Niels Henrik Abel

“Biographies and handwritten manuscripts from the Abel Prize website”

“Abel proposed himself the problem of finding all equations solvable by radicals, and succeeded in solving all equations with commutative groups, now called Abelian equations. Among Abel's other achievements are the discovery of the elliptic functions and their fundamental properties, his famous theorem on the integration of algebraic functions [and] theorems on power series.”

“After the solutions of the third and fourth degrees had been found by Cardano and Ferrari, the problem of solving the equation of the fifth degree had been the object of innumerable futile attempts by mathematicians of the 17th and 18th centuries. Abel's paper shows clearly why these attempts must fail, and opens the road to the modern theory of equations, including group theory and the solution of equations by means of transcendental functions.”

“Niels Henrik Abel... wrote a series of mathematical papers that secures him a position among the greatest mathematicians of all time. In his Mémoire sur les équations algébriques... Abel proves the impossibility of solving general equations of the fifth and higher degrees by means of radicals. The paper was published at Oslo in 1824 at Abel's own expense. In order to save printing costs, he had to give the paper in a very summary form, which in a few places affects the lucidity of his reasoning.”

“He appears to have fully developed in his own mind the subject of the separation of symbols of operation and quantity, not indeed to the extent of founding its results upon an algebraical theory, but to that of giving the theory a wider amount of application. He was a daring generalizer, and sometimes went too far: had he lived, he would have corrected some of his writings. And yet he appears to have been deeply impressed with the notion that a great part of mathematical analysis is rendered unsound by the employment of divergent series.”

“In the obituary published by Crelle, in his "Journal," he states distinctly that the large number of important memoirs which Abel had ready for publication was the immediate reason of the "Journal" being undertaken.”

“He was sent in 1815 to the cathedral school of Christiania, where he did not show any remarkable sign of progress, until 1818 when M. Holmboe, a newly-appointed professor of mathematics, afterwards the writer of Abel's life, and editor of his works, discovered his talent for mathematics, and aided him in pursuing those sciences beyond the elements.”

“In Leibnitz's day... equations of the 2d, 3d, and 4th degrees were reduced to pure equations, but the reduction of equations of higher degrees than the 4th remained an unsolved problem, on which mathematicians spent much labor, until Niels Henrik Abel... a Norwegian mathematician of great ability and acuteness, demonstrated (1824) that the quintic equation and a fortiori the general equation of any order higher than five, is incapable of solution by radicals. Cf. Abel, Démonstration de l'impossibilité de la résolution algébrique des équations générates qui passent le quatriéme degré”

“Rigorous analysis begins with the work of Bolzano, Cauchy, Abel, and Dirichlet and was furthered by Weierstrass.”

“It is most remarkable that two men as different in character and outlook as Abel and Galois should have been interested in the same problem and should have attacked it by similar methods. Both approached the problem of the quintic equation in the conviction that a solution by radicals was possible; Abel at eighteen, Galois at sixteen. In fact, both thought for a while that they had discovered such a solution; both soon realized their error and attacked the problem by new methods.”

“Other mathematicians confess that they have been unable to understand this proof and some have made the correct observation that Ruffini, perhaps by proving too much, had proved nothing in a satisfactory manner. Monsieur Abel has shown by a more penetrating analysis that there can be no algebraic [radical] roots, but he does not deny the possibility of transcendental roots. We recommend this problem to the attention of mathematicians specializing in this field.”

“Abel's theorem was pronounced by Jacobi the greatest discovery of our century on the integral calculus. The aged Legendre, who greatly admired Abel's genius, called it "monumentum aere perennius." During the few years of work allotted to the young Norwegian, he penetrated new fields of research, the development of which has kept mathematicians busy for over half a century.”

“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 extraordinary success in mathematical study led to the offer of a stipend by the government, that he might continue his studies in Germany and France. ...Encouraged by Abel and Steiner, Crelle started his journal in 1826.”

“The name of Abelian function has been given to the higher transcendents of multiple periodicity which were first discussed by Abel. The Abelian functions connected with a curve f(x,y) are if the form \int u\,dx, where u is a rational function of x and y. The theory of Abelian functions has been studied by a very large number of modern writers.”

“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 masters and not the pupils.”

“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 nobody wants to learn. The most absolute egotism reigns everywhere. The only thing the French look for in strangers is the practical; no one can think except himself, he is the only one who can produce anything theoretical. This is the way he thinks and so you can understand it is really difficult to be noticed, particularly for a beginner.”

“On the whole, Divergant series are the work of the Devil and it's a Shame that one dares base any Demonstration on them. You can get whatever result you want when you use them, and they have given rise to so many Disasters and so many Paradoxes. Can anything more horrible be conceived than to have the following oozing out of you:0 = 1 - 2n + 3n - 4n + etc.where n is a whole Number? Risum teneatis amici. [Laughter retains friends]”

“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, they have not as yet succeeded. I therefore dare hope that the mathematicians will receive this memoir with good will, for its purpose is to fill this gap in the theory of algebraic equations.”