Elliptic Integral: Why Is It Called That?

[LagrangeEuler]

Why this integral is called elliptic?
$$I(k)=\int^{\frac{\pi}{2}}_0(1-k^2\sin^2 \varphi)^{-\frac{1}{2}}d \varphi$$

 

[Delta2]

Because it arises when you try to calculate the length of an arc of an ellipse.