Planimeter

Planimeter is an instrument by means of which the area within any closed curve can be practically measured. Several devices have been introduced for this purpose, one of the simplest being Amsler's planimeter. This consists of two arms, o J, T w, jointed at J and capable of moving quite freely in one plane. The end o of one arm is fixed, while the extremity T of the other arm can be made to exactly move over the curve whose area is required. The arm T J passes through the centre of a wheel w, and serves as its axis of rotation. The revolutions of the wheel measure the area traced out by the point t, and are independent of the position of the wheel on its axis, T J. However the tracing-point may move, its motion can be resolved into two components: one along the arm T j, and the other at right angles to it. It is obvious that the first will not cause the wheel to rotate; hence the wheel only records the motion which is perpendicular to its axis. After T has traced out the entire curve, the arms will obviously return again to their original position. It can then be mathematically proved that the area of the curve is equal to T j x s where s is the length recorded by the wheel.