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 analog instrument uses integration 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. It was particularly useful for calculations of rates of flow and total flow of liquids. The planimeter has a 4" black plastic 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 chart board; and a white metal recording unit with white plastic measuring wheel and vernier and a metal registering dial. The recording unit is marked: LASICO (/) USA. Underneath the curved metal arm is a serial number: 97661.
- A black plastic case is lined with black and red foam padding. A label inside the lid is marked: DESIGNED AND BUILT BY (/) MAXIMILIAN BERKTOLD, CALIFORNIA – USA. An extra tracing lens is inside the case. The bottom of the case is marked with the recycling symbol around the number 2 and above the letters HDPE. The instrument was received with instructions, 2011.0043.01.01.
- Georg Lory (1897–1968) worked for several German instrument manufacturers, including R. Reiss, before he immigrated to the United States in 1925 and worked for the Eugene Dietzgen Company in its San Francisco office. In 1929 he established the Los Angeles Scientific Instrument Company in Los Angeles. He repaired and made equipment including surveying instruments, planimeters, and pantographs. In 1944 he applied for a patent on the square root planimeter, receiving it in 1949. According to the donor, who joined the company in 1950, this example was made in 1990 as model 2000-V. It sold for $865.00 and was intended for use with Foxboro Chart 898418. For an earlier example, see 2011.0043.02. Lasico closed in 2008, although Absolute Accuracy, a successor firm in the same location, continues to distribute models 10, 20, and 30.
- References: Georg Lory, "Square Root Radial Averager" (U.S. Patent 2,458,009 issued January 4, 1949); accession file.
- 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