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Top: Animation of Pythagoras' proof that
for any right triangle, the sum of the squares
of the 2 short sides equals
the square of the long side.
When Pythagoras proved this
around 550 BC, he sacrificed a herd
of perfect white oxen to Apollo.
Professor Foote makes a specialty/hobby of one of the world's niftiest gizmos, the polar planimeter, and tells you all you could ever hope to know about how the polar planimeter works. (I am convinced it works on Magickal Principles.)
But here's Larry Leinweber's Java Applet polar planimeter that you can play with for hours. Try everything in all the drop-down menus. I mean you can't break it or break your computer no matter what forbidden or impossible thing you try to make it do. (You can break a real polar planimeter, though. But please don't.)
Any schmuck can figure out the Area of a square or a rectangle or a perfect circle or a triangle or a rhombus or a quadrilateral or a regular dodecagon. (See Fig. 1.)
But if you have a map or a satellite image of the USA state of Maine, how do you figure out the surface area of the really squiggly and irregular-shaped Sebago Lake?
Well, you just reach for your Polar Planimeter, that's how. You put the tracer arm anywhere on the shoreline, and then trace the entire shoreline until you come back to where you started, and Holy Shit!!! There's the fucking Area of Sebago Lake on little number wheels (like your car's odometer)!!! No batteries! No software! No math even! You just trace the outline of any irregularly (or regularly) shaped Area, and wham! There's your Area in hectares or square kilometers or acres or whatever the hell you use for surface area.
Goddam it, I just want one. How much do they cost?
Larry Leinweber has this site, Larry's Cerebral Snack Bar / Games and Puzzles to Feed your Head, and it has lots of screwy things.
Larry's giving away his Java Polar Planimeter for free, I think. Anybody who makes a virtual Polar Planimeter and gives it away for free is a Spiritually Advanced Sort of Person. He makes Earth a Better Planet.