One hundred years have passed since the birth of Niels Henrik Abel; as I am now about to perform the glorious task, which has been thrust upon me, to honor the memory of our great compatriot by briefly singling out his importance for Science, for our University, and for our country, my thoughts are brought back to a great era, filled with great social and political transformations, but also rich in brilliant scientific results. Applied mathematics has reaped great and important triumphs, in particular in the fields of astronomy and physics. But just at the same time mathematics, essentially represented by two towering geniuses - Gauss and Cauchy, has started to turn its gaze back to the pure and abstract theories. They initiated thereby that great movement, which has run through the whole of the previous century, and which has reformed higher mathematics from its foundations at the same time as it has enriched it with new theories, which have encompassed a huge area, of which we formerly could have had no inkling. It was in this movement that Niels Abel took such a significant part that he will forever be counted as one of the greatest mathematicians ever.
As a student at the Cathedral school of Christiania he caught the attention of his teacher by the ease with which he solved mathematical problems; Abel had to have his own individual problems, would there be any point in subjecting him to such tasks at all. "His eyes were opened," as he at some other occasion has put it himself. He assimilated in short time everything, which his school could offer him as far as mathematics was concerned, and asked his teacher, later professor Holmboe, for a continuing education. Holmboe, who was later to be one of his closest friends, taught him a course in higher mathematics, and afterwards pursued together with him three extensive volumes by one of the greatest mathematicians of the past - Euler. But the teacher was not able to keep up with his student for long. In 1821 he matriculated and soon started to work with great problems not yet solved by science. Not even the university could now teach him anything in his chosen field.
Our university had just been founded, being just ten years old, when it became fortunate to claim this great thinker into its midst. When it was to be set up, our country did not possess a collection of scientists out of which one could select its teachers; one had to be content to choose the most learned men one could find. It is hence not so remarkable, that it became difficult for the two or three men, who were able to appreciate the unusual talent of the modest student, to persuade their colleagues and the higher authorities of his worth. It was his fate to be treated like the promising beginner, whom one advised to prepare for his future studies through improving his command of the classical languages, while he in fact was busy extending the boundaries of science. This is surprising for us to learn; Abel knew as a matter of fact enough Latin to be able to read Euler and Gauss. Finally it was arranged that the poor student, along with other promising youths, were to be financially supported by the State to educate themselves through a two-year visit to France and Germany. In order to better show the contrast between the opinions held by the highest authorities and that of reality, I would like to point out some words written by Abel in a letter to professor Hansteen concerning his first sojourn at Berlin: he says: "I guess I have not learned anything from others on this trip, but this I have not considered to be the purpose of my travels. Contacts ought to be the main thing for the sake of the future." Events would show, that Abel was right. In Berlin he got to know Crelle and found in him a friend. That man decided at that very time to start a mathematical journal, the for all mathematicians so well-known "Journal für die reine und angewandte Mathematik" to whose first and most eager contributors Abel belonged. It turned out to be of pivotal importance for him, that he in Crelle’s Journal could present his discoveries to the mathematical world.
The first months of his stay in Germany also constituted an important moment in Abel’s development. It is certainly so, that much of what he has done, has its roots in an earlier period of his life, but there was now added a very important thing: critique and self-criticism. Once again, "his eyes were opened," as he has written in his letters, but this time not to his own giftedness, but to the flaws of traditional mathematics. He threw himself headlong in the already started reform movement; he wanted to bring about a greater stringency and a more scientific method. When he treated a long since known result - the binomial formula of Newton, he was astonished that the question of its convergence had not yet been fully treated. He filled this gap, and computed the sum of Newton’s series in the greatest possible generality. He thereby gave important contributions to the theory of infinite series in general and power series in particular; he hence prepared in an exceedingly valuable way for the growth of the modern theory of functions. In my opinion his letters show, when taken together with his notes, that he had anticipated a theory like that of Weierstrass’ Theorie der analytischen Functionen.
Abel’s favorite theme was, however, the theory of algebraic equations. Also here he had worked from first principles. Gauss and Cauchy had given proofs of the Fundamental Theorem of Algebra, to which later mathematicians only have had little to add. Gauss had in addition exhaustively treated the equations connected to the problem of circular division into equal parts, Abel proved the impossibility of a general method of solving equations of degree higher than four by radicals, and thereby brought the theory to a rather new level. He then set out to determine those equations which can be solved in that way, and discovered the most important general results in this new field. But death prevented him to present his findings, so that his successor Galois, one of the most outstanding minds of the past century, had to redo those discoveries once again: because Galois died before the collected works of Abel were published for the first time. Furthermore it was Abel, who first taught the mathematicians to use the auxiliary tool, which now has been named the Galois resolvent; Galois himself expressively announced that the idea was Abel’s. Finally Abel learned to solve that class of equations, which now bears his name. His other theories gave him rich opportunity to apply this discovery and show its worth. But much more important is that the two latter discoveries: Galois’ Resolvent and the theory of Abelian equations, were the two most important tools for Galois, when he gave the theory of equations its final form and thereby gave the foundation for the rise of our contemporary theory of groups.
One now knows with complete certainty, that Abel was not the first who discovered elliptic functions. Gauss had already made the discovery by the time Abel was born, but had not communicated it. Abel made the discovery anew, and he turned out to be the one, to whom the scientific world attributes its knowledge of those functions. Already in his first publications of this, he presented all the fundamental parts of the theory. Here Abel did not get, as in the case of the theory of equations, an immediate successor, but a real rival, in the two years younger Jacobi, who at the same time published his first results on the transformations of elliptic functions, a subject Abel had not yet treated. One year only lasted that peculiar race, which thus had been initiated; because longer did not the life of Abel turn out to be. The genius of Abel brought forward words of admiration from his rival; in a letter to the old mathematician Legendre, whose works had instigated those discoveries, Jacobi wrote of one of Abel’s works: "It is beyond my praise, as it is above my own works". In fact, none of the mathematicians of the time praised Abel stronger than Jacobi. The two great men knew to cherish each other, because also on the part of Abel there are words of high appreciation. By the work of the two there resulted during the span of less than two years an important and extensive theory, one of the most beautiful in mathematics, which additionally has been of inesteemable importance for the development of the general theory of functions.
At the end I will talk about the calculus of integration. It became early on a subject for Abel’s research. Repeated statements in his letters show that he had in mind the publication of a singular work on this, but the circumstances, under which he lived, and his untimely death, did not allow him to execute his plan. In his last work on elliptic functions, he has inserted some of those investigations; some others are to be found in some works before his travels, yet some more in later notes. Also here Abel was a pioneer, showing others the way.
But one of this discoveries he had communicated in full; I am referring to the theorem of Abel. With this he has created a memorial, more permanent than any monument. This is in fact the words of old Legendre, who called it a "monumentum aere perennius". When I mention this theorem I have recalled the difficult investigations thereto connected, and which only a much later period has managed to complete, but to which Abel himself gave the first and important contributions. Similarly I have recalled the vast theory, which is founded on that theorem as a basis: the theory of Abelian functions.
All those foundational discoveries Abel conveyed to the scientific world during the span of three years, and before he had turned 27 himself. A series of distinguished mathematicians have largely gained their fame by completing what Abel had not concluded, and by building on the fundaments that he had laid. Restricting myself to mention the most outstanding of those, whose works already are concluded, Jacobi, Galois, Riemann, Weierstrass, Hermite, Kronecker, have fully testified to the fertility of Abel’s ideas.
With his genius Abel combined a high degree of personal amiability. His undemanding modesty was a prominent feature of his character. Crelle, who knew him so well, says in his obituary, that such a great modesty does maybe not fit in this world. It may also have been seen as a weakness. But notwithstanding the fact that he during his whole life had to fight against almost desperate economical circumstances on the one hand, and on the other had to suffer from a lack of understanding of his worth here at home, he has walked straight ahead on his road, without letting himself be discouraged. I find, that he thereby has shown a strength of character that deserves the greatest acknowledgment. His warm feelings for his country and his loving care for his relatives, come across clearly in his letters. In spite of Crelle’s repeated attempts to keep him in Berlin, he was not to be prevailed upon to stay. Only when it became clear, that he could not expect a permanent position at our university, did he reluctantly give permission to receive an offer from Berlin. The announcement, that the offer was to be made, was written two days after his death. Thus became his fate. But he is the greatest scientist in Norway, and the greatest glory of our young university. His reputation has gone far beyond that of his country and his times. This numerous assembly of the most outstanding scientists from all over the world, who have done us the great honor to meet here to join us in honoring his memory, is a dear sign to us, of how high his name and his accomplishments still are held.
Already some years ago it was decided at a meeting of Scandinavian natural scientists, that a monument to Abel should be erected at his centennial. Admittedly it did not succeed in realizing this decision by the desired time, but the erection of the monument has been assured. In due time a visible memory of Abel will rise as an inspirational sign for future generations.