The Wikipedia ellipse article borrows (with the Goldwynism "For computational purposes a much faster series where the denominators vanish at a rate ...") a series from http://www.iamned.com/math/ which offers "A Rapidly converging formula for the circumference of an elipse that gives log (1024/27((a+b)/(a-b))^8) digits per term : ", a formula satisfyingly symmetrical in a and b, but which unfortunately vanishes (instead of 4a) for b=0, and generally fails for b<a. Does anybody know what this should be? The Wikipedia article actually corrects an obvious asymmetry in the iamned.com source, but the Wikigrapher must not have tested it. --rwg Here is what I typed: 8*\[Pi]/Q^(5/4)* Sum[Pochhammer[1/12, n]*Pochhammer[5/12, n]*(v1 + v2*n)* r^n/n!^2, {n, 0, \[Infinity]}] /. r -> 432*(a^2 - b^2)^2*(a - b)^6*b*a/Q^3 /. Q -> b^4 + 60*a*b^3 + 134*a^2*b^2 + 60*a^3*b + a^4 /. v1 -> a*b*(15*b^4 + 68*a*b^3 + 90*a^2*b^2 + 68*a^3*b + 15*a^4) /. v2 -> -a^6 - b^6 + 126*(a*b^5 + b*a^5) + 1041*(a^2*b^4 + a^4*b^2) + 1764*a^3*b^3 A correct formula for a>=b is elliplen[a_, b_] := 4*a*EllipticE[Sqrt[1 - b^2/a^2]]