On almost all early planimeters (and some later ones), the motion of the tracer point was transferred to a wheel that rolled around the surface of a cone, disc, sphere, or cylinder while the wheel also slid along a track. The dual motion recorded both the distance the tracer point moved along the x axis of the drawing and the distance the tracer point moved along the y axis of the drawing. The turning of the cone generated a series of angles. The sum of these angles is represented by an integral, taken (in the clockwise direction) along the boundary of the region being measured, multiplied by a constant based on the dimensions of the planimeter. The user read the result of the integral from numbered dials on the instrument. For instance, G. A. Carse and J. Urquhart described the operation of Oppikofer's planimeter with the equation:
where ω is the angle turned through by the cone; α is the central angle of the cone; r is the radius of the cone; and r' is the radius of the wheel. Modern mathematicians recognize the integrals calculated by various planimeters as versions of Green's Theorem, which expresses the relationship between a line integral around a simple closed curve and a double integral over the plane region bounded by that curve.
|Ad for a rolling planimeter similar to the instruments by Sang and Coradi in the mathematics collections, from Catalogue of Keuffel & Esser Co., 33rd ed. (New York, 1909), 325.|
These forms of planimeters were heavy and expensive, so they were never popular in the United States. The collection contains examples acquired from Europe that were designed by members of the second generation of planimeter inventors, Scottish civil engineer John Sang and Swiss engineer and railway designer Kaspar Wetli, and a later device developed by German geomagnetist Adolf Schmidt. Some of the major manufacturers of polar planimeters also offered earlier forms of planimeters, as is illustrated by a rolling sphere planimeter made in the Zurich workshop established by Gottlieb Coradi. Finally, in the second half of the 20th century, the American company Lasico sold a wheel and disc planimeter for finding the average of the square roots of all the radial distances from the zero circle to a given line on a circular chart.
"Planimeters - Cone/Disc/Cylinder" showing 1 items.
- In the 1810s and 1820s, three Europeans independently invented the planimeter, an instrument for finding the area bounded by a closed curve. The third of these devices, by the Swiss Johannes Oppikofer, was the first to be commercially manufactured, by Ernst of Paris in 1836. In 1849 Kaspar Wetli, a Swiss engineer and railway designer, received an Austrian patent for replacing the cone used by Ernst with a wheel and disc. This increased the accuracy with which the instrument traced a curve when the surface being measured was uneven. Georg Christoph Starke made improvements to the device, which was described by Simon Stampfer in 1850. The planimeter won a prize at the 1851 Crystal Palace Exhibition.
- Starke's workshop in Vienna manufactured at least 100 of the instruments over a ten-year period. (The Science Museum in London estimates the date for serial number 103 as around 1860.) This example is serial number 44. Hermann, Adolf, and Robert Schlagintweit, German brothers and explorers, took this instrument on their travels in India and the Himalayas between 1854 and 1857. They were sponsored by the East India Company on the recommendation of Alexander von Humboldt.
- The planimeter is in a large wooden case that has two spools of replacement wire in the lid. The instrument has a rectangular brass base. A brass wheel mounted on the base is divided by degrees and marked by tens from 0 to 90 three times. A wheel inside this wheel is marked from 0 to 110. The larger wheel is attached by a metal shaft to a metal wheel. When the metal wheel turns, measurements are recorded with a pointer and the smaller wheel.
- A second part of the instrument has a tracer arm with brass and ivory handles at one end. When the handle is moved, a wire running the length of the arm turns a porcelain disc. This whole assembly slides perpendicularly onto brass rods on the base of the first part of the instrument. Thus, the arm rotates the porcelain disc, which rotates the metal wheel, which triggers the recording pointer and wheels. The arm is engraved: Patent von Wetli & Starke No 44. The arm is also engraved: Schlagintweit Ho.
- The case also holds two large points, which may be meant to serve as styluses but which do not fit the instrument; a brass calibration disc engraved with two circles, marked 49.88 c.m. and 35.63.c.m.; three brass tacks; and a key on a blue ribbon. The instrument was received with documentation (1986.0633.02 and 1986.0633.03).
- References: Louis F. Drummeter Jr., "Starke & Kammerer in San Francisco," Rittenhouse 5 (1990): 25–32; Peggy Aldrich Kidwell, "Planimeter," in Instruments of Science: An Historical Encyclopedia, ed. Robert Bud and Deborah Jean Warner (London: Garland Publishing, 1998), 467–469; Charles Care, "Illustrating the History of the Planimeter" (Undergraduate 3rd Year Project, University of Warwick, 2004), 8, 29–37; accession file.
- Currently not on view
- date made
- Wetli, Casper
- Starke, Christoph
- ID Number
- catalog number
- accession number
- Data Source
- National Museum of American History, Kenneth E. Behring Center