By Gunter's Scale. Extend the compasses, on the line of sines, from 90° to 53° 8', that extent will reach, on the line of numbers, from 480 to 384 the perpendicular BC.

Extend the compasses, on the line of sines, from 90° to 36° 52', the complement of the angle A; that extent will reach, on the line of numbers, from 480 to 288, the base AB.

3. In the triangle ABC, are given the angle A = 79* 23', the angle B -- 54° 22', and the side BC = 125; required AC and AB. Ans. AC = 103.4, and AB = 91.87.

4. In a right-angled triangle, there are given the angle A = 56° 48', and the base AB = 53.66, to find the perpendicular BC and hypothenuse AC. Ans. BC = 82 and AC = 98.

5. In the right-angled triangle ABC, are given the angle A = 39° 10', and the perpendicular BC = 407.37, to find the base AB and hypothenuse AC. Ans. AB = 500.1, and AC = 645.

CASE 2. Two sides and an angle opposite one of them being given,

to find the other angles and side.

RULE.

As the side opposite the given angle,
Is to the other given side,
So is the sine of the angle opposite the former,
To the sine of the angle opposite the latter. *

* This is evident from the demonstration of the rule in the preced

Add the angle thus found to the given angle, and subtract their sum from 180°, the remainder will be the

third angle.

After finding the angles, the other side may be found by Case 1.

Note.-The angle found by this rule is sometimes ambiguous, for the operation only gives the sine of the angle, not the angle itself; and the sine of every angle is also the sine of its supplement.

When the side opposite the given angle is equal to, or greater than the other given side, then the angle opposite that other given side is always acute; but when this is not the case, that angle may be either acute or obtuse, and is consequently ambiguous.

EXAMPLES.

1. In the triangle ABC, are given the angle C = 33° 21', the side AB=.98 and the side BC=.7912; required the angles A and B, and the side BC.

By Construction, Fig. 43. Make BC = .7912 by a scale of equal parts, and draw CA, making the angle C= 33° 21'; with the side AB=.98, in the compasses, taken from the same scale of equal parts, and B as a centre, describe the arc ab, cutting AC in the point A, and join BA; then is ABC the triangle required: the side AC, measured by the scale of equal parts will be 1.54, and the angles A and B, measured by a scale of chords will be 26° 21' and 120° 18'

G

Here the arc ab cuts AC in one point only, because AB is greater than BC; therefore the angle A is acute, and not ambiguous.

To the angle C = 33° 21' add the angle A = 26° 21', and the sum is 590 42 which subtracted from 180°, leaves the angle B= 120° 18'.

By Gunter's Scale. 1. Extend the compasses from .98 to .79 on the line of numbers, that extent will reach from 33° 21' to 26° 21', the angle A, on the line of sines.

2. Add the angle A= 26° 21' to the angle C=33° 21', and the sum will be 59° 42'; then extend the compasses from 33° 21' to 59° 42', on the line of sines, that extent

3. In the triangle ABC, are given the angle C = 33° 21', the side BC = 95.12 and the side AB = 60, to find the angles A and B, and the side AC.

By Construction, Fig. 44. This is constructed in the same manner as the preceding example; only AB, being shorter than BC, the arc ab cuts AC in two points on the same side of BC; hence the angle A may be either acute or obtuse. The side required has also two values as AC and AC.

The sum of the angles C and A subtracted from 180° leaves the angle B = 86° 1' if A be acute, or 27° 17' if A be obtuse.

To find the side AC answering to the acute value of the angle A.

To find the side AC, answering to the obtuse value of the angle A.

4. In a triangle ABC, the side AB is 274, AC 306, and the angle B 78° 13'; required the angles A and C, and the side BC. Ans. A = 40° 33', C = 61° 14', and BC= 203.2.

5. In a right angled triangle, there are given the hypothenuse AC = 272, and the base AB=232; to find the angles A and C, and the perpendicular BC. Ans. A=31° 28', C= 58° 32' and BC = 142.

6. In a right angled triangle ABC, the hypothenuse AC is 150 and one side BC 69; required the angles and other side. Ans. C = 62° 37', A = 27° 23' and AB 133.2.

CASE 3.

Two sides and the included angle being given, to find the

other angles and side.

RULE.

Subtract the given angle from 180°, and the remain