analogies.

By the equation (1), page 77, we have

Similarly,

sin. c cos. A cos. a sin. b- sin. a cos. b cos. C.

sin. c cos. B cos. b sin. a- sin. b cos. a cos. C

.. sin. c (cos. A+ cos. B) sin. (a+b) (1—cos. C) . . . . (1). Now from the equations (3), page 77, we have

sin. A sin. c sin. a sin. C

sin. B sin. c = sin. b sin. C

.. (sin. Asin. B) sin. c = (sin. a± sin. b) sin. C..

Dividing (2) by (1) there results

....

(2).