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 moves on two German silver rollers. The roller on the left rotates a steel wheel that in turn rotates an axle, which turns the measuring wheel and registering dial. The measuring wheel has a vernier. All three parts are made of white plastic. A piece of leather on a string is placed between the steel wheel and the axle when the instrument is stored.
- The twelve-inch rectangular German silver tracer arm is attached to a bronzed brass carriage below the measuring wheel and between the rollers. It has a brass tracer with steel point and support. The length of the arm is adjustable, and it is evenly divided to 0.5mm and numbered from 10 to 64. An extension for the tracer arm adds ten inches to its length and is numbered from 65 to 110.
- Above the roller on the left is marked: F. Weber & Co (/) Philadelphia. Above the roller on the right is marked: G. Coradi, Zürich (/) Switzerland (/) No 3811. An oblong German silver testing rule is marked for 0", 1", 2", 3", and 4". It is also marked: G. Coradi. Zurich.
- A fitted wooden case covered with black morocco leather is lined with purple velvet. A brush is in the corner of the case. A printed calibration chart glued inside the lid has columns for Scales, Position of the vernier on the tracer bar, and Value of the unit of the vernier on the measuring roller. The values in the Position column are handwritten. A paragraph explains how to effectively use and care for the instrument. The date on the chart indicates that the Coradi firm made serial number 3,811 on May 14, 1914. Another piece of paper glued inside the lid explains how to safely remove the instrument from the case. The case's key is on a string inside the case.
- The Zurich workshop of Gottlieb Coradi (1847–1929) made a variety of planimeters, beginning in the early 1880s, with the rolling sphere form debuting around 1900. According to a 1915 catalog, Coradi sold this size of rolling sphere planimeter as model 32. F. Weber & Company was founded in Philadelphia in 1853 and took that name in 1887. It is best known for manufacturing and distributing art products. Other American firms, such as Keuffel & Esser, also distributed Coradi's precision disc planimeter. K&E sold this size as model 4262 for $95.00 from 1900 to 1915. Compare to MA*333660. 1977.0112.02 is an instruction manual.
- References: J. W. Beardsley, "Description and Theory of Coradi's Rolling Ball Planimeter," Journal of the Association of Engineering Societies 28 (1902): 67–77; J. Y. Wheatley, The Polar Planimeter and Its Use in Engineering Calculations (New York: Keuffel & Esser, 1903), chapter 10, http://www.leinweb.com/snackbar/planimtr/wheatley/s10-6.htm; Mathematical-Mechanical Institute of G. Coradi, Catalogue of Mathematical Precision Instruments (Zurich, 1915), 13–17; "About Us," Martin/F. Weber Co., http://www.weberart.com/about/index.html; Catalogue of Keuffel & Esser, 30th ed. (New York, 1900), 308; Catalogue of Keuffel & Esser, 35th ed. (New York, 1915), 317.
- Currently not on view
- date made
- F. Weber & Co.
- Coradi, Gottlieb
- ID Number
- catalog number
- accession number
- Data Source
- National Museum of American History, Kenneth E. Behring Center