E28  Specimen de constructione aequationum differentialium sine indeterminatarum separatione
(Example of the construction of equations)
Summary:
In this paper, Euler investigates a differential equation that he encountered in finding the arc length of an ellipse. This differential equation cannot be solved by separation of variables, as is indicated in the title of the article. Euler first develops a formula for the arc length of an ellipse by cleverly manipulating a binomial series, then shows that this formula satisfies the desired differential equation. Integrating factors make a brief appearance.
According to Eneström, this paper was probably presented to the St. Petersburg Academy on January 9, 1733.
Publication:

Originally published in Commentarii academiae scientiarum Petropolitanae 6, 1738, pp. 168174

Opera Omnia: Series 1, Volume 20, pp. 1  7
 Reprinted in Comment. acad. sc. Petrop. 6, ed. nova, Bononiae 1743, pp. 156162 + 1 diagram
[E28a]
Documents Available:
 Original publication: E028 (in the Commentarii)
 Latin transcription and English translation by Tom Polaski: E28 (English) E28 (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 E28 include:
 Deakin Mab., “Euler invention of integraltransforms.” Archive for History of Exact Sciences, 33 (4), pp. 307319 (1985).
 Maltese G., “On the relativity of motion in Leonhard Euler's science.” Archive for History of Exact Sciences, 54 (4), pp. 319348 (2000).
 Sandifer E., "Arc length of an ellipse." How Euler Did It. (Published online
by the MAA.)
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