E189  De serierum determinatione seu nova methodus inveniendi terminos generales serierum
(On the determination of series, or a new method for finding the general terms of series)
Summary:
This paper attempts to determine f(x), given f(1), f(2), f(3), etc. Euler starts with an example: f(n) = n and tries f(x) = x + S_{n=1}^{∞}b_{n} sin(npx). Note that f(x)  x must be periodic and that f(x+1) = f(x). Then he gets f(x+1) by Taylor series at f(x) and gets a differential equation of infinite order. A few paragraphs later, we have a perfect Fourier series.
According to the records, it was presented to the St. Petersburg
Academy on September 21, 1750.
Publication:

Originally published in Novi Commentarii academiae scientiarum Petropolitanae 3, 1753, pp. 3685

Opera Omnia: Series 1, Volume 14, pp. 463  515
Documents Available:
 Original Publication:
E189,
from the Göttinger DigitalisierungsZentrum, at GeorgAugustUniversität Göttingen
 Alexander Aycock has made draft translations into German and English, available online at EulerKreis Mainz.
 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 E189 include:
 Ferraro G., “Some aspects of Euler's theory of series: Inexplicable functions and the EulerMaclaurin summation formula.” Historia Mathematica, 25 (3), pp. 290317 (Aug 1998).
 Heideman MT, Johnson DH, Burrus CS., “Gauss and the history of the fast Fouriertransform.” Archive for History of Exact Sciences, 34 (3), pp. 265277 (1985).
 Wilson C., “The great inequality of Jupiter and Saturn  from Kepler to Laplace.” Archive for History of Exact Sciences, 33 (13), pp. 15290 (1985).
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