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Mathematical Treasure: Polar Planimeter Invented by Jacob Amsler

Amy Ackerberg-Hastings (University of Maryland University College)

Amsler Polar Planimeter, Type 2

Amsler Type 2 Polar Planimeter, Sold by Crosby Steam Gage & Valve, 1880s, Smithsonian Institution negative number SIA-73-1252.

Swiss mathematician Jacob Amsler (1823–1912) was aware of other efforts to develop planimeters, mathematical instruments that allow users to determine the area of a curved region by tracing its boundary. By 1854, he had simplified earlier designs into two arms connected with a pivot. One arm was anchored at the end away from the pivot, and one arm traced the drawing. Because the pivot and both arms can move around the anchor, the motion of the tracer arm is graphed with polar coordinates instead of linear xy-axes. Amsler's device therefore was called a polar planimeter.

Amsler then established a workshop in Switzerland to manufacture planimeters. By the 1910s, his firm had sold 50,000 instruments, both directly to users and to retailers such as Crosby Steam Gage & Valve of Boston. In the United States, polar planimeters were particularly popular for measuring the areas of diagrams produced by steam engine indicators, devices attached to steam engines to measure their horsepower and identify potential problems inside the engine. Charles W. Batchelor (1845–1910) used the planimeter pictured here in his work as one of Thomas Alva Edison's chief assistants.

Instructions for Amsler Polar Planimeters, ca 1912

Brochures that functioned both as advertisements and instruction cards were often distributed with Amsler polar planimeters. This example depicts the nine types of polar planimeter manufactured by Amsler's firm in Schaffhausen, Switzerland, and was printed after 1912. Smithsonian Institution negative number AHB2013q009202.

This object and other planimeters from the Smithsonian's National Museum of American History are now shown and described at the website

Index of Mathematical Treasures

Index of Mathematical Objects


Amy Ackerberg-Hastings (University of Maryland University College), "Mathematical Treasure: Polar Planimeter Invented by Jacob Amsler," Convergence (September 2014)

Mathematical Treasures: Smithsonian National Museum of American History Object Groups