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Uploaded on Nov 8, 2007
We show how the basic notion of rational trigonometry---quadrance---arises from the geometry of the ancient Greeks. The little-known sister theorem to Pythagoras features prominently, and is closely related to a theorem of Archimedes.
For someone brought up on classical trigonometry, quadrance is the square of the distance. But future generations will come around to the Euclidean realization that it is quadrance which is actually the fundamental quantity: distance is the square root of quadrance--assuming (and it is a BIG assumption) that you have a viable theory of square roots around.
Pythagoras' theorem is matched up with the Triple quad formula (TQF) ---in my opinion these are the two most important theorems in mathematics. The reach of the TQF is long indeed.