In the introduction of Abel (1828), an addition theorem is stated; an informal paraphrase of this theorem is given by Kleiman (2004, p. 308): “the integrals of arbitrary algebraic differentials form a very large class of transcendental functions with an addition formula; if an arbitrary sum of functions arising from the same differential cannot be given by a single function of the same sort, as is the case for elliptic functions, at least the sum can be given by a sum of a specific number of them, plus certain algebraic and transcendental terms”.