"In 1829, the year that Abel died, Carl Gustav Jacob Jacobi published his "Fundamenta nova theoriae functionum ellipticarum." Jacobi based his theory of elliptic functions on four functions defined by infinite series and called theta functions. ...The addition theorems of elliptic functions can also be considered as special applications of Abel's theorem on the sum of integrals of algebraic equations. The question now arose whether hyper-elliptic integrals could be inverted in the way elliptic integrals had been inverted to yield elliptic functions. The solution was found by Jacobi in 1832 when he published his result that the inversion could be performed with functions of more than one variable. Thus the theory of Abelian functions of p variables was born, which became an important branch of Nineteenth century mathematics."