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Wikipedia attributes Leibniz's formula to Madhava of Sangamagrama, James Gregory and Gottfried Leibniz. But what were their proofs?

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sorry about the format –  Alex Nov 12 '12 at 3:15
    
@ Alex - I don't see a proof in the article on Madhava of Sangamagrama. Thanks for the other reference. I wish we had a reference that tied Gregory's proof down to a specific historical source. –  Jonah Sinick Nov 12 '12 at 3:23
    
en.wikipedia.org/wiki/… –  Alex Nov 12 '12 at 3:26

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A good reference for this question is Ranjan Roy: The Discovery of the Series Formula for $\pi$ by Leibniz, Gregory and Nilakantha. Mathematics Magazine, Vol. 63, No. 5, December 1990, pp. 291-306. The article is reprinted in Berggren et al.: Pi: A Source Book.

Gregory did not publish his results on series, but he corresponded with John Collins in London from 1669 until his (Gregory's) death in 1675, and the arctan-formula is contained there. The correspondance was published by Turnbull in 1939.

In the title of his article, Ray has chosen to name Nilakantha rather than Madhava who lived a century earlier. The reason is that we do not have the original text of Madhava; while he is credited by Nilakantha, it is not clear if this also includes the arctan-formula.

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Thanks Per Manne! –  Jonah Sinick Nov 14 '12 at 5:01
    
The article is available online here. –  ShreevatsaR Feb 26 at 2:10

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