E74 -- De variis modis circuli quadraturam numeris proxime exprimendi

(On various methods for expressing the quadrature of a circle with verging numbers)

Summary:

In this paper, adjacent to E72, Euler studies methods for approximating \(\pi\). He presents various clever ways to decompose \(\tan^{-1} 1 = \pi/4\) as the sum of terms of the form \(\tan^{-1}(1/p)\), then suggests using Leibniz' series to get a good approximation for \(\pi\). He also introduces a way to calculate the number of terms necessary to produce an approximation accurate to a given number of decimals. He closes the paper with the "not inelegant" theorem that \(\frac{\sin x}{x} = \cos(x/2)\cos(x/4)\cos(x/8)\cos(x/16)\ldots\).

According to the records, it was presented to the St. Petersburg Academy on February 20, 1738.

Publication:

Originally published in Commentarii academiae scientiarum Petropolitanae 9, 1744, pp. 222-236
Opera Omnia: Series 1, Volume 14, pp. 245 - 259
Extract published in Arch. der Math. 26, 1856, pp. 350-351 (Grunert) [E74a]
A handwritten French translation of this treatise can be found in the library of the observatory in Uccle, near Brussels.

Documents Available:

Original publication: E074 (in the Commentarii)
Tom Polaski has completed an English translation with accompanying Latin transcription: E74 English, E74 Latin
The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E74 include:
- Tweddle I., “Machin, John and Simson, Robert on inverse-tangent series for p.” Archive for History of Exact Sciences, 42 (1), pp. 1-14 (1991).

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