#
PlanimetersPolar–Amsler

About the same time that Sang and Wetli were working on planimeters, the Swiss mathematician Jacob Amsler was figuring out how to make the instrument smaller, less expensive, and easier to use. By 1854, he had eliminated most of the complex mechanisms typical of earlier designs, reducing the planimeter to 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 *x-y *axes. Thus, this form of planimeter is called a polar planimeter. Since the pivot moved back and forth, it traced an area whose net measurement was zero. This meant that the area traced by the tracer point exactly equaled the area of the closed curve the user was measuring. This area was equivalent to 2π multiplied by the product of the length of the tracer arm, the radius of the counting wheel on the tracer arm, and the number of revolutions completed by the counting wheel. Again, mathematicians today work out the detailed mathematics of the operation of Amsler's planimeter with Green's Theorem.

By 1857, Amsler opened a highly successful workshop that sold 50,000 polar planimeters of six different types all over the world in its first 60 years. Other manufacturers of scientific instruments also copied his design, including Hine & Robertson of New York City and American Steam Gauge & Valve Manufacturing Company of Boston. A prominent Swiss modification of the Amsler planimeter is featured on the next page. Planimeters directly from Amsler or on his design were sold by American retailers to universities, engineers, and factory managers until at least the 1930s.

"Planimeters - Polar–Amsler" showing 1 items.

## Instructions for Amsler's Polar Planimeter

- Description
- In 1854 Jacob Amsler, a Swiss teacher and mathematician, devised a planimeter that did not need the cones or wheel-and-disc constructions of earlier instruments such as 1983.0474.02 and 1986.0633.01. His smaller and simpler device also used polar coordinates rather than the Cartesian coordinate system. Amsler established a workshop to produce polar planimeters, and he built a network of agents in Europe and the United States to distribute the instrument. Over 50,000 polar planimeters of at least six different types were sold by the time Amsler died in 1912, and the firm continued under his son's name (Alfred J. Amsler & Co.) until at least 1960.

- This three-page leaflet was printed for one of Amsler's agents, Amsler & Wirz. Charles T. Amsler, a Swiss immigrant, and possibly a relative of Jacob Amsler, began to sell European instruments in Philadelphia in 1848 and briefly partnered with A. H. Wirz from 1855 to 1857. In 1861 C. T. Amsler sold his business to William Y. McAllister and returned to Switzerland.

- The leaflet shows a Type 3 Amsler polar planimeter and explains how to use the instrument. Amsler & Wirz sold it for $20.00, filled orders within six weeks, and recommended the planimeter to draftsmen, engineers, surveyors, ship builders, architects, and machinists. The year 1856 is written in pencil at the top of the first page, and the top left corner is embossed with the words "Turkey Mill" and a ship, presumably referring to the English paper manufacturer. The leaflet was found in the Museum before 1984.

- References: Peggy Aldrich Kidwell, "Planimeter," in
*Instruments of Science: An Historical Encyclopedia*, ed. Robert Bud and Deborah Jean Warner (London: Garland Publishing, 1998), 467–469; Michael S. Mahoney, "Amsler (later Amsler-Laffon), Jakob," in*Dictionary of Scientific Biography*, ed. Charles Coulston Gillispie (New York: Scribner, 1970), i:147–148;*C. T. Amsler's Illustrated Catalogue of Optical, Mathematical, and Philsophical Instruments*(Philadelphia, 1855); C. T. Amsler and A. H. Wirz, advertisement,*Ohio Journal of Education*4, no. 12 (December 1855): 411; C. T. Amsler, advertisement,*Scientific American*13, no. 1 (September 12, 1857): 7; William Y. McAllister,*A Priced and Illustrated Catalogue of Mathematical Instruments*(Philadelphia, 1867), 9–11.

- Location
- Currently not on view

- date made
- 1856

- maker
- Amsler and Wirz

- ID Number
- 1987.0107.10

- accession number
- 1987.0107

- catalog number
- 1987.0107.10

- Data Source
- National Museum of American History, Kenneth E. Behring Center