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 1851 Scottish civil engineer John Sang (1809–1887) exhibited a form of rolling planimeter at the Crystal Palace Exhibition in London. He called the instrument a "planometer," which he changed to "platometer" when he described the instrument to the Royal Scottish Society of Arts a few months later. Like all planimeters, this object measures the area bounded by a closed curve. Sang's device is also significant because it inspired James Clerk Maxwell to work on planimeters, which in turn gave James and William Thomson ideas that helped them develop a mechanical integrator.
- This example is an improved version of Sang's original instrument. A brass cone is on a steel rod that connects two brass rollers. An open brass frame surrounds the rod. It has four brass rollers that slide along a brass base to which the rod is anchored. The frame has a tracer with an ivory handle, a silver measuring wheel that rolls against the side of the cone, and a small magnifying glass. The handle on the tracer arm and the construction of the measuring wheel are changed from Sang's original design.
- The measuring wheel rotates only when the tracer arm's movement is perpendicular to the axis of the cone. The rate at which the wheel moves depends on its distance from the vertex of the cone. For example, when the tracer arm moves a distance S perpendicular to the axis, its reading changes by an amount equal to the area of a rectangle with sides equal to S times the distance from the vertex. The instrument is in a wooden case.
- References: John Sang, "Description of a Platometer, an Instrument for Measuring the Areas of Figures Drawn on Paper," Transactions of the Royal Scottish Society of Arts 4 (1852): 119–129; "Description of Sang's Platometer, or Self-Acting Calculator of Surface," Journal of the Franklin Institute 23 (1852): 238–241; Charles Care, "Illustrating the History of the Planimeter" (Undergraduate 3rd Year Project, University of Warwick, 2004), 39–44; Charles Care, "A Chronology of Analogue Computing," The Rutherford Journal 2 (2006–2007), http://www.rutherfordjournal.org/article020106.html.
- Currently not on view
- date made
- Sang, John
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- accession number
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- Data Source
- National Museum of American History, Kenneth E. Behring Center