Book: An Elementary Treatise on Elliptic Functions

Overview
Arthur Cayley presents a compact, systematic introduction to elliptic functions as a natural extension of classical trigonometric theory. The text frames elliptic functions as inverses of elliptic integrals and develops their fundamental properties, periodicity in two directions, algebraic relations, and addition formulas, at a level intended to be accessible to readers familiar with advanced calculus and algebra. Historical threads linking the work of Abel, Jacobi, and Weierstrass are acknowledged indirectly through the choice of themes and the techniques adopted.
The narrative emphasizes concrete manipulations and explicit formulae rather than abstract existence proofs. Cayley favors algebraic derivations that illuminate computational aspects of elliptic functions while maintaining a concise mathematical style. The book serves both as an introduction for students encountering the subject and as a reference for researchers seeking effective calculational tools.

Main Themes and Structure
The core development begins with elliptic integrals and proceeds to their inversion, which yields the central class of doubly periodic functions now recognized as elliptic functions. Cayley systematically treats canonical forms of elliptic integrals, reduction processes, and the passage from integrals to functions with prescribed periods. Attention is paid to the classification of singularities, the role of branch points, and the geometric picture of periods as lattice vectors in the complex plane.
Following foundational material, the text explores addition theorems and algebraic relations among elliptic functions, showing how these mimic and extend familiar trigonometric identities. Transformations of modulus and period, which underpin the modular behavior of elliptic functions, are treated through explicit algebraic substitutions. The exposition closes with applications that illustrate how elliptic functions solve inversion problems for algebraic integrals and connect to the theory of algebraic curves.

Key Concepts
Periods and the lattice structure are presented as the organizing notion that distinguishes elliptic functions from single-period trigonometric functions. Cayley emphasizes how the doubly periodic character leads to a rich algebra of meromorphic functions on the torus, including rational expressions in basic elliptic functions and identities arising from pole and zero configurations. Addition formulas are developed not only as computational tools but as reflections of underlying group-like structures on the curve.
Elliptic integrals receive careful treatment as analytic expressions whose inversion produces the elliptic functions; the classification of integrals according to their algebraic form and branch structure guides the construction of canonical functions. The connection between these analytic constructions and algebraic properties of plane cubic curves appears as an implicit geometric theme, foreshadowing later more explicit algebraic-geometric formulations.

Method and Style
Cayley's method combines classical analysis with algebraic manipulation, favoring symbolic and determinant-based techniques familiar from his work in invariant theory. Arguments are concise and computationally driven, with emphasis on deriving explicit formulae and reducing general cases to standard forms. Proofs are economical, often relying on algebraic identities and clever substitutions rather than lengthy analytic estimates.
The style is formal and brisk, suited to readers who appreciate succinct derivations and are comfortable filling intermediate steps. Pedagogical passages organize material in a logical progression from elementary examples to more involved transformations, enabling a reader to build technique alongside conceptual understanding.

Legacy and Importance
The treatise played a role in making the subject of elliptic functions more approachable to English-speaking mathematicians of the late 19th century by synthesizing analytic and algebraic viewpoints. Cayley's algebraic emphasis helped highlight connections between elliptic functions and invariant theory, and his concrete computations influenced later treatments of elliptic curves and modular relations. Though later developments framed the theory in more abstract language, the book remains valuable historically as a snapshot of the transition from classical analysis to the algebraic and geometric perspectives that define modern theory.