# How did Ramanujan empirically obtain these errors?

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?

• Im going out on a limb here but I guess it's a remainder of a known calculation. – user3298 Nov 29 '15 at 1:46
• I think you'll get very good explanations if you post this to MSE. – Kushal Bhuyan Nov 29 '15 at 2:03

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).