12. Viewing a ship at anchor from T, the top of a tower, which stood on a hill near the sea shore, I measured the angle of depression, and found it to be 4° 52'. I then descended to the bottom of the tower, and there, at B, took the angle of depression of the ship, and found it to be 4° 2'. From these two observations, the horizontal distance of the vessel is required, and also the height of the tower above the level of the sea, allowing the height of the tower to be 54 feet.

50'; also the angle STB=90°-4°52′=85°8′.

13. Wanting to know the height of a castle standing

on an inclined plane AB; I measured from the bottom of

the building a distance BD= 40 feet; at D I took the angle CDB=41°; from D I measured 60 feet to A, where

B

D

A

I measured CAB=23°45′. The height of the castle

is required.

In the triangle ADC, we have ACD=BDC-A= 41°-23° 45′-17° 15′.

Then to find CB, the height of the castle:

14. Wanting to know the distance of an object at D, from two others A and B, and also the distance between A and B, I set up a pole at C in a right line with AB, at CI measured the angle ACD, and found it to be 57°; then I measured CD=784.8 yards; at D I measured the angles CDA and ADB, and found them to be 14° and 41° 30′ re

ACD+ADC=57°+14° 71°; therefore

30'.

In the triangle ADB,

The angle B=CAD-ADB=109°-41°30′=67°

15. Wanting to know the height of an object on the other side of a river, and also its distance from me; the object was on the same plane with the spot on which I stood by the side of the river; and not having room to go backward on the same plane, on account of the immediate rise of a hill which lay to my back, I placed a mark where I stood, and measured in a direct line from the object, up the hill, whose ascent was so regular that I might consider it a straight line, to the distance of 132 yards, where I perceived that I was above the level of the top of the object. I then took the angle of depression of the mark by the river side equal 42°; I also took the angle of depression of the bottom of the object, equal 27°, and of its top 19°. Required the height of the object, and the distance of the mark from its bottom.