Vega's Second Calculation of Pi

In 1794 Vega’s second estimate of was published [1]. Vega used Euler’s formula of the form . The picture from [1] ("results from Thesaurus") shows that he calculated to 143 decimal places and the other values to 144 places. In fact he took the calculation of from his first calculation of π, which was correct to 137 decimals.
To calculate , he summed positive and negative terms separately:
. Here , , , , ….
So Vega's calculation is presented as , where and are the sums of the positive and negative terms of the series for .
Comparing exact Mathematica calculations with [1] shows that Vega calculated the positive part correctly to 139 decimals, the negative part correctly to 142 decimals, and the final estimate of was correct to 136 decimals.

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DETAILS

In 1789, Vega used the formula , and the sum was not calculated again. We included it in this Demonstration for completeness. The denotations for terms of this sum are from [1–5].
References
[1] J. B. Vega, Thesaurus Logarithmorum Completus (logaritmisch-trigonometrischer Tafeln), Leipzig, 1794, p. 633.
[2] J. B. Vega, "Détermination de la démi-circonférence d'un cercle, dont le diamétre est=1," Nova Acta Academiae Scientiarum Imperialis Petrapolitanea for 1790, Vol. 9, 1795 pp. 41-44.
[3] W. W. Rouse Ball and H. S. M. Coxeter, Mathematical Recreations and Essays, 13th ed., New York: Dover Publications, 1987 pp. 356–357.
[5] I. Hafner, "Vega's Calculation of Pi" from the Wolfram Demonstrations Project—A Wolfram Web Resource. (Jul 10, 2013) demonstrations.wolfram.com/VegasCalculationOfPi.
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