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.
- This planimeter is designed to find the average of the square roots of all the radial distances from the zero circle to the record line on a circular chart that rests on a turntable underneath the instrument. The planimeter has a 4" green metal angled arm with an adjusting screw and tracer lens; a 6" curved metal arm with a mounting screw and a slotted follower that is inserted in the turntable; and a green metal recording unit with white plastic measuring wheel and vernier and a metal registering dial. The recording unit is marked: LASICO. Underneath the curved metal arm is a serial number: 66607.
- The circular white paper chart is marked: Taylor (/) ROCHESTER, N.Y., U.S.A. (/) CHART OP2750 (/) METER ORIFICE (/) TAYLOR INSTRUMENT COMPANIES. The square root symbol is in front of the word CHART. The hours of the day are printed around the outside of the chart. The turntable is black plastic. It is fastened to a white plastic rectangular base. Underneath the base is marked: TURNTABLE 1 06. It is also marked: MADE IN USA (/) MRC R 20 FF.
- According to the donor, who worked for the Los Angeles Scientific Instrument Company from 1950 until it closed in 2008, this example was made in 1955 as model 2000-V. It replaced model 125, which had a tracer point instead of a tracer lens. Model 2000-V sold for $550.00 in 1955. For a later example and company history, see 2011.0043.01. The turntable was model 1006 and sold for $225.00.
- Taylor Instrument Companies copyrighted the OP2750 chart on June 13, 1933. The firm has a history of selling and making thermometers, barometers, and other instruments that dates back to 1851.
- References: Accession file; "The Taylor Story," http://www.taylorusa.com/taylor-about; Library of Congress, Catalog of Copyright Entries . . . for the Year 1933, part 4, n.s., vol. 28, no. 2 (Washington, D.C.: Government Printing Office, 1934).
- Currently not on view
- date made
- Los Angeles Scientific Instrument Company
- ID Number
- accession number
- catalog number
- Data Source
- National Museum of American History, Kenneth E. Behring Center