Prytz Planimeters

This type of planimeter was invented in about 1875 by Holger Prytz, a Danish mathematician and cavalry officer, as a simple and economical alternative to Amsler's polar planimeter.
It is a very simple device, consisting of a rod with its ends bent at right angles.  One end, the tracer point T, is sharpened to a point, while the other end, C, is sharpened to a chisel edge parallel to the rod. The chisel edge is usually slightly rounded, making it look similar to a hatchet, and consequently the device is also known as a "hatchet planimeter."  Prytz referred to it as a "stang planimeter," "stang" being Danish for "rod." 

When the tracer point T moves along a line, the chisel edge C follows a tractrix.  Note that the path of the chisel is always tangent to the planimeter.

Here is a Prytz planimeter traversing an ellipse.
The tracer point goes around the ellipse.  Note that the chisel point does not return to the point where it started.  The angle between the initial and final positions of the chisel is roughly proportional to the area of ellipse.  The constant of proportionality is the square of the length of the planimeter.  The longer the planimeter, the more accurately the area is measured.

A very simple Prytz planimeter can be made from a two-bladed pocketknife. A description of this is given in the Amateur Scientist column "An Excursion into the Problem of Measuring Irregular Areas"  by C.L. Stong in Scientific American, August 1958, pages 107-114. This was reprinted in "The Scientific American book of projects for the amateur scientist" by C.L. Stong, and more recently in Scientific American's "The Amateur Scientist": The Complete 20th Century Collection on CD-ROM.
Differential geometers will be interested to know that the Prytz planimeter is a simple example of parallel translation on an S1 bundle governed by a connection on an SU(1,1) principal bundle. For details, see Geometry of the Prytz Planimeter.
Pictures of Prytz planimeters
Animations of Prytz planimeters
How a Prytz planimeter measures area
Differential Geometry Day Presentation
Kalamazoo College presentation

Main Planimeter Page
Last update 1 August 2001
Robert Foote