On various methods for expressing the quadrature of a circle with verging numbers
In this paper, adjacent to E72, Euler studies methods for approximating π. He presents various clever ways to decompose arctan(1) = π/4 as the sum of terms of the form arctan(1/n), then suggests using the Leibniz series to get a good approximation for π. He also introduces a way to calculate the number of terms necessary to produce an approximation accurate to a given number of decimals. He closes the paper with the "not inelegant" theorem that sin(x)/x = cos(x/2)∙cos(x/4)∙cos(x/8)∙cos(x/16)....
Original Source Citation
Commentarii academiae scientiarum Petropolitanae, Volume 9, pp. 222-236.
Opera Omnia Citation
Series 1, Volume 14, pp.245-259.