Therefore AB=BE+AE=225.7+5=230.7, the

height required.

9. Wanting to know the height and distance of an inaccessible object; at the least distance from it, on the horizontal plane, I took its angle of elevation equal to 58°, and going 100 yards directly from it, found the angle then to be only 32°; required its height and distance from the first station, the instrument being 5 feet above the ground at each observation.

Here the angle ABC=BCD-A=58°-32°=26°.

Again, in the triangle BCD, we have given CB,

10. Wanting to know the distance between two forts, which were separated from me by a large river, I measured a convenient base AB of 300 yards. Now from

the extremities of the base line AB, I took the following 0', and HBM=45° 15′. Required the distance HM etween the forts.

1. HAM-37° 0′

BAM-58° 20′

ABH-53° 30′

Sum 148 50-HAB+ABH.

Therefore, 180°-148°50′-31° 10' AHB.

2. BAM-58°20′

ABH-53° 30′

HBM-45° 15′

Sum 157° 5'-BAM+ABM.

Therefore, 180°-157°5′-22° 55′-AMB.

45'.

3. ABH+HBM ABM 53° 30′+45° 15′-98°

so is sin. ABM 98° 45′, or 81° 15′ 9.99492

12.47204

9.59039

2.88165

...

to AM 761.4655

3. In the triangle AMH,

AM+AH=761.4655+465.9776=1227.4431
AM-AH-761.4655-465.9776-295.4879

Then 71° 30′-35° 44′-35° 46′ AMΗ.

Note. In a similar manner may the distances be taken between any number of remote objects posited round a convenient station line, which is often of very great use in extensive surveys; as we may determine the sides and angles of large tracts of land, from any station, whether we be within or without it, provided we can command a view of the angles from the station. By this method also a ship at sea may determine the distance of visible ports or headlands.

In this manner we are enabled to take plans of coasts, harbours, cities, towns, fleets, fortifications, &c.

11. Being at sea, I observed a point of land to bear east by south, and after sailing north-east 12 miles, I observed it again, and found its bearing to be southeast by east. How far was I from the point of land, when I made the last observation?

Let A be the ship's place at the first observation; then, by the question, the angle A

B

A

is 5 points, or 56° 15′, every point of the compass being 11° 15'; and the angle B is 9 points, or 101° 15′: therefore the angle C is 22° 30'; and the side AB