“In its form, the contents of the memoir [Mémoire sur une propriété générale... (1826)] belongs to the integral calculus. Abelian integrals depend upon an irrational function y which is connected with x by an algebraic equation F((x,y))=0. Abel's theorem asserts that a sum of such integrals can be expressed by a definite number p of similar integrals, where p depends merely on the properties of the equation F((x,y))=0. It was shown later that p is the deficiency of the curve F((x,y))=0. The addition theorems of elliptic integrals are deducible from Abel's theorem. The hyperelliptic integrals introduced by Abel, and proved by him to possess multiple periodicity, are special cases of Abelian integrals whenever p = or > 3. The reduction of Abelian to elliptic integrals has been studied mainly by Jacobi, Hermite, Königsberger, Brioschi, Goursat, E. Picard and O. Bolza...”
“The mathematicians have been very much absorbed with finding the general solution of algebraic equations, and several of them have tried to prove the impossibility of it. However, if I am not mistaken...”
Niels Henrik Abel
“On the whole, I do not like the French as well as the Germans; the French are extremely reserved toward strangers... Everybody works for himself without concern for others. All want to instruct, and n...”
Niels Henrik Abel
“It is readily seen that any theory written by Laplace will be superior to all produced of lower standing. It appears to me that if one wants to make progress in mathematics, one should study the maste...”
Niels Henrik Abel
“Like Jacobi and many other young men who became eminent mathematicians, Abel found the first exercise of his talent in the attempt to solve by algebra the general equation of the fifth degree. ...His ...”
Niels Henrik Abel
“Abel had sent to Gauss his proof of 1824 of the impossibility of solving equations of the fifth degree, to which Gauss never paid any attention. This slight, and a haughtiness of spirit which he assoc...”
Niels Henrik Abel