A well-known method of measuring the distance across a stream is illustrated in the figure below, where the distance from A to some point P is required.
Run a line from A to C by standing at C in line with A and P. Then run two perpendiculars from A and C by any of the methods already given,—sighting on a protractor or along the edge of a book if no better means are at hand. Then sight from some point D, on CD, to P, putting a stake at B. Then run the perpendicular BE. Since DE : EB = BA : AP, and since we can measure DE, EB, and BA with the tape, we can compute the distance AP.
There are many variations of this scheme of measuring distances by means of similar triangles, and pupils may be encouraged to try some of them. Other figures are suggested on [page 244], and the triangles need not be confined to those having a right angle.
A very simple illustration of the use of similar triangles is found in one of the stories told of Thales. It is related that he found the height of the pyramids by measuring their shadow at the instant when his own shadow just equaled his height. He thus had the case of two similar isosceles triangles. This is an interesting exercise which may be tried about the time that pupils are leaving school in the afternoon.
Another application of the same principle is seen in a method often taken for measuring the height of a tree.
The observer has a large right triangle made of wood. Such a triangle is shown in the picture, in which AB = BC. He holds AB level and walks toward the tree until he just sees the top along AC. Then because