Elliptic Functions: An Elementary Text-Book for Students of Mathematics

Elliptic functions occupy a rare place in mathematics: they are at once ancient and startlingly modern, the product of centuries of contemplation about the areas of ellipses and the paths of swinging pendulums, yet today they form the backbone of elliptic curve cryptography and the proof of Fermat's Last Theorem. This elementary text-book, first published in the 1890s, guides students through this elegant branch of mathematics with clarity and historical awareness. Baker begins with the profound discoveries of Euler, who in 1761 achieved the complete integration of an equation involving elliptic functions, introducing a revolutionary method of comparison between functions of different moduli. The reader then follows the work of Landen, whose celebrated theorem revealed that any arc of a hyperbola may be measured by two arcs of an ellipse, and Lagrange's completion of the comparison problem in 1784. The volume culminates with Legendre's monumental synthesis, which organized the entire field into an independent science and produced the numerical tables that made the theory practically useful. This text remains valuable for the student who wants to understand elliptic functions as they emerged historically, before the language of complex analysis became standard. TheLanden transformation serves as a recurring thread, showing how these seemingly disparate results cohere into a unified and beautiful theory.