it in B; then ABC is the triangle required. The angles measured by a scale of chords (S. 27.) will be A = 32°. 15′, B=114°. 24' and c=33°. 21'.

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

AB=98

To find the angles in the right angled triangles ADC and CDB.

: radius, sine 90°

:: AD=82.883

: sine DBA=57°. 45′

Then DBA +DBC=57°. 45′ + 56°. 39′=114°, 24′ ABC,

OR THUS,

The remaining angles may be found by Rule I.

(P) 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

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 angle 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 (X. 23, and note.)

(Q) 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. (D. 47.) will solve all the cases in right-angled triangles (except the 6th), and the first and second case in oblique-angled triangles: Rule II. (F. 48) will solve the 6th case in right-angled triangles and the 3d case in oblique; and Rule III. (G. 48.) will solve the last case in oblique triangles.

CHAP. III.

THE APPLICATION OF PLANE TRIGONOMETRY TO THE MENSURATION OF DISTANCES, HEIGHTS, &c.

(R) The mensuration of distances and heights depends upon

the

the rules of plane trigonometry already explained, together with the use of certain instruments for taking angles.

(S) 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 34 inches, the observed angles may be measured to half a minute, or the 120th part of a degree.

(T) 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.

(V) 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.

(W) The use of instruments must be acquired under the direction of a person well skilled in their several adjustments, as but little information can be obtained from written description*, and even the most expert observer will find it necessary, in several cases, to apply corrections to the different angles according to the situations of the objects. See Chapter IV.

* Copper plates, well executed, of theodolites, sextants, quadrants, and other instruments for taking angles, are given in Adams' Geometrical and Graphical Essays, edited and published by Mr. Wm. Jones, mathematical instrument maker in Holborn, London.

(X) Or, produce AB till Ba be equal to it, and CB till the angle a be equal to the angle A; then will the distances Bc and ac, be equal to the distances BC and AC...oy 218 2

(Y) Or, without any instrument to measure an angle, the distances AC and BC may be found. Make AF of any length in a line with AC, and BG in a line with BC; measure AF, AB, BG, FB and AG; then the three sides of the triangles BAF and ABG will be given, with which the angles BAF and ABG may be found; their supplements will give the angles CAB and CBA, and hence the distances AC and BC may be found as in the example.

(Z) The perpendicular distance DC may be readily determined, for having found, BC, the angle B and BC will be given; or find AC, then AC and the angle A will be given to find pc. The perpendicular distance Dc may be found without any instrument for taking angles. With a cross-staff fixed at D observe the object at C, and set up a staff in a line with it at E, and another at B at right angles to CE, move the cross-staff from D to E, and set up a pole or staff at H perpendicular to EC. Measure the distances DB, DE, and also EG in the direction EH till G, B, and C make one straight line. Then

EG-DB: DE EG EC from which take DE.

OR, EG-DB: DE::DB: DCI e

The four following examples are referred to the foregoing figure, and serve to exercise the above observations.

EXAMPLE II.

An engineer wanted to know the breadth of a river, (over which the general intended to pass the whole army,) in order to determine how many pontoons of 5 feet broad and six feet asunder, he should have occasion to make use of. Perceiving an object (c) on the opposite bank of the river close to the edge of the water, he measured 144 paces or 360 feet along the edge. of the river (as AB). The angle at a between the object and the base line was 83°. 57', and at B it was 80°. 32'; required the perpendicular breadth (DC) of the river, and the number of pontoons sufficient to form a passage for the troops?

Answer. The perpendicular breadth of the river, is 528 paces or 1320 feet, a military pace being 24 feet; and 120 pontoons will answer the purpose.

EXAMPLE IN.

Two ships of war intending to cannonade a fort at c, sepa

rated

rated from each other 500 yards, and coming as near to the shore as possible without being in danger of running a-ground, the officers in each ship observe the angles formed between the other ship and the fort, and find them, viz. A=38°. 16′, and B=37°.9′. Are the ships at a proper distance for commencing a cannonade, the most convenient distance being about 300 yards? ... L

Answer. A is 312 yards from the fort, and в 320.

EXAMPLE IV.

Wanting to know my distance from an object at c, I measured from E to D in a line with the object 29 poles, from D, I measured DB at right angles to ED=43 poles, and coming to E I measured EG in the direction EH=54 poles, what is my distance DC from the object?

Answer. 113 poles.

EXAMPLE V.

Wanting to know the distance of two objects A and B (which are 500 yards from each other,) from a tower at c on the opposite side of a river, and, having no other instrument than a chain of 22 yards long, I measured AF, in a straight line with AC, and found it to be 300 yards long, I likewise measured FB and found it to be 649-2 yards, I then measured BG=600 yards and AG 1000 yards, required the distances AC and Be? Answer. AC=455.8 yards, and BC=577·8.

EXAMPLE VI.

B

Suppose I wanted to know the distance between two places A and B, accessible at both ends of the line AB, and that I measured Ac=735 yards, and BC=840, also the angle ACB=55°. 40′. What is the distance between A and B?

A

(A.) If the lines AC and BC be produced till DC be equal to

CB and EC to AC, then will the distance DE be equal to the distance AB without calculation.

EXAMPLE