Euler's Estimate of Pi

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

In [1] Euler derived the formula . He claimed that his formula was better for calculation than the Leibniz–Gregory formula , since for , the factor in the series has values , which are simpler to calculate with. He illustrated this with the formula . He calculated eight terms of the sum for each of the arc tangents on the right to 27 decimal places each and concluded that . On the next page he calculated terms 9–16 of the first part and terms 9–10 of the second part and concluded that . To eighteen places, . To 30 places, .

Contributed by: Izidor Hafner (June 2013)
Open content licensed under CC BY-NC-SA


Snapshots


Details

See [2] for a history of Euler's attempts to calculate an approximation to . The calculations from [1, p. 141] are added.

References

[1] L. Euler, "Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae," Nova Acta Academiae Scientarum Imperialis Petropolitinae 11, 1798, pp. 133-149. www.math.dartmouth.edu/~euler/tour/tour_08.html.

[2] E. Sandifer. "How Euler Did It: Estimating ." MAA Online. Feb 2009. (Jun 20, 2013).



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send