According to a Greek text that was overwritten with Christian prayers, Archimedes worked out some of the principles of calculus over 1900 years before Newton and Leibniz. He called it The Method.
In The Method, Archimedes was working out a way to compute the areas and volumes of objects with curved surfaces, which was also one of the problems that motivated Newton and Leibniz. Ancient mathematicians had long struggled to "square the circle" by calculating its exact area. That problem turned out to be impossible using only a straightedge and compass, the only tools the ancient Greeks allowed themselves. Nevertheless, Archimedes worked out ways of computing the areas of many other curved regions.
Much more is explained in the book The Archimedes Codex. The entire text is available for free on Google Books (you might need this). (via long now)
This question posed to Cecil at The Straight Dope has occupied most of my day today:
Here's the original problem essentially as it was posed to us: "A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?"
I'll give you a few moments to think about that before discussing the answer...
Cecil says that the obvious answer -- that the plane does not take off because it remains stationary relative to the ground and the air -- is wrong. The plane, he says, can take off:
But of course cars and planes don't work the same way. A car's wheels are its means of propulsion--they push the road backwards (relatively speaking), and the car moves forward. In contrast, a plane's wheels aren't motorized; their purpose is to reduce friction during takeoff (and add it, by braking, when landing). What gets a plane moving are its propellers or jet turbines, which shove the air backward and thereby impel the plane forward. What the wheels, conveyor belt, etc, are up to is largely irrelevant. Let me repeat: Once the pilot fires up the engines, the plane moves forward at pretty much the usual speed relative to the ground--and more importantly the air--regardless of how fast the conveyor belt is moving backward. This generates lift on the wings, and the plane takes off. All the conveyor belt does is, as you correctly conclude, make the plane's wheels spin madly.
After reading the question this morning and discussing it with Meg for, oh, about 3 hours on and off, I was convinced that Cecil was wrong. There's no way that plane could take off. The conveyor belt keeps pace with the speed of the plane, which means the plane remains stationary from the POV of an observer on the ground, and therefore cannot lift off.
Then I read Cecil's answer again this evening and I've changed my mind; I'm fairly certain he's right. For a sufficiently long conveyor belt, that plane is taking off. It doesn't matter what the conveyor belt is doing because the airplane's energy is acting on the air, not the belt. I had better luck simplifying the problem like so: imagine instead of a plane, you've got a rocket with wheels sitting on that belt. When that rocket fires, it's eventually going to rocket off the end of that belt...which means that it doesn't remain stationary to the ground and if it had wings, it would fly.
What do you think? Can that plane take off?
See also Feynman's submerged sprinkler problem and an old argument of Newton and Huygens: can you swim faster through water or syrup?
Update: Well, that got out of control in a hurry...almost 300 comments in about 16 hours. I had to delete a bunch of trolling comments and it's not productive to keep going, so I closed it. Thanks for the, er, discussion and remember, the plane takes off. :)