crds@NCoast.ORG (Glenn A. Emelko) (01/07/91)
Oh a big foobar did I make; the inversion of the summation of the squares of
the natural numbners was accomplished by Leanord Euler in 1736, sorry Euler!
Euler also took the same approach as Machin but with better "terms" and came
up with:
Pi = 20 * arctan(1/7) + 8 * arctan(3/79)
then used:
to find arctan(x), let y = x^2/(1+x^2) and solve:
y 2y 2*4y^2 2*4*6y^3
arctan(x) = - * ( 1 + -- + ------ + -------- + ....)
x 3 3*5 3*5*7
and using this computed Pi to 20 decimal places in about 1 hour, bravo Euler!
Carl Friedrich Gauss (sheesh I even had his name wrong) was only at the
periphery of the work done to evaluate Pi, being associated with many of
the people mentioned in both the book as well as the Scientific American
article, including Johann Dase, who was a calculating prodigy, and who worked
for Gauss during the mid 1800's. Hope I have it straight this time!
Anyhow, do enjoy!
Glenn A. Emelko, corrections pointed out and made by same.
PS: Finally I can flame myself first!