Kepler: so, you see, the orbit of a planet is elliptical. To find where the Earth is, we need a method to calculate the arc length of an ellipse.Newton: This is a fluent problem, and, as usual, it can be solved with an infinite series expansion.Leibniz: Hold my beer, what's a fluent? The arc length of an ellipse is an integral problem. You want to compute $\int f(x)dx$ where $f(x) = \sqrt{\frac{1 - k^2x^2}{1 - x^2}}$ for a certain constant $k$. For that, you have to find a closed form function such that $F'(x) = f(x)$.100 years had passed. The search for Leibniz's closed-form solution for the elliptic integral, that is $f(x) = \int_{c}^{x} R \left(t, \sqrt{P(t)} \right) dt$ where $R$ is a rational function and $P$ is a polynomial of degree 3 or 4, had been fruitless.By the end of the 17th century, Bernoulli -- which one is left as an exercise -- conjectured that the task is impossible. It was finally confirmed by Liouville in the 19th century who proved that elliptic integ…