E74 -- De variis modis circuli quadraturam numeris proxime exprimendi

(On various methods for expressing the quadrature of a circle with verging numbers)


Summary:

In this paper, adjacent to E72, Euler studies methods for approximating \(\pi\). He presents various clever ways to decompose \(\tan^{-1} 1 = \pi/4\) as the sum of terms of the form \(tan^{-1}(1/p)\), then suggests using Leibniz' series to get a good approximation for \(\pi\). He also introduces a way to calculate the number of terms necessary to produce an approximation accurate to a given number of decimal places. He closes the paper with the "not inelegant" theorem that \(\frac{\sin x}{x} = \cos(x/2) \cos(x/4) \cos(x/8) \cos(x/16) \ldots\).

According to the records, it was presented to the St. Petersburg Academy on February 20, 1738.

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