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Perplexed comments on Generalizing From One Example - Less Wrong
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Check out the Feynman lecture #22 - the one in which he starts with the laws of algebra and ends up with de Moivre's theorem. With a calculation of pi/2 = 1.5709 along the way. Prettiest thing I've ever seen.
Incidentally, Feynman did it the hard way, since he didn't have computers. You can compute pi on a spreadsheet simply by simulating a harmonic oscillator.
Before anyone else complains: yes, there were computers in 1961, and had been for over twelve years, but Feynman doesn't use any in the lecture. And certainly Henry Briggs, who calculated the first fourteen-place common log tables and whom Feynman cites in the relevant section, didn't use any in 1620, and the results Feynman presents are far less precise.
And Lecture #22 - "Algebra" - is a thing of beauty. Anyone who likes mathematics will like it.