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I always had this question, but was unable to get a definitive answer to it.

There is the theorem of division of the arc length of the lemniscate with ruler and compass. So I always wondered, is it possible to reconstruct the theory of elliptic functions from purely geometric considerations? If this is possible, then to what extent?

Geometric means geometric constructions to prove main formulas but without use of functions of complex variables.

Note that I don't imply that this geometric theory would be practical. From modern standpoint, this construction probably will be considered as a waste of time. I'm interested in the theoretical possibility of such a development. Also I'm interested in understanding the intuition behind such a development, why it is possible?

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    $\begingroup$ The Jacobi elliptic functions $sn$., $cn$, and $dn$ are constructed geometrically from an ellipse of excentricity $k$ in exactly the same ways as $\sin$ and $\cos$ from a circle. $\endgroup$
    – Henri Cohen
    Commented Sep 16, 2021 at 15:12

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One possible source is the book

  • Lawden, D. F. Elliptic functions and applications, New York, NY etc.: Springer-Verlag, 1989, doi:10.1007/978-1-4757-3980-0,

(esp. Ch. 4 Geometrical Applications). In partiqular it contains the proof of addition theorem for Jacobi elliptic functions based on the formulae of spherical trigonometry. Probably many such gems are collected in old books. In partiqular this (Legendre's) proof is presented in the book

  • Cayley, A. An elementary treatise on elliptic functions 2nd ed. Dover Publications, Inc., New York, 1961 (p. 27) (Internet Archive version).

One more collection is the dissertation

Also there is a special book

  • Dragović, V. & Radnović, M. Poncelet porisms and beyond. Integrable billiards, hyperelliptic Jacobians and pencils of quadrics. Basel: Birkhäuser, 2011, doi:10.1007/978-3-0348-0015-0

about geometry, billiards and (hyper)elliptic curves.

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Some elementary parts of the theory of elliptic functions can indeed be developed in this way. To those books listed by Alexey Ustinov I can add a large treatise by G. Halphen, Traité des fonctions elliptiques et de leurs applications (JFM 22.0447.01), (BnF) in 3 volumes, published in 1886-1890.

He never misses a opportunity to give a geometric proof of some theorem, when available.

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