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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.

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I don't understand French, but it seems that Legendre is the key person who coined these terms. You can use DeepL (www.deepl.com) to translate French. So if one can decipher the definition of le module then l'angle du module will begin to make sense.

Module itself is from Latin so there is no difference in French/German/ or English. Can we make sense if we take module as a "measure". This is how the unabridged Oxford Dict. tells us:

"< classical Latin modulus standard unit of measurement, rhythmic measure, pipe for controlling flow of water < modus mode n. + -ulus -ule suffix. Compare Middle French, French module (1547 in sense 3b, 1611 in Cotgrave in sense 4, 1762 in sense 6), Italian modulo (14th cent. in sense 3b, 1806 in sense 6, 1837 in sense ‘unit of measurement of running water’: compare sense 13)."

From his 1825 book, Traité des fonctions elliptiques et des intégrales Eulériennes, (available from Google books) shows the following:

Book image

"From now on we will call transcendental functions or elliptical functions, the integrals included in this formula. The transcendental H will be assumed to vanish or begin when phi=0; its extent will be determined by the variable phi, which we will call the amplitude; the constant c, always smaller than unity, will be called the modulus; we can always represent c by sin-theta, and theta will be what we call the angle of the modulus."

Translated with www.DeepL.com/Translator (free version)

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