PRESTIGIOUS PLANIMETERS: The A. Ott (Kempten)
Compensating Polar Planimeter impacts the field of mathematics.
Photo courtesy of Wikipedia.org
OOOOH NOOO!!!! I spilled my La Colombe black medium cold brew coffee onto my letter-size (8.5in x 11in) printer paper while reaching for a soft pretzel! I need at least 54.5 square inches to complete my proof for the rectangular case of Green’s Theorem. If only there were a tool that I could use to find (with less than 5% error) the amount of usable paper left.
Powerfully, polar planimeters precisely produce perimeter-based predictions, predicting plot proportions. In other words, one can use a polar planimeter to trace out the perimeter of a region and read an output of the area of said region. Upper School math teacher Keenan Friend comments, “I spent a few hours reading through old polar planimeter papers last night.”
Aside from simply measuring the area of coffee spilt while reaching for a soft pretzel, or measuring the area of the shadow of a soft pretzel itself, planimeters have proven greatly useful in measuring the power output and efficiency of engines. Provided a pressure-volume graph of an engine, commonly presenting an arbitrary closed loop, a planimeter proves useful in finding the area of the enclosed region, which represents the work done by the engine.
But how, one may ask, is it even possible for such a perfect device to exist? One can say it’s truly a miracle of mathematics. The workings of the device boil down to a theorem in Multivariable Calculus known as Green’s Theorem. George Green was a self-taught English mathematician who worked at his father’s windmill in Nottingham. In 1828, despite receiving no formal education and not holding a true educational position, he published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, in which he included what would later become known as Green’s Theorem.
Green’s Theorem states that the circulation density of a vector field integrated over a simply connected region with a closed boundary is equal to the line integral of the vector field over the positively-oriented boundary. If the vector field is one such that the circulation density is equal to one, a much more intuitive case emerges that even you can understand: namely, that it is possible to find the area of a region using only measurements taken of the perimeter. While performing such measurements theoretically is possible with elementary calculus, it would take a great man to invent a great innovation to perform these measurements in the real world.
1854—Schaffhausen, Switzerland. A mathematician by the name Jakob Amsler-Laffon developed his model of the polar planimeter, renowned for its precision. Within the first 60 years of pristine polar planimeter production, profits peaked, with 50,000 units of their six varieties flying off the shelf. Alfred Amsler, the son of Jakob Amsler-Laffon, took over in 1885. Even though the business started to fade after Amsler-Laffon’s death in 1912, the company was able to stay afloat until the 1960s.
But Amsler’s reign could not last forever. In the summer of ’67 (the nineteenth-century one), an American company emerged. Keuffel and Esser (K&E) was founded in New York City and quickly became popular in the U.S. Renowned for its drafting and engineering tools, K&E found special success with its slide rules. However, the company found a new market in reselling imported Amsler planimeters. But this was not enough. Although they had originally sold Amsler planimeters, K&E began to produce planimeters of its own, directly competing with the Swiss company. Suddenly, competition in the highly valuable planimeter market was wide open.
Following K&E’s ascent in the planimeter market by initially reselling Amsler’s fixed arm polar planimeters, it became necessary to innovate and adapt to changing market demands. Needing a producible design that could outperform Amsler and K&E, they looked no further than Gottlieb Coradi’s world-famous work on integraphs and the compensating polar planimeter from 1880 Zürich, Switzerland. Coradi’s design was significantly less sensitive to set-up conditions and could produce more accurate readings with improved arm geometry and hinge mechanics. Without the need for any more consideration, K&E polarly pivoted to producing Coradi’s style-compensating polar planimeters, a move that cemented the company into every history book of American Business.
Now you may be wondering why your Kueffel and Esser polar planimeter clearly reads “K+E,” while the original abbreviation of the company was “K&E.” The change in typography reflects the company’s shift in mindset from a traditional branding to a more mathematical, minimalist, and modern maxim. So next time you spill a La Colombe black medium cold brew coffee onto a letter-size (8.5in x 11in) printer paper while reaching for a soft pretzel and ponder how to properly calculate perimeter-based plot predictions, remember the planimeter.
