The polar planimeter

Two polar planimeters: Siemens & Halske (made by A. Blankenburg), Germany ~1930 and Allbrit/Stanley, UK 1950.

The planimeter, invented by Jacob Amsler (1823-1912) was used for computing the area enclosed by a closed curve.
figure 1

The arm OB (first figure below) has a pivot point in O which is fixed to the paper. Usually, this is implemented as a sharp point pressed into the paper by a weight.
Point B, which can describe a circle around O, rests in turn on a point of the arm AC. Point A is is moved by the user along the curve drawn on the paper. The other end of this arm rests on the paper with a steel wheel W, the axis of which is accurately parallel to AC. (In the Siemens planimeter, W is on the other side of A; also there is a supporting wheel perpendicular to W). The rotation of W can be measured accurately with the help of a division on its rim, and a revolution counter. If AC is moved in its own direction, W slips over the paper without turning. However any movement component n normal to AC leads to a proportional rotation of W.
figure 2

If the original position of the arm is AB (next figure), and if it is moved to the nearby position A'B', then we have
figure 3

if higher order terms are neglected (l is the length of AB).
If A makes a complete roundtrip over the closed curve K, then the surface covered by AB is equal to
figure 4

This area is equal to the area enclosed by K because the area between the circle and K is covered twice in opposite directions. If O is outside K, then
figure 5

Hence the area of K is proportional to the rotation of W, the constant of proportionality being determined by the dimensions of the planimeter.
figure 5

In the Siemens planimeter drawing, part 4 is an accessory used for calibrating the instrument.

Reiss compensation polar planimeter 3005 (DDR, 1969)

The explanation was adapted from the ENSIE encyclopedia (Amsterdam 1949). The top drawing was taken from the Siemens Spezial-Planimeter manual (Berlin 1928).
Prof. Dr. Joachim Fischer (München) provided a number of corrections and additions.