Annotations to a certain passage of Descartes for finding the quadrature of the circle
According to Jacobi, a treatise with this title was read to the Berlin Academy on July 20, 1758. According to St. Petersburg Academy records, it was presented to that Academy on October 15, 1759.
Euler shows that 4Ï = tan(Ï /4) + 1/2∙tan(pi/8) + 1/4∙tan(Ï /16) + 1/8∙tan(Ï /32) + …. Combining this with the integral of arctangent, he gets an expression for Ï /4 that converges rapidly. For n=5, it gives 12 decimal places of accuracy.
Original Source Citation
Novi Commentarii academiae scientiarum Petropolitanae, Volume 8, pp. 157-168.
Opera Omnia Citation
Series 1, Volume 15, pp.1-15.