Wikipedia attributes Leibniz's formula to Madhava of Sangamagrama, James Gregory and Gottfried Leibniz. But what were their proofs?
Early proofs of Leibniz's formula
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0http://en.wikipedia.org/wiki/Madhava_of_Sangamagrama#The_value_of_.CF.80_.28pi.29 – 2012-11-12
1 Answers
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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0$T$he article is available online [here](http://www.maa.org/programs/maa-awards/writing-awards/carl-b-allendoerfer-awards/the-discovery-of-the-series-formula-for-pi). – 2014-02-26