E74 -- De variis modis circuli quadraturam numeris proxime exprimendi

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


This is an earlier paper, adjacent to E72, in which Euler calculates p several ways, starting with arctan. He notes that arctan(x) + arctan(y) = arctan((x+y)/(1-xy)), and uses that to say that p/4 = 4 arctan(1/5) - arctan(1/70) + arctan(1/99). He gets a more general version that allows him to write p/4 lots of ways, then sets out expanding arctan several ways. Then we get that sin(x)/x= cos(x/2) cos(x/4) cos(x/8) cos(x/16) …

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

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