2. Take two stakes, _X_, _Y_, in a field, preferably two or three hundred feet apart, always marked on top with crosses so as to have exact points from which to work. Let it then be required to stake out or "range" the line from _X_ to _Y_ by placing stakes at specified distances. One boy stands at _Y_ and another at _X_, each with a plumb line. A third one takes a plumb line and stands at _P_, the observer at _X_ motioning to him to move his plumb line to the right or the left until it is exactly in line with _X_ and _Y_. A stake is then driven at _P_, and the pupil at _X_ moves on to the stake _P_. Then _Q_ is located in the same way, and then _R_, and so on. The work is checked by ranging back from _Y_ to _X_. In some of the simple exercises suggested later it is necessary to range a line so that this work is useful in making measurements. The geometric principle involved is that two points determine a straight line. [Illustration] [Illustration] 3. To test a perpendicular or to draw one line perpendicular to another in a field, we may take a stout cord twelve feet long, having a knot at the end of every foot. If this is laid along four feet, the ends of this part being fixed, and it is stretched as here shown, so that the next vertex is five feet from one of these ends and three feet from the other end, a right angle will be formed. A right angle can also be run by making a simple instrument, such as is described in Chapter XV. Still another plan of drawing a line perpendicular to another line _AB_, from a point _P_, consists in swinging a tape from _P_, cutting _AB_ at _X_ and _Y_, and then bisecting _XY_ by doubling the tape. This fixes the foot of the perpendicular. [Illustration] 4. It is now possible to find the area of a field of irregular shape by dividing it into triangles and trapezoids, as shown in the figure. Pupils know from their work in arithmetic how to find the area of a triangle or a trapezoid, so that the area of the field is easily found. The work may be checked by comparing the results of different groups of pupils, or by drawing another diagonal and dividing the field into other triangles and trapezoids. These are about as many types of field work as there is any advantage in undertaking for the purpose of securing the interest of pupils as a preliminary to the work in geometry. Whether any of it is necessary, and for what pupils it is necessary, and how much it should trespass upon the time of scientific geometry are matters that can be decided only by the teacher of a particular class. [Illustration] [Illustration] [Illustration] [Illustration] [Illustration] [Illustration] [Illustration]