E74 -- De variis modis circuli quadraturam numeris proxime exprimendi
(On various methods for expressing the quadrature of a circle with verging numbers)
This is an earlier paper, adjacent to E72,
in which Euler calculates p several ways, starting with arctan. He notes that arctan(x) +
arctan(y) = arctan((x+y)/(1-xy)), and uses that to say that p/4 =
4 arctan(1/5) - arctan(1/70) + arctan(1/99). He gets a more general version that allows him to write
p/4 lots of ways, then sets out expanding arctan several ways. Then we get that sin(x)/x=
cos(x/2) cos(x/4) cos(x/8) cos(x/16) …
According to the records, it was presented to the St. Petersburg Academy on February 20, 1738.
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.
- Original publication: E074
(in the Commentarii)
- 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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