How did Ramanujan empirically obtain these errors?

Question

In one of Srinivasa Ramanujan's writings, he discusses the perimeter of an ellipse, $p$. He finds two approximations (page 39):

16. The following approximations for $p$ were obtained empirically: $$p=\pi\left[3(a+b)-\sqrt{(a+3b)(3a+b)}+\epsilon\right]\tag{49}$$ where $\epsilon$ is about $ak^{12}/1048576$; $$p=\pi\left\{(a+b)+\frac{3(a-b)^2}{10(a+b)+\sqrt{a^2+14ab^+b^2}}+\epsilon\right\}\tag{50}$$ where $\epsilon$ is about $3ak^{20}/68719476736$.

Here, $a$ and $b$ are the semi-major and semi-minor axes, and $k$ is the eccentricity.

How did Ramanujan "empirically" find these error values? Did he use geometric measurements, numerical analysis, or something completely different?

Answer 1

According to Villarino's paper Ramanujan's Perimeter of an Ellipse "Ramanujan never explained his “empirical” method of obtaining this approximation, nor ever subsequently returned to this approximation, neither in his published work, nor in his Notebooks. Indeed, although the Notebooks does contain the above approximation (see Entry 3 of Chapter XVIII) the statement there does not even mention his asymptotic error estimate..."

Almkvist and Berndt in Gauss, Landen, Ramanujan, the Arithmetic-eometric Mean, Ellipses, $\pi$, and the Ladies Diary (1988) speculated that Ramanujan developed a continued fraction expansion for the perimeter based on the infinite series found by Ivory in 1796. That produces his error estimates (p.601).

As for proofs, already back in 1933 Watson claimed "I have proved that in these formulae the errors (which are zero in the limiting case of the circle) are in defect and that they increase when the eccentricity is increased". But alas like Fermat, and Ramanujan, he withheld the method. Villarino gave a proof of a strengthened version of Watson's claim in 1978.