# 3D Collection – Measuring Areas using the Orthogonal Planimeter

«Probably never before have so many surface areas of flat figures needed to be calculated, while we are occupied with the most detailed land surveys and the most significant road constructions.»

Carl Maximilian von Bauernfeind: Die Planimeter von Ernst, Wetli und Hansen, welche den Flächeninhalt ebener Figuren durch das Umfahren des Umfangs angeben, München 1853, pg. 1.

Words of Karl Maximilian Bauernfeind, German geodesist and bridge engineer, written in 1853. The 19th century, the age of surveying, saw a great need to simplify surveying and replace it with a mechanical method. The devices invented for this purpose are called planimeters. The first planimeter was invented by Swiss engineer Johannes Oppikofer in 1827. Because it was incomplete, Oppikofer never made it known to the public. It was only after a revision by Heinrich Rudolf Ernst that the device became famous and was awarded a prize by the Paris Academy of Sciences in 1836.1 Based on the principle of a rotating cone, this planimeter could mechanically indicate the area of a surface simply by travelling around its circumference. This was all the more astonishing because “the area of a figure is known not to have any mathematical relationship whatsoever to the size of the circumference”2.

A new planimeter structure by Kaspar Wetli

The orthogonal planimeter, which is now available as a 3D model, was developed by Kaspar Wetli (1822-1889). It is an enhancement of Ernst’s planimeter. However, in his orthogonal planimeter, Wetli replaced the rotating cone with a flat disc. This is ultimately a special case of a cone with an opening angle of 180°.

This enabled the measuring wheel to be moved beyond the zero point and thus the counter to be turned in the opposite direction, which corresponds to the integration of negative values.3 Thus, in the words of Simon Stampfer, who was professor at the Imperial and Royal Polytechnic Institute in Vienna at the time, Wetli’s planimeter was “way ahead” of all previous planimeters.4  In addition: The instrument “always calculates the area of any rectilinear or curvilinear figure using the same extremely simple operation, namely outlining its circumference using a pointed object […]”5. Stampfer’s 1850 writing on Wetli’s planimeter was published in four journals and led not only to the spread of knowledge on mechanical integration, but also to great renown for Kaspar Wetli.6

Wetli was considered one of the most significant engineers and surveyors of his time. Wetli was cantonal engineer of Zurich at the time when he invented the orthogonal planimeter. In this role, along with Professor Johannes Wild and Heinrich Pestalozzi, Wetli was jointly responsible for the new map of the canton of Zurich. The new planimeter enabled areas to be measured with a margin of error of less than one per mille. This high precision earned the instrument a prize at the first World’s Fair in London in 1851. Upon the invention of the planimeter, Wetli became widely known and was promptly inundated with orders. His obituary stated:

«The very invention of his planimeter was bound to afford him a distinguished position among his contemporaries, and ensures that he will continue to be remembered in the future.»

Obituary of Wetli, Kaspar, in: Schweizerische Bauzeitung [Swiss Construction Journal], 13/14 (1889), no. 15, pg. 89.

Goldschmid, Starke and Ausfeld – the three manufacturers of the Planimeter

Wetli commissioned the first orthogonal planimeters made by Zurich mechanic Jakob Goldschmid (1815-1876) starting from 1849. This included the instrument that was exhibited at the World’s Fair and became part of the collection of scientific instruments and teaching aids. Goldschmid completed an apprenticeship with well-known Zurich mechanic Johann Georg Oeri and, after a period of travelling, returned to his workshop.7 By marrying Oeri’sdaughter, Goldschmid became co-owner of his workshop and then took it over completely upon Oeri’s death in 1852. Like Oeri before him, Goldschmid became a sought-after mechanic and instrument maker. He became known mainly for his aneroid barometers and free-floating pantographs.

Almost at the same time, Wetli also commissioned the production of his planimeter in the workshop of Georg Christoph Starke in Vienna, where the instrument was built until about 1870.8 The Wetli-Starke system was subsequently patented. Wetli’s planimeter was manufactured at another workshop, Ausfeld in Gotha / Thuringia, from 1850/51 onwards. Very little is known about these two workshops.

The instruments were expensive, but they were in demand and according to Wetli, about 150 planimeters were produced in total.9

The big turning point came shortly thereafter: Inspired by the functioning of Wetli’s planimeter, various mechanics and scientists took it upon themselves to find simpler mechanical constructions. Thus, the much simpler polar planimeter was invented by Swiss mathematician and physicist Jakob Amsler-Laffon  in 1854; and sold commercially starting from 1856. Independently of Amsler, Austrian mining engineer Albert Miller Ritter von Hauenfels  and Pavel Alexseevich Zarubin from Russia also came up with the same design in 1855. Swiss companies Amsler, G. Coradi and A. Ott, located in Schaffhausen, Zurich and Kempten respectively, later became important manufacturers.

# 3D Digitalisation

Object selection
Around 150 Wetli planimeters were produced in total, only a few of which were made by Zurich mechanic Goldschmid: the orthogonal planimeter in our collection is thus an extremely rare instrument, if not the only one still in existence. We have not come across any other collection in possession of a Wetli-Goldschmid planimeter to date. Any information regarding other surviving Wetli planimeters made by Goldschmid would be gratefully received: scientific-instruments@library.ethz.ch.
The 3D digitalisation serves, on the one hand, to (digitally) preserve and document this rare instrument. In addition, researchers in the field of science history can examine the object from any location. In the field of education, the animation provides an understanding of how the mechanical surface area calculation works; without the need to touch the object.

Photogrammetry and modelling
The glass panes and all animated structures were created manually from the digital copy. In contrast, the rest of the instrument was easy to digitise.

1. Bauernfeind, Carl Maximilian von: Die Planimeter von Ernst, Wetli und Hansen, welche den Flächeninhalt ebener Figuren durch das Umfahren des Umfangs angeben. München: Joh. Palm's Hofbuchhandlung, 1853. ETH-Bibliothek Zürich, Rar 15756, Vorwort S.III-IV, https://doi.org/10.3931/e-rara-63964 / Public Domain Mark.
2. Bauernfeind (1853), pg. 6.
4. Stampfer, Simon: Ueber das neue Planimeter des Caspar Wetli, S. 2, https://doi.org/10.3931/e-rara-63966
5. ibid.
6. Gessner, Samuel; Hashagen, Ulf; Peiffer, Jeanne; Tournès, Dominique; Mathematisches Forschungsinstitut Oberwolfach (Hg.): Report 58/2017: Mathematical Instruments between Material Artifacts and Ideal Machines: Their Scientific and Social Role before 1950, 2017, pg. 35.
7. For example, the dioptric prismatic compass, which is also available as a 3D digital copy, comes from Oeri's workshop, compare: https://etheritage.ulapiluh.myhostpoint.ch/2023/05/31/vermessungskompass/
9. Gessner (2017), pg. 35.