The Theory of the Polar Planimeter: An Application of the Method of Barriers

Extract

Many instruments have been devised for mechanically recording the area of a closed curve when a tracing point is carried round its boundary, and, of these, probably the best known is the Amsler Planimeter invented in 1854 by Professor J. Amsler of Schaffhausen.

It belongs to the class known as polar planimeters, so called because the tracing arm turns about a fixed centre or pole.

The theory of the instrument has been given in various ways by different writers, but perhaps the neatest explanation is that given by, Professor Henrici in the British Association Report, 1894. As this is probably pretty well known, only the briefest resumé will be given here in order to make the extension of it and the present article readily intelligible. The explanation depends on the general theorem that the area, reckoned algebraically, swept out by a line of fixed length in a complete cycle, is equal to the difference of the areas described by its ends. In the Amsler Planimeter, one end of the tracing arm moves over the circumference of the circle described by the end of the pole arm while the tracing point at the other end describes the boundary of the area.