To ascertain the height of a tree, tower, etc., fold a square of paper across, and glancing along the hypothenuse (longest side) of the right angle so found, ascertain at what point your line of sight just catches the top of the object. Then its height is the same distance as the distance from where you stand to its foot, or the length of a plumb line falling from its summit, together with the height of your eye above the ground, added.
Another method is to measure the shadow of the object on a level surface, next measure your own. Then
As your shadow is to your height so is the shadow of the object to its height.
The area of a square is equal to the square of one of its sides.
The area of a triangle is equal to the base multiplied by half the height.
The areas of figures containing more than three sides may always be found by resolving such figures into a series of right angled triangles.
Very frequently the surveyor is called upon to measure an inaccessible line. There are many ways of solving such a problem, but one of the simplest is as follows:
Supposing the required distance is that from bank to bank of a river (Y-X). Then lay off the base line Y-M, driving stakes at each end; make M-P at right angles to Y-M. Sight from P to X, and drive in a stake at Z. Then:
Z M : M P :: Z Y : Y X.