E275  Annotationes in locum quendam Cartesii ad circuli quadraturam spectantem
(Annotations to a certain passage of Descartes for finding the quadrature of the circle)
Summary:
Euler shows that 4p = tan(p/4) + ½ tan(p/8) +
¼ tan(p/16) + 1/8 tan(p /32) + ….
Combining this with the integral of arctan, he gets an expression for p/4 that converges rapidly.
For n = 5, it gives 12 decimal accuracy.
According to C. G. J. Jacobi, a treatise with this title was read to the Berlin Academy on
July 20, 1758.
According to the records, it was presented to the St. Petersburg Academy on
October 15, 1759.
Publication:

Originally published in Novi Commentarii academiae scientiarum Petropolitanae 8, 1763, pp. 157168

Opera Omnia: Series 1, Volume 15, pp. 1  15
 A handwritten French translation of this treatise can be found in the library of the observatory in
Uccle, near Brussels.
Documents Available:
 Original publication: E275
 A translation of E275 into English has been done by Jordan Bell of Carleton University (Ontario, Canada).
 German translation (Artur Diener and Alexander Aycock): E275
Return to the Euler Archive