If the first equation in (W) be divided by the first in

(V'), we shall have,

tan.‡a = {

-cos.S cos.(S-A)

cos.(S-B) cos.(S—C)

༽

And corresponding expressions may be obtained for tan.b and tan.c.

NAPIER'S ANALOGIES.

If the value of cos.c, expressed in the third equation of group (S), Prop. 7, be substituted for cos.c, in the second member of the first equation of the same group, we have,

cos.acos.a cos.2b+sin.a sin.b cos.b cos. C+ sin.b sin.c cos.A; which, by writing for cos.b its equal, 1—sin.b, becomes, cos.a=cos.a-cos.a sin. 2b+sin.a sin.b cos.b cos. C+sin.b sin.c cos. A. Or, 0 ——cos.a sin.b+sin.a sin.b cos.b cos. C+sin.b sin.c cos.A. Dividing through by sin.b, and transposing, we find, cos.A sin.c cos.a sin.b-sin.a cos.b cos. C;

hence, cos.A=

cos.a sin.b-sin.a cos.b cos.C

sin.c

(1)

By substituting the value of cos.c, in the second of the equations of group (S), Prop. 7; or, merely writing B for A, and interchanging b and a, in the above value, for cos. A, we obtain,

Adding equations (1) and (2), member to member,

we have,

cos.A+cos.B — sin.(a+b)—sin.(a+b) cos. C

=

sin.c

=

;

by remembering that sin.a cos.b+cos.a sin.b sin.(a+b). (See Eq. (7), Sec. I, Plane Trig.).

sin.(a+b). (3)

Whence, cos. A + cos.B=(1—cos. C') sin.c

In any spherical triangle we have, (Prop. I), sin. A sin.B: sin.a sin.b;

And therefore, sin.A + sin.B : sin.B :: sin.a+sin.b :

Dividing equation (4) by equation (3), member by

member, we obtain,

sin. A+ sin.B

=

sin.C sin.a+sin.b cos.A+cos.B 1-cos. C sin.(a+b)

(5)

Comparing this equation with Equations (20) and (26), Sec. I, Plane Trigonometry, we see that it can be re

duced to

sin.a+sin.b

tan. (A+B) = cot.C×

(6)

sin.(a + b)

Again, from the proportion,

sin.A : sin.B :: sin.a: sin.b,

we likewise have,

sin.A-sin.B sin.B: sin.a-sin.b: sin.b;

Dividing this equation by equation (3), member by member, we obtain,

Comparing this with Equations (22) and (26), Sec. I, Plane Trigonometry, we see that it will reduce to

sin.a-sin.b

tan.§(A — B) = cot.§×sin.(a + b) (7)

Now, sin.a + sin.b = 2sin. (a + 3)

Sec. I, Plane Trig.).

and, sin. (a + b)

5);

; Eq. (15),

= 2sin.(@ + *) cos.(a + Š), Eq. (30), b)

Sec. I, Plane Trig.).

2

2

Dividing the first of these by the second, we have

Writing the second member of this equation for its first member in Eq (6), that equation becomes

tan. †(A + B) = cot. C
cot. cos. (ab)

cos. (a+b)*

(8)

By a similar operation, Eq. (7) may be reduced to

tan. †(A — B) = cot. sin. (a+b)

cot. §αsin. §(a — b) (9)

Equations (8) and (9) may be resolved into the proportions

cos. (a+b): cos. (a - b) :: cot. C : tan. (A + B); sin. (a + b): sin. †(a —b) :: cot. C: tan. (A — B). These proportions are known as Napier's 1st and 2d

Analogies, and may be advantageously used in the solution of spherical triangles, when two sides and the included angle are given.

When expressed in language, these proportions furnish the following rules:

1. The cosine of the half sum of any two sides of a spherical triangle is to the cosine of the half difference of the same sides, as the cotangent of half the included angle is to the tangent of the half sum of the other two angles.

2. The sine of the half sum of any two sides of a spherical triangle is to the sine of the half difference of the same sides, as the cotangent of half the included angle is to the tangent of the half difference of the other two angles.

The half sum, and the half difference of two angles of a spherical triangle, may be found by these rules, when two sides and the included angle are given; and by adding the half sum to the half difference, we get the greater of these two angles, and by subtracting the half difference from the half sum, we get the smaller. The third side may then be found by proportion.

We have analogous proportions applicable to the case in which two angles and the included side of a spherical triangle are given.

To deduce these, let us represent the angles of the triangle by A, B, and C, and the opposite sides by a, b, and c; A', B', C', a', b', c', denoting the corresponding angles and sides of the polar triangle.

Now, Eq. (9) is applicable to any spherical triangle, and when applied to the polar triangle, it becomes

tan. (A' — B') = cot. C'

But by Prop. 6, Sec. I,

Spherical Geometry, we have

A' 180°-a, B' 180°-b, C180°-c, = =

=

a' = 180° — A, b' — 180° — B, c' = 180° — C

A+B

Whence, (A'B')=1(b—a), §(a' + b′) = 180°. 2

}(a' — b') — }(B — A), †C′ — 90° — Jc.

1

By the substitution of these values in Eq. (n), that

— — tan. (a —b), and sin. (B—A)=

since tan. (6— a)

- sin. (AB).

=

By applying Eq. (8) to the polar triangle, and treating the resulting equation in a manner similar to the above, we find

cos. (AB) tan. c,
cos. (A + B)

(g)

Equations (p) and (2) may be resolved into the fol

lowing proportions.

sin. (A + B) : sin.

cos. (A + B): cos.

(A — B) :: tan. c: tan. (a — b); (A — B) :: tan. c: tan. (a + b).

These proportions are called Napier's 3d and 4th Analogies, and when expressed in words become the following rules:

1. The cosine of the half sum of any two angles of a spherical triangle is to the cosine of the half difference of the same angles, as the tangent of half the included side is to the tangent of the half sum of the other two sides.

2. The sine of the half sum of any two angles of a spherical triangle is to the sine of the half difference of the same angles, as the tangent of half the included side is to the tangent of the half difference of the other two sides.

The half sum, and the half difference of two sides of a spherical triangle, may be found by these rules, when two angles and the included side are given; and by adding the half sum to the half difference, we get the greater of these sides, and by subtracting the half difference from the half sum, we get the smaller.