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Vega's Calculation of Pi

In 1776, Hutton derived a formula for . In 1789, Vega used that formula as , to calculate . Vega did not know Machin's formula from 1706, which is better than Hutton's. Vega calculated arctan combining each positive term with the negative term following it. In this way he got two series:
,
where , , , , …,
where , , , , ….
For each series, Vega first calculated the coefficients , , , … and , , , …, then the terms of the series, and finally he summed the series. Then he added the two and multiplied by 8.
The smaller and larger parts were correct to 137 and 127 decimal places, respectively, so his calculation of was correct up to 126 decimals. Had Vega made all the calculations correctly, he would have calculated to 140 decimals.

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Vega sent his paper together with the calculation to Nova Acta in 1789.
References
[1] G. Vega, "Détermination de la circonférence d'un cercle," Nova Acta Academiae Petropolitanae, IX, 1795, p. 41.
[2] W. W. Rouse Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover Publications, 1987 pp. 356–357.
[3] The MacTutor History of Mathematics Archive. "Georg Freiherr von Vega." (Jul 2, 2013) www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Vega.html.
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