Planimeters

A planimeter with a closed curve.

Situations frequently arise in which a person needs to measure the area of a two-dimensional space bounded by a closed curve. For instance, cartographers draw topographical maps that show the contours of different elevations or attempt to represent parts of the landscape with irregular boundaries, such as forests. Medical professionals measure an image of a tumor or internal organ. Biologists observe irregularly-shaped natural phenomena, such as the parts of plants or animals. When most industrial machinery was powered by steam engines, the engines were equipped with devices called indicators that recorded changes in pressure within the engine while its pistons moved through a cycle. Monitoring these changes helped operators know when the engine was in danger of breaking or was using fuel inefficiently. The readings taken by indicators generated curved diagrams, similar to the drawing in the banner image on this page.

Although there are numerous applications that produce drawings of curved areas, measuring those areas is not easy. Historically, people divided up a curved area into rectangles and triangles, whose areas could be easily calculated, and then added up all of the pieces. Since this was both time-consuming and inexact, practitioners, mathematicians, and mathematical instrument makers tried to come up with a machine that would measure a curved area automatically. In 1818, Johann Martin Hermann, a Bavarian surveyor, built the first planimeter, a device that traces a curve with a needle point connected to a measuring wheel that converts the length of the tracing to the result of an integral function. Italian mathematician Tito Gonnella and Swiss inventor Johannes Oppikofer, who may have been aware of Gonnella's work, constructed their own planimeters in 1825 and 1826, respectively. The following pages show how objects in the Smithsonian's collections illustrate further developments in the history of planimeters.

Acknowledgement

The digitization of this group of artifacts was made possible through the generous support of Edward and Diane Straker.