9464

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.

SNAPSHOTS

  • [Snapshot]

DETAILS

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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+