# Etymology of certain terms in the theory of elliptic integrals

In the theory of elliptic integrals, one encounters the terms "amplitude" and "modular angle" in relation to incomplete integrals of the first kind, which are two variables that denote the upper limit of the elliptic integral (the amplitude) and a certain parameter of the integral (the modular angle). I guess the naming of the term "amplitude" is remniscent of the origins elliptic integrals have in physical problems- in particular, the problem of arbitrary amplitude pendulums (not under the restriction of small angles). According to this interpretation, the amplitude of an incomplete elliptic integral of the first kind corresponds directly to the amplitude of the corresponding pendulum oscillation.

The problem is that this interpretation doesn't fit the formulas; according to several sources, the amplitude of the elliptic integral that calculates the period of a pendulum is $$\pi/2$$ regardless of the pendulum's amplitude.

In addition, i've no guess about the naming of the term "modular angle", so i'd also like to know the etymology of that.