Abel’s Approach to Elliptic Integrals

The theory of elliptic integrals and functions was a major research topic during the nineteenth century. Great mathematicians such as Euler, Lagrange, and Legendre made important contributions in this field. The Norwegian mathematician Niels Henrik Abel revolutionized this theory as he introduced novel ideas and approaches. Specifically, he proved what is called Abel’s addition theorem which is a sweeping extension of previous addition theorems for elliptic integrals obtained in the eighteenth century. This chapter provides an elementary review of addition theorems for elliptic integrals before 1830 with special focus on Abel’s addition theorem. An important aim of the chapter is to bring out the intuition behind the various methodological approaches. The chapter also contains a short review of Abel’s life with special reference to his contribution to the theory of elliptic integrals and beyond.