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Planimeters · Volume 2

A Short History of the Planimeter

Measuring an area was a real problem for surveyors and engineers. A field on a map, a cross-section of an embankment or a ship’s hull, the area inside an engine’s pressure curve: these are irregular shapes with no formula. The old ways were to divide the figure into triangles and strips and add them up, or to lay it over squared paper and count. The planimeter, an instrument that measures an area simply by tracing round it, was invented several times in the first half of the nineteenth century, perfected by Jakob Amsler in 1854, and made in hundreds of thousands until computers replaced it.

Figure 1 — An Amsler polar planimeter in the Norwegian Mapping Authority's map museum at Hønefoss. Amsler's design of 1854, a pole arm, a tracer arm and one wheel, is the ancestor of every polar planimeter.
Figure 1 — An Amsler polar planimeter in the Norwegian Mapping Authority's map museum at Hønefoss. Amsler's design of 1854, a pole arm, a tracer arm and one wheel, is the ancestor of every polar planimeter. — Photo: Ssu, CC BY-SA 4.0, via Wikimedia Commons.

2.1 Before Amsler

  • Johann Martin Hermann (1785–1841), a Bavarian land surveyor, devised a planimeter using a wheel running on a cone in 1814. It was never published and no example survives; Amsler himself credited Hermann with the idea, and the Deutsches Museum marked “200 years of the planimeter” in 2014. Some sources give 1818.
  • Tito Gonnella of Florence reinvented the cone-and-wheel mechanism in 1824, realised the cone could be replaced by a disc, and showed his instrument at the Great Exhibition of 1851.
  • Johannes Oppikofer of Switzerland built a cone planimeter, made by Ernst of Paris from 1836.
  • Kaspar Wetli and Georg Christoph Starke of Vienna made a wheel-on-disc planimeter, patented in Austria in 1849.
  • John Sang of Scotland made a rolling “platometer” in 1851, and James Clerk Maxwell, then 23, described a new form of it to the Royal Scottish Society of Arts in January 1855, though he seems never to have built it.

These were all orthogonal or linear planimeters: a carriage moved on rails, and a wheel on a cone or disc integrated the area strip by strip. They were large, expensive and delicate.

Figure 2 — Tito Gonnella's planimeter as he published it in 1841: a carriage running on rails, with a wheel driven by a disc.
Figure 2 — Tito Gonnella's planimeter as he published it in 1841: a carriage running on rails, with a wheel driven by a disc. — Gonnella, 1841; public domain, via Wikimedia Commons.

2.2 Jakob Amsler and the polar planimeter

Jakob Amsler (1823–1912) was a Swiss mathematician and physicist, a doctor of Königsberg under Franz Neumann, teaching mathematics at the gymnasium in Schaffhausen. In 1854 he devised a far simpler instrument: two arms hinged together, one swinging about a fixed pole, the other carrying the tracer, and a single wheel whose roll gives the area. It needed no rails and no disc, it could be carried in a small case, and it was accurate. He published it in 1856 in the Vierteljahrsschrift of the Zürich natural history society, under the title “On the mechanical determination of the area, static moments and moments of inertia of plane figures”.

Amsler set up a workshop in Schaffhausen in 1854 and soon gave up teaching for instrument making. By May 1856 he had made more than 100 planimeters, and by his death in 1912 about 50,000. The earliest survivor, serial number 5 of 1854, is in the Deutsches Museum. A pole weight appears from about serial 543, and the firm also made integrators that measured moments as well as areas, used for ships and for earthworks on the railways. His son Alfred Amsler (1857–1940) carried the firm on. The instrument was soon copied: one unauthorised Berlin maker, A. Blankenburg, made about 7,500 before the First World War.

Figure 3 — Jakob Amsler (1823–1912), inventor of the polar planimeter.
Figure 3 — Jakob Amsler (1823–1912), inventor of the polar planimeter. — Portrait, public domain, via Wikimedia Commons.
Figure 4 — An Amsler polar planimeter: the classic fixed-arm instrument, with the pole weight at the end of its arm and the wheel and dial on the tracer arm.
Figure 4 — An Amsler polar planimeter: the classic fixed-arm instrument, with the pole weight at the end of its arm and the wheel and dial on the tracer arm. — Photo: Nol Aders, CC BY-SA 3.0, via Wikimedia Commons.
Figure 5 — An Amsler type 4 planimeter with an adjustable arm, sold by Salmoiraghi of Milan.
Figure 5 — An Amsler type 4 planimeter with an adjustable arm, sold by Salmoiraghi of Milan. — Photo: Ancelli, CC BY-SA 3.0, via Wikimedia Commons.

K&E sold “Planimeter, Amsler’s Patent, of German Silver, for computing small areas, in case, with directions” for $33 in its 1876 catalogue.

2.3 Coradi and the compensating planimeter

Gottlieb Coradi (1847–1929) opened a workshop in Zürich in 1880 and began making Amsler-type planimeters. In 1894 he introduced the compensating polar planimeter, based on a design patented by Lang in Switzerland in 1893: the pole arm rides above the tracer arm and the two meet in a ball-and-socket joint, so the instrument can be set up with the joint on either side, and readings taken both ways can be averaged to cancel its errors. By the 1930s the compensating type had all but replaced Amsler’s original design. Coradi also made precision rolling planimeters, the integraph and the Henrici–Coradi harmonic analyser.

Figure 6 — A Coradi compensating polar planimeter of about 1901, signed "G. Coradi Zürich", in its case. K&E's first 4242s were Coradi instruments.
Figure 6 — A Coradi compensating polar planimeter of about 1901, signed "G. Coradi Zürich", in its case. K&E's first 4242s were Coradi instruments. — Photo: Vitaliy VK, CC BY-SA 4.0, via Wikimedia Commons.

2.4 The German makers

  • Gebrüder Haff, of Pfronten in the Allgäu, was founded in 1835 by the brothers Heinrich, Fritz and Karl Haff, sons of a clockmaker. Haff made drawing instruments and planimeters, supplied K&E unsigned from 1867 to 1986, and is still in business, still making planimeters.
  • A. Ott of Kempten, founded by Albert Ott in 1873, was one of the major industrial makers of planimeters and integrators, and made at least one model for K&E.
  • Dennert & Pape of Altona, better known for Aristo slide rules, made planimeters for a time.
  • In Britain, W. F. Stanley of London sold Amsler planimeters and made his own; in the United States, the Los Angeles Scientific Instrument Company (Lasico), founded in 1928, made planimeters when German imports were cut off in the Second World War.

2.5 K&E

Keuffel & Esser, founded in New York in 1867 and later at Hoboken, New Jersey, sold a full range of planimeters for engineers and surveyors: fixed-arm polar planimeters (4236, 4236M), adjustable compensating planimeters (4240, 4242, 4242½), rolling and disc planimeters (4260, 4262), a radial planimeter (4215) for circular recorder charts, and later digital models. K&E published a 32-page manual, Compensating Polar Planimeters (1949, revised 1957), and in 1903 a full book by J. Y. Wheatley, The Polar Planimeter and its Use in Engineering Calculations.

Figure 7 — A polar planimeter as drawn in K&E's catalogue of 1908.
Figure 7 — A polar planimeter as drawn in K&E's catalogue of 1908. — K&E catalogue, 1908; public domain, via Wikimedia Commons.
Figure 8 — K&E's 1915 catalogue: the compensating planimeters 4240 and 4242. The 4242's pole arm "bears index marks for the different settings … so that when the instrument is used with the pole inside of a f…
Figure 8 — K&E's 1915 catalogue: the compensating planimeters 4240 and 4242. The 4242's pole arm "bears index marks for the different settings … so that when the instrument is used with the pole inside of a figure, the constant is a round number, 20,000, for any setting." — K&E catalogue, 1915; scan via Clark McCoy, mccoys-kecatalogs.com.
Figure 9 — The 1930 K&E price list, where the 4242½ first appears among the "new items": "similar to 4242, but with one testing rule and table of settings for the English system only".
Figure 9 — The 1930 K&E price list, where the 4242½ first appears among the "new items": "similar to 4242, but with one testing rule and table of settings for the English system only". — K&E price list, 1930; scan via Clark McCoy, mccoys-kecatalogs.com.
Figure 10 — The 1936 catalogue: the 4242 "Paragon" and the 4242½ "Anvil", "similar to No. 4242, but with one testing rule and table of settings for the English System only, and without provision for finding th…
Figure 10 — The 1936 catalogue: the 4242 "Paragon" and the 4242½ "Anvil", "similar to No. 4242, but with one testing rule and table of settings for the English System only, and without provision for finding the mean height of indicator diagrams". — K&E catalogue, 1936; scan via Clark McCoy, mccoys-kecatalogs.com.
Figure 11 — The 1954 catalogue: the 4242½ reintroduced, "with slightly lower degree of precision, yet adequate to meet Government specifications". My planimeter is probably of this generation.
Figure 11 — The 1954 catalogue: the 4242½ reintroduced, "with slightly lower degree of precision, yet adequate to meet Government specifications". My planimeter is probably of this generation. — K&E catalogue, 1954; scan via Clark McCoy, mccoys-kecatalogs.com.
Figure 12 — The 1962 catalogue, after the renumbering: the 4242 became the 62 0015 and the 4242½ the 62 0020.
Figure 12 — The 1962 catalogue, after the renumbering: the 4242 became the 62 0015 and the 4242½ the 62 0020. — K&E catalogue, 1962; scan via Clark McCoy, mccoys-kecatalogs.com.

The 4242 cost $36 in 1901, $46.50 in 1909, $55 in 1921 and $98 in 1949. No price for the 4242½ has been found.

2.6 What planimeters were used for

  • Steam-engine indicator diagrams. An indicator drew the pressure in a cylinder against the piston’s stroke; the area inside the loop, divided by its length and multiplied by the indicator spring’s scale, gave the mean effective pressure, and from it the engine’s horsepower. Many small planimeters were sized for indicator cards.
  • Surveying and maps: the areas of fields, estates, lakes and catchments, read directly in acres or hectares by setting the tracer arm to the map’s scale.
  • Earthworks and quantities: cross-sections of cuttings and embankments for railways, roads and canals, and the capacities of reservoirs from their contour maps.
  • Recorder charts: flow and temperature charts, with special radial planimeters for circular charts.
  • Science: leaf areas in botany, mineral proportions in rock sections, meteorological records, and in the 1970s the acreages of insect-damaged forest on aerial survey maps.
Figure 13 — A steam-engine indicator of 1904. Its pencil drew the cylinder pressure against the stroke, and the planimeter measured the area of the loop.
Figure 13 — A steam-engine indicator of 1904. Its pencil drew the cylinder pressure against the stroke, and the planimeter measured the area of the loop. — From The New Catechism of the Steam Engine, 1904; public domain, via Wikimedia Commons.

2.7 The end of an instrument

Electronic planimeters, with an encoder in place of the wheel and a digital readout, appeared in the later twentieth century from Tamaya, Koizumi (Placom), Ushikata, Haff and Lasico, and computer drawing and GIS software finally made the planimeter unnecessary. Haff still makes mechanical and digital planimeters today.

Figure 14 — A Tamaya Planix 7 digital planimeter of the late 1990s: a rolling instrument with an electronic readout.
Figure 14 — A Tamaya Planix 7 digital planimeter of the late 1990s: a rolling instrument with an electronic readout. — Photo: Tilt u, public domain, via Wikimedia Commons.

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