be used with greatest advantage, and when one or more of the angles is nearly 90°, we ought to employ the group (3). The group (7) may be made use of in any case.

Case 6. Given the three angles A, B, C, required the three sides a, b, c.

It is manifest that this case does not admit of solution, for any number of unequal similar triangles may be constructed, having their angles equal to the angles A, B, C.

We shall conclude this chapter by giving one or two numerical examples.

Example 1. Given A = 68° 2′ 24′′, B = 57° 53′ 16′′-8, a = 3754 feet, required C, b, c.

Example 2. Given a = 145, b = 178°3, A = 41° 10′, required B, C

This example belongs to case 4, and since the given angle A is acute, and the side b opposite to the required angle B greater than the side a, the solution will be ambiguous.

We have

log. sin. Blog. sin. A + log. b

- log. a

log. sin. A = 9.8183919

log. b = 2.2511513

12.0695432

log. a = 2.1613680

log. sin. B = 9.9081752

The angle in the tables corresponding to this logarithm is 54° 2′ 22′′, but we cannot determine a priori whether the angle sought be this angle, or its supplement 125° 57′ 38′′.

If we take the 1st value,

C = 84° 47′ 38′′ and the triangle required is ABC

If we take the second value,

see last figure.

C = 12° 52′ 22′′ and the triangle required is AB'C Example 3. Given a = 1783, b = 145, A = 41o 10', required B.

This example also belongs to case 4, but since the given angle A is acute, and the side b opposite the required angle B less than the side a, it follows that the angle B must be an acute angle, and the solution will not be ambiguous.

We have

But

log. sin. B = log. sin. A + log. b — log, a

The angle in the tables corresponding to this logarithm is 32° 21′ 54′′, and since, in the present instance, the supplement of 32° 21′ 54′′ cannot belong to the case proposed, the solution is not ambiguous.

Example 4. Given a=3754, b=3277-628, and the included angle 57° 53′ 16′′.8; required A, C, b.

The angles A and B being determined, the side c may be readily found from

log. c = log. a + log. sin. C-log. sin. A

Example 5. Given a = 33, b = 42.6, c = 53·6, required A, B, C.
Taking the formula marked (§) in chap. III. we have

log. sin. A

=log. R+log. 2+}{log.s+log.(s—a)+log.(s—b)+log.(s—c)}—{log.b+log.c}

log. sin. B

=log.R+log.2+3{log.s+log.(s—a)+log.(s—b)+log.(s—c)}—{log.a+log.c}

log. sin. C

=log.R+log.2+${log.s+log.(s—a)+log.(s—b)+log.(s—c)}—{log.a+log.b}

} {log.s+log.(s—a)+log.(s—b)+log.(s—c)} = 2·8468675 ..log.R+log.2+3{log.s+log.(s—a)+log.(s—b)+log.(s—c)} =13·1478975

Subtracting from this number the values log. 6+ log. c; log. a + log. c; log. a + log. b; in succession we find,

Having determined A and B by the above method, we find the above accurate value of C, by subtracting the sum of A and B from 180°. If, however, it had been required to determine C alone (being an angle nearly equal to 90°) we could not have found its value with sufficient accuracy from the common tables, for it will be seen, upon referring to them, that the number 9.9999740 may be the logarithm of the sine of any angle from 89° 22′ 20′′ up to 89° 22′ 25′′, consequently the above method cannot be applied with propriety to determine the exact value of C, unless we previously determine A and B.

The angle C may however be determined directly, and with great accuracy, from any of the three formulæ (c), (3), (~), in Chap. III.

1·3424227

2-8421098

C = 89° 22′ 20′′

= log. R+} {log.(s—a)+log.(s—b)}— {log.s+log.(s—c)}

1·4996871

{ log. (s—a)+log. (s—b)) = 1·4210549 log. R=10.

11.4210549

··. 1⁄2 {log. s + log. (s—c)} = 1.4258126

Subsidiary Angles are angles which, although not immediately connected with a given problem, are introduced by the computist in order to simplify his calculations. Their use, and the method in which they are employed, will be understood from what follows.

When two sides of a triangle, and the included angle, are given, according to the method pursued in the last chapter, we must determine the two remaining angles before we can compute the third side. It frequently happens, however, in practice, that the side only is required, and it therefore becomes desirable to have some direct method of computing the side independently of the two angles.

Suppose that a, b, C are given, and c is required. By chap. III. prop. 4, c3 = a2 + b3 2 ab cos. C

the side c is determined theoretically at once by this expression, but the formula

is not adapted to logarithmic computation, and would, if employed practically, lead to a very tedious and complicated calculation. We can, however, put this expression under a form adapted to logarithmic calculation, by having recourse to an algebraical artifice, and introducing a subsidiary angle.

‚ c* = a* + b1 2 ub cos. C

b)* (1 + tan. Q)

= (a - b) sec. 2

c = (a - b) sec.

log.clog. (a — b) + log. sec. — log. R

The angle is known from the equation.

log. tan. = log. 2 + (log. a + log. b) + log. sin.

being thus determined, log. sec. can be found from the tables, and the value of c becomes known.

The angle, which is introduced into the above calculation, in order to render the expression convenient for logarithmic computation, is called a subsidiary angle.

The above transformation may be effected in a manner somewhat different, as before.

Assume

c' a' + b' — 2 ab cos. C

=