E52 -- Solutio problematum rectivicationem ellipsis requirentium

(Solution of a problem requiring the rectification of an ellipse)

Summary:

Euler starts with integrals of a certain form, which are really elliptical integrals, and derives second-order ordinary differential equations using the so-called “Modular equation” whose solution can be put back through the given integral. Then several geometric problems are solved, which cause special cases of derived differential equations to appear.

According to the records, it was presented to the St. Petersburg Academy on June 9, 1735.

Euler had solved the problem dealt with in this treatise by the end of 1734. (See his letter to Daniel Bernoulli, Bibl. math. 73, 1906/7, p. 140)

Publication:

Originally published in Commentarii academiae scientiarum Petropolitanae 8, 1741, pp. 86-98
Opera Omnia: Series 1, Volume 20, pp. 8 - 20
Reprinted in Comment. acad. sc. Petrop. 8, ed. nova, Bononiae 1752, pp. 77-90 + 1 diagram [52a]

Documents Available:

Original publication: E052 (in the Commentarii)
German translation (Artur Diener and Alexander Aycock): E052

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