From Mathologer.
Today’s mission is to reveal a surprisingly simple trick that lets you use any old ax(e) to calculate the area of plane shapes. More broadly, we’ll explore planimeters — mechanical instruments that compute area by tracing a shape’s edge. Real mathematical magic in action!
00:00 Intro (a mathematical fairy tale)
07:16 Cool machines (history and types of planimeters)
09:05 Green’s theorem (the most common high-level calculus explanation)
13:05 Easy peasy (the easiest and at the same time pretty rigorous explanation)
20:30 Full sweep (the best but somewhat handwaivy explanation)
26:50 AX (making sense of the ax planimeter)
31:45 Cute stuff
33:26 Thank you!
Things to check out:
Probably the best exposition of the handwaivy part of this video.
Robert L. Foote & Ed Sandifer, Area Without Integration: Make Your Own Planimeter, https://persweb.wabash.edu/facstaff/footer/papers/foote.sandifer.reprint.pdf
In general, check out Robert L. Foote’s website, lots of planimeter and other gold 🙂
Robert L. Foote, How planimeters work, https://tinyurl.com/yc53vzan
https://persweb.wabash.edu/facstaff/footer/HOME.HTM
Richard Courant textbook
Differential and Integral Calculus (Volume II)-Blackie & Son, 1936
Nice exposition of why area = arm x roll starting with tracing a rectangle and a linear planimeter.
Paul Kunkel, The planimeter, https://web.archive.org/web/20021201115325/http://www.nas.com/~kunkel/planimeter/planimeter.htm
One of my favourite serious maths popularisations. My video follows the chapter on planimeters quite closely.
John Bryant and Christopher Sangwin, How round is your circle, Princeton Uni Press, 2011.
Moment planimeters, a really really nice twist to the planimeter story.
Christopher Sangwin, Locating the centre of mass by mechanical means
https://tinyurl.com/mr2dje5
That’s the nice survey of maths machines that I mentioned in this video.
Fr. A Willers, Mathematische Maschinen und Instrumente, Akademie Verlag, Berlin, 1951.
Various takes on the Green’s theorem explanation of planimeter math(s).
Tanya Leise, As the Planimeter’s Wheel Turns: Planimeter Proofs for Calculus Class, https://tleise.people.amherst.edu/HomePage/LeisePlanimeter.pdf
Oliver Knill, The Planimeter and the Theorem of Green
https://people.math.harvard.edu/~knill/teaching/math21a2000/planimeter/index.html
Bill Casselman and John Eggers, The Mathematics of Surveying: Part II. The Planimeter
https://www.ams.org/publicoutreach/feature-column/fcarc-surveying-two
Stan Wagon, Mathematica in action, Springer, 2010
Stan Wagon, Polar planimeter Wolfram demonstration project
Some animations in this video were produced using a customised version of the code underlying this demonstration project.
https://demonstrations.wolfram.com/ThePolarPlanimeter/
Nitty-gritty analysis of the hatchet planimeter.
Theory of the hatchet planimeter, Simon Farthing
https://www.ias.ac.in/public/Volumes/sadh/008/04/0351-0359.pdf
Bicycle tyre tracks, a different angle from which to look at the hatchet planimeter.
Robert Foote, Mark Levi, and Serge Tabachnikov, Tractrices, Bicycle Tire Tracks, Hatchet Planimeters, and a 100-year-old Conjecture
An old newpaper article by Marty and me about hatchet planimeters.
Marty Ross and Burkard Polster, Hatcheting a plan. The AGE, https://www.qedcat.com/archive_cleaned/161.html
Calculus without calculus
https://en.wikipedia.org/wiki/Visual_calculus
The book I mentioned at the end.
Tom Apostol and Mamikon Mnatsakanian. New Horizons in Geometry. Mathematical Association of America, 2012
The Haff company’s website
The manual for using my analog polar planimeter.
https://www.haff.de/wp-content/uploads/2018/05/manual317_315_313.pdf
An old planimeter applet (you need to figure out a way to run it, I use CheerpJ Applet Runner. Use at your own risk! 🙂
https://web.archive.org/web/20161011062058/http://www.leinweb.com/snackbar/planimtr/planimtr.htm
T-shirt: Label your axes https://www.teepublic.com/t-shirt/12403182-label-your-axes-funny-math
Soundtrack: Hourglass by Yehezkel Raz and The Prophets Song by Ian Post
Enjoy!
Burkard
