Reconstruction of Vega's First Calculation of Pi

In 1789, Vega sent his first calculation of to the Petersburg Academy. He used the formula , where is the smaller part and is the larger part. Vega calculated arctan by combining each positive term with the negative term following it. In this way, he got two series:
,
where , , , …;
and ,
where , , , ….
For each series, Vega first calculated the coefficients , , , … and , , , …, then the terms of the series, and finally he summed the series. The smaller and larger parts were correct to 137 and 127 decimal places, respectively, so his calculation was correct up to 126 decimals. Had Vega done all the calculations correctly, he would have calculated to 140 decimals.
This Demonstration locates Vega's mistakes. The coefficients were all correct, but the terms were not. The crucial point was the calculation of , which was correct only to 127 decimals.

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References
[1] G. Vega, "Détermination de la circonférence d'un cercle," Nova Acta Academiae Scientiarum Imperialis Petropolitanae, IX, 1795 p. 41.
[2] W. W. R. Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover, 1987 pp. 356–357.
[3] The MacTutor History of Mathematics Archive. "Georg Freiherr von Vega." (Jan, 2012) www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Vega.html.
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