"His memoirs on elliptic functions, originally published in Crelle's Journal (of which he was one of the founders), treat the subject from the point of view of the theory of equations and algebraic forms, a treatment to which his researches naturally led him. The important and very general result known as Abel's theorem, which was subsequently applied by Riemann to the theory of transcendental functions, was sent to the French Academy in 1826, but was not printed until 1841: its publication then was due to inquiries made by Jacobi, in consequence of a statement on the subject by B. Holmboe in his edition of Abel's works issued in 1839. ...Abel's theorem ...may be described as a theorem for evaluating the sum of a number of integrals which have the same integrand, but different limits—these limits being the roots of an algebraic equation. The theorem gives the sum of the integrals in terms of the constants occurring in this equation and in the integrand. We may regard the inverse of the integral of this integrand as a new transcendental function, and if so the theorem furnishes a property of this function. For instance, if Abel's theorem be applied to the integrand (1 - x^2)^\frac{-1}{2} it gives the addition theorem for the circular (or trigonometrical) functions."