3. A man travels from A to B, 3·7 miles, then turning a little to the left hand goes from в to C, which is 4.7 miles: at c he observes that a and в make an angle of 29° 16'. What is the distance from A by the shortest cut?

Answer. The side AB, opposite the given angle c, being greater than BC, the triangle belongs to the ambiguous case: and AC is either 7 or 2 miles.

(137) CASE III. Given two sides and the angle contained between them, to find the rest.

Given The side AB=98

The side BC=95.12

Required the angles a and

The angle B=114°24′ C, and the side AC.

1. Extend the compasses from 193.12 to 2.88 on the line of numbers, that extent will reach from 32° 48′ to 0° 33′ on the line of tangents. This is the method of working such examples as this; but so small an angle as 33' is not contained on the scale.

2. Extend from 33° 21′ to 65° 36′ on the line of sines, that extent will reach from 98 to 162 on the line of numbers.

(138) CASE IV. Given the three sides, to find the angles.

The side AB=98

Given The side BC=95.12

{

The side AC-162.34

Required the angleɛ, A, B, and c.

BY CONSTRUCTION.

Draw the longest side AC = 162.34 from a scale of equal parts; with AB=98 in your compasses (taken from the same scale) and one foot in a describe an arc; with BC=95.12 in your compasses cross

ED

it in B; then ABC is the triangle required. The angles measured by a scale of chords (64) will be a=32° 15′, B=114° 24′, and c=33° 21'.

Let E be the middle of the base AC, and BD perpendicular

Then &AC + ED=81·17+1·713=82.883 AD the greater segment, and ACED=81·17-1·713-79.457 = V=DC the less segment.

To find the angles in the right-angled triangles ADB and CDB.

BY GUNTER'S SCALE, according to the first method.

1. Extend the compasses from 324.68 to 193-12 on the line of numbers, that extent will reach from 2.88 to 1.713 the distance of a perpendicular from the middle of the base.

2. Extend from 98 to 82.883 on the line of numbers, that extent will reach from 90° to 57° 45′ on the line of sines.

3. Extend from 95·12 to 79.457 on the line of numbers, that extent will reach from 90° to 56° 39' on the line of sines.

BY GUNTER'S SCALE, according to the second method. 1. Extend the compasses from half the sum of the three sides 177.73 to one of the containing sides AB=98, that extent will reach from Ac=162.34 the other containing side, to a fourth number 89.5, on the line of numbers.

2. Extend the compasses from this fourth number 89.5 to (the difference between the half sum of the three sides and the side opposite to the angles sought) 82.61 on the line of numbers, that extent will reach from 90° on the line of sines to the required angle 32°, on the line of versed sines, immediately under the line of sines.

This is derived from the proportions in the investigation of Gunter's Rule (46 and note).

SCHOLIUM.

(139) There are some authors and teachers of trigonometry, who make no distinction of cases between right and obliqueangled triangles, but divide the whole into three cases; because the three rules necessary for solving the problems that occur in oblique trigonometry, are sufficient for solving those which occur in right-angled trigonometry. For instance, Rule I. (131) will solve all the cases in right-angled triangles (except the 6th), and the first and second cases in obliqueangled triangles: Rule II. (133) will solve the 6th case in right-angled triangles and the 3d case in oblique; and Rule III. (134) will solve the last case in oblique triangles.

CHAP. III.

THE APPLICATION OF PLANE TRIGONOMETRY TO THE MENSURATION OF HEIGHTS, DISTANCES, ETC.

(140) THE mensuration of heights and distances depends upon the rules of plane trigonometry already explained, together with the use of certain instruments for taking angles.

(141) Horizontal and vertical angles are usually measured with a theodolite furnished with one or two telescopes, and a vertical arc; and if the horizontal and vertical arcs of the instrument be described with a radius of not less than 31 inches, the observed angles may be measured to half a minute, or the 120th part of a degree.

(142) Angles which are oblique to the horizon are generally taken with a sextant, which must be held in such a position, that its plane may coincide with the two objects and the eye. When vertical angles are taken with this instrument, an artificial horizon must be used, and the reflected image of the object from the glasses of the sextant must be brought to coincide with the reflected image of the same object in the artificial horizon.

(143) Base lines are generally measured with rods, or the four pole Gunter's chain; but common tape of 50 or 100 feet in length is often preferred both for accuracy and expedition: especially if it be kept dry, and the ground be tolerably level.

(144) The use of instruments must be acquired under the direction of a person well skilled in their several adjustments,