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 instrument consists of a brass drum and disc (covered with paper numbered in pencil by ones from 1 to 25) mounted on metal stands that are fastened to a wooden base. U-shaped metal bars form a frame on two sides of the base. A series of brass gears beneath the disc are turned by a crank at the side of the instrument. A magnifying glass, measuring dial, and registering dial are attached to the center of the disc.
- A rectangular shaft connects the lens and dials to a carriage that moves along a track next to the drum. The carriage has a pointer that moves on the surface of the recording drum, above a metal scale with a vernier. A wooden platform next to the carriage track folds out on two supports to make a surface for writing. A nameplate at the center front of the wooden base is marked: Otto Toepfer & Sohn (/) Werkstätten (/) für wissenschaftslische Instrumente (/) POTSDAM.
- A drawing of a curve was placed on the disc and traced with the pointer. The rotations of the disc and shaft were recorded by the drum to provide a measurement of the space enclosed by the curve. For an earlier, similar instrument, see 1986.0633.01.
- In 1905 Adolf Schmidt (1860–1944), director of the geomagnetic observatory in Potsdam, Germany, from 1902 to 1928, described his invention of this planimeter. It worked by converting a first-order derivative to linear equations. Schmidt argued that his device was more controlled and accurate over repeated observations than similar instruments. The Potsdam firm of Otto Toepfer & Sohn (1873–1919), which also manufactured Schmidt's other inventions and the instruments used by the observatory, sold the instrument as model 53 for 850 DM in 1907. This example was found in the Museum collections around 1982.
- References: H. Podewski, "Geheimrat Professor Dr. Adolf Schmidt," http://www-app3.gfz-potsdam.de/obs/niemegk/en/geschichte/geschichte_obs/aschmidt/adsmi_en.html; J. Bartels, "Adolf Schmidt, 1860–1944," Terrestrial Magnetism and Atmospheric Electricity 51 (1946): 439–447; Adolf Schmidt, "Ein Planimeter zur Bestimmung der mittleren Ordinaten beliebiger Abschnitte von registrierten Kurven," Zeitschrift für Instrumentenkunde 25 (1905): 261–273; Otto Toepfer & Sohn, Erdmagnetische Variations-Instrumente (Potsdam, 1907), 68–69, http://www.astropa.unipa.it/biblioteca/Strumenti/e-catalogues/Toepfer1907/Catalogo.html.
- Currently not on view
- date made
- ca 1910
- Otto Toepfer & Sohn
- ID Number
- accession number
- catalog number
- Data Source
- National Museum of American History, Kenneth E. Behring Center