To find the sum of ang. A & C. To find the angles A and C.

180°
69

2)111 sum

55° 30' half sum of

11,46390

ang. A and C. Istotan.ang. A--C4° 23′ 8.88412

The three fides of any triangle being given, to find the angles.

Fig. 57. plate 4.

BC 80 Required all the angles.
AC 60

AB:

AB: AC+BC:: BC-AC: BD-AD

۲

100 : 140::20:28

In all cases of this kind, the | To the base greater segment is adjacent Add + difference to the greater side, and vice

verfa.

The greater feg.

14

64

50 14

To fec. ang. A 53° 8' 10.22185 | Tofec.ang. B 36° 52′ 10.09691

Angle C may be found thus: From 180, fubtract the sum of angles A and B, the remainder will give angle C. Or add the complements of the angles A and B together, and the fum is ( angle C.

:

The preceding problem is frequently wrought according to the following Rule.

Add the three fides together, and, from half the sum, subtract the fides severally; then add the complements of the logarithms of the half sum, and of the difference between the halfsum, and the fide oppofite to the angle fought, to the logarithms of the differences of the two other fides and half sum; and half their fum will be the tangent of half the angle required. Thus, let angle A be required:

:

53° 8' The angle CAB.

The angles BC may be found by problem 1. of oblique angled trigonometry.

We come now to the application of trigonometry, to the mensuration of heights and distances,

MENSU

MENSURATION OF HEIGHTS AND
DISTANCES.

THE instruments commonly made use of in measuring heights and distances, are the Geometrical Quadrant, the Theodolite and the Geometrical square.

The Geometrical quadrant is used for investigating vertical angles; whether they be angles of * altitude, or angles of depreffion.

The Theodolite serves for measuring angles on a horizontal plane, or on an inclined plane.

A vertical plane, is that which is at right angles with the horizon. A horizontal plane, is that which is parallel to the horizon.

The Geometrical quadrant, is the fourth part of a circle, and is divided into 90°, to which two fights are adapted, and a plumb line suspended from the centre ; it is commonly made of brass or wood. Fig. 1. plate 4.

The