Niels Henrik Abel
… But we say: Young scientists, do not listen to anything but your own
inner voice
which tells the tasks best suited for your own inclinations and abilities.
Read and contemplate the writings of the men of genius,
But never become pliable pupils,
or egotistical admirers.
The device must be:
Objectivity toward the facts and freedom in the choice of views.
Guglielmo Icilio Bruto Timoleone Libri
The greatest Norwegian mathematician was born to a Lutheran minister in Finnoy, Norway, on August 5, 1802. The father, Pastor Soren Georg Abel named his second son, the future mathematician, Niels Henrik after the maternal grandfather, Niels Henrik Simonsen according to their custom. The Pastor’s family consisted of seven children in a time when Norway was desperately poor as a result of wars with England and Sweden. They were borderline poverty stricken but somehow they manage to support themselves.
Pastor Abel was in charge of his children’s education at home. However, in the fall of 1815, he decided to send his two oldest sons, Hans Mathias and Niels Henrik, to school in Oslo. Niels Henrik was only thirteen years of age, and this was younger than usual for a boy to be sent away from home. Niels Henrik and his brother were admitted to the Cathedal School in Oslo, an institution founded around 1250 AD as a Latin school for the children of government officials. The first year Pastor Abel paid tuition for his two sons but later they were awarded fellowships by the school. The school recently had lost most of its teachers to the new University of Oslo. The quality of the facility was unsatisfactory, and the school was staffed with inexperienced and flat out incompetent instructors. Due to these reasons, Niels Henrik’s academic performance in the first two years was not impressive.
At the same time, Pastor Abel was slowly loosing the trust of his community and the Church. He was the center of public controversies. The discussion reached the level of the ridiculous when someone published a personal attack against Pastor Abel. This became a preoccupation for young Niels, and increased the difficulties on the young man’s mind. In addition to this, his mathematics teacher sometimes would beat him. Niels Henrik pretended to be the happiest among his classmates at parties. However, when he was alone he was sad and depressed. His greatest amusement was going to the theater. Anytime he could scrap his scarce money together, he would be in the audience. Ever since this time theaters were his favorite leisure until death.
When Niels Henrik’s first mathematics instructor was forced to resign for beating a student to death, he was replaced by Bernt Michael Holmboe, who was an assistant to Christopher Hansteen at the university. Holmboe was later to edit the first edition of Niels Henrik’s collected works in 1839. Hansteen became Niels Henrik’s faculty adviser when Niels Henrik entered to the University of Kristiania. The two soon became close friends and strong supporters of Niels Henrik. This inspiring teacher discovered Niels Henrik’s mathematical talent, and at the age of sixteen, had him devoting almost all his time to it. Holmboe was excited over the great products of Niels Henrik’s progress, and praised him to the other teachers and the university professors.Pastor Abel died in 1820 at the age of forty-eight; Niels Henrik was eighteen. Ironically, when he was young and full of ambition, he taught the peasants in his community to stay away from the disease of that time; alcoholic spirits. However, about two years before his death, he started drinking and the habit slowly got the better of him. It was clear that his death was approaching. For Niels Henrik, it was the end of his childhood. The care of his mother and six siblings fell on Niels Henrik’s shoulders. Niels Henrik’s older brother, Hans Mathias, by this time showed strong symptoms of insanity and needed his mother’ care. In his superficial and irresponsible family there was no one to share his task.
Heroically, Niels Henrik never complained about having to support his family. By the time of his father’s death he had only the scholarship from the Cathedral School yet, somehow, he was able to assist two of his closest siblings, the fourth brother Peder Mandrup Tuxen, and his sister Elisabeth Magdalene to receive an education at Oslo.
Niels Henrik’s first main contribution to mathematics came before entering college. It was finding the general solution of the quintic (fifth degree) equation such as ax5 + bx4 + cx3 + dx2 +ex + f = 0. Niels Henrik thought he had developed the answer. Holmboe and Hansteen sent the paper to the mathematician Ferdinand Degen in Denmark since they knew that there was nobody in Norway with the ability to understand if the answer was correct. Before receiving an answer from Degen, Niels Henrik discovered a mistake in his paper and questioned if there was an answer. He eventually proved that an algebraic solution to the quintic equation was impossible. However, Degen suggested Niels Henrik to take the subject of elliptic integrals, which would become the focus of his work and the source of his fame.In July of 1821 Niels Henrik graduated from the Cathedral School and in the following month he passed the entrance examination for the University of Kristiania with mediocre marks in all subjects except mathematics. The University had no funds for fellowships but assisted some needy students by providing free room, light, and firewood in the university dormitory. Niels Henrik applied for this aid. For the student who had absolutely no resources, a free room was not sufficient. Therefore he also applied for a cash grant. Holmboe had done everything possible to relieve Niels Henrik’s poverty, convincing his colleagues that they too should subscribe to make it possible for Niels Henrik to continue his mathematical researches. Some of the university professors took an exceptional step and they personally contributed money for the fellowship. Niels Henrik was able to start his study in a more free environment with this remarkable financial arrangement.
In June 1822, when Niels Henrik was nineteen, he completed his preparatory examinations at the University of Kristiania, and was entitled to use the imposing title Candidatus Philosophiae. He was rewarded with the highest marks in theoretical physics and mathematics. He was left on his own to study. It was the beginning of his productive period. There were some of his works that he felt worthy of publication but there was no periodical for scientific publication in Norway.
Niels Henrik, Baltazar Mathias Keilhau, and Christian Peter Boeck, who had both graduated from the Cathedal School two years before Niels Henrik recognized that a periodical was of prime necessity for scientific progress in Norway. During the fall of 1822, prospective patrons throughout Norway were asked to support the establishment of the nation’s first scientific periodical, and almost 200 subscribers were found. In early 1823, the first number of the new “Magazine for the Natural Sciences” (Magazin for Naturvidenskaben ) was printed. Niels Henrik’s work was printed for the first time in the next issue of the Magazin. It was a paper on the so-called functional equations, which is a topic that would only much later develop into a very important field of mathematics. One newspaper reviewed the Magazin in detail and commented about Niels Henrik’s paper: ”A memoir which justifies the greatest expectation for the young mathematician.”
In the summer of 1823, Niels Henrik took a trip to Copenhagen by the invitation of Professor Soren Rasmussen who was impressed with Niels Henrik’s creative works. This was the first time he met Christine Kemp who was the daughter of War Commissar Christian Kemp, who had been long dead. She was not wealthy, but making her own living which was a little unusual for young women of the day. Niels Henrik described Christine to his friend Keilhau: “She is not beautiful, has read hair and freckles, but she is a wonderful girl.” This description later changed the lives of both Keilhau and Christine.
Around the university and in town it was well known that Niels Henrik was a genius who was supported by professors, and those professors expected great mathematical works from him. But he was modest, friendly and optimistic, almost immature in some respects. He helped his friends and family willingly and competently with understanding and thoughtfulness. He was friendly outwardly but felt lonely and craved sympathy on the inside. Almost all people who have some associations with Niels Henrik were drawn into friendship with him. It is remarkable how willing his friends were to speak for him and assist him whenever he was in need. Niels Henrik’s mathematical and astronomical friends persuaded the university to appeal to the Norwegian Government to subsidize him for a grand mathematical tour of Europe. To impress the authorities, Niels Henrik submitted an extensive memoir of his work. Unfortunately the university has having a severe financial struggle of it own, and the memoir was lost. In 1825, the Government finally agreed to the grant to travel Europe for two years, which made the Government nearly bankrupt.
Niels Henrik was forced to wait almost two years. However, he did not waste this time. First studying French, the German, and working incessantly at his mathematics. With his optimism, he also able to work an engagement to Christine Kemp. He was twenty-three years old.
The start of this trip was a dart premonition of things to come. Niels Henrik first stopped in Denmark to meet with Degen, only to discover that he had died three days before his arrival. His next stop was Berlin. Here he met August Leopold Crelle, an influential engineer and amateur mathematician who was pondering the creation of a journal dedicated to new mathematical ideas. Crelle first suspected that Niels Henrik was a candidate for the trade school where he worked. After struggling over language differences, Niels Henrik replied that he was only interested in mathematics. Soon Crelle understood that Niels Henrik was a genius.
The two took a trip to Gottingen to meet Carl Friedrich Gauss “Prince of Mathematics” as he sometimes has been named. Niels Henrik had studied one of Gauss’s books about number theory, Disquisitiones Arithmeticae, in detail during his first year at the university. Unfortunately, when the two arrived in Gottingen, Gauss was old and cared little for students or pupils. He was indifferent in regard to honors, but from time to time he brought forth works which were mathematical masterpieces. Gauss duly received Niels Henrik’s paper, Memoir on a general property of a very extensive class of transcendental functions. Without reading the masterpiece, he tossed it aside with the disgusted exclamation “Here is another of those monstrosities!” Abel decided not to call on Gauss.
During this trip Crelle became a scientific Holmboe to Niels Henrik, and had far more weight in the mathematical world than Holmboe ever had. If Crelle helped to make Niels Henrik’s reputation, Niels Henrik more than paid for the help by making Crelle’s. When he left Germany to visit Italy, Crelle offered him the editorship of his mathematical journal, Journal fur die reine und angewandte Mathematik.
Although, Niels Henrik enjoyed his fellows, sight seeing and comedy theaters on the way to France, so far, his trip had no positive outcome for his academic career. His resources were scarce, and he had yet to make a name for himself. The destination after Germany was Paris. Paris abounded with famous mathematicians and physicists at this time. Niels Henrik mentioned a considerable number of them, occasionally with an important observation to characterize them. He was obviously disappointed that many were so old that only their fame remained.
When he arrived to Paris, he managed to have his “masterpiece”, the paper on elliptic functions and integrals, which included Abel’s theorem, presented to the French Academy of Sciences. He had confidence on the paper. He wrote to Holmboe on October 24, 1826: “ Without boasting, I dare say it is good, and I am curious to see what judgment will be given at the Academy.” If the work were accepted, he would have been made. Unfortunately, the Academy picked Adrien Marie Legendre and Augustin Louis Cauchy as referees to judge it. Legendre, who was in his seventies, claimed that he could not read the handwriting and left all the work to young Cauchy. He was thirty-nine years old.
Cauchy was the leading active mathematician in Paris. His scientific production was enormous. For long periods he appeared before the Academy once a week with his new paper, so the Academy introduced a rule restricting the number of articles a member could request published in a year. Among his colleagues he was not always popular, often considered hypocritical and egotistical. In his preoccupation with his own work he showed little interest in others’ work. Niels Henrik did not seem to have liked Cauchy personally, but he admitted the fact that he considered Cauchy to be the only one who knew how mathematics should be presented and demonstrated.
While Cauchy “misplaced” Niels Henrik’s memoir with a small motive of jealousy, Niels Henrik was waiting for the judgement of his paper from the Academy, and improving his French by subletting a room in one French family at Paris. Furthermore, he sent his mathematical works in order to make Crelle’s Journal internationally known.
Niels Henrik had waited to hear from the Academy for ten months while in Paris. He was optimistic enough to hope that the commendatory decision of the Institute would not be long delayed. While he waited in Paris he had too much leisure time to meditate upon his own worries. He had enjoyed the theater many nights in Paris. From Oslo, he had hardly heard from his family. His relation with his future wife grieved him more. She moved to Denmark to live with her sister who was married to a physician. At times he was hoping to receive the lectureship in Oslo after this journey. However, he had not received any good news regarding this from Oslo. While in Paris he consulted good physicians for what he thought was merely a persistent cold. He was told that he had tuberculosis. However, he refused to believe the doctors’ diagnosis.
When months went by without any call from the Academy, and his finance never seemed any worse, Niels Henrik was convinced that there was no reason to stay in Paris anymore. Between Christmas and New Year’s he left Paris without sorrow. He stopped by in Berlin before he returned to Oslo to see Crelle, his fatherly friend. The life in Berlin was more comfortable and friendly to him. However, his purse was almost empty which forced him to leave his friend to journey back to his homeland. Before he returned to Oslo, he made one more stop in Denmark to see Christine Kemp.
On May 20, 1827, Niels Henrik arrived in Oslo in failure. Not only was he unable to get the recognition he deserved and the professorship he desperately needed, he was in debt and had contracted tuberculosis. In addition to the failure he had been passed over to fill a vacancy in the mathematics department at the university. The position had instead been given to his friend Holmboe. Holmboe did not want to take this position to offer a teaching position to Niels Henrik. However, he had no choice but accepting this offer when they threatened to give the job to a foreigner if he did not agree to take it. Niels Henrik survived on grants from the university and gifts from his friends.
The first summer since Niels Henrik was back from Paris, he was ill, coughed, was feverish and stayed at his mother’s house. In the middle of August 1828, he returned to Oslo for his class. He also wrote mathematical papers almost continuously since he returned from his journey. One of the papers, a continuation of his “knockout” of Jacobi, was sent to Schumacher. Otherwise all his works went to Crelle. The volumes of the Journal for 1828 and 1829 are filled with his work.
During the fall of 1825, Niels Henrik left Oslo on his journey, another mathematician, who was two years younger than Niels Henrik, the Privatdocent Carl Gustav Jacob Jacobi, began his first series of lectures at the university in Berlin. When Niels Henrik was traveling southward, Jacobi had already obtained a better paid position at the University of Konigsberg. It was assumed that the two young mathematicians must have seen each other at the time.
In the summer of 1827, while Niels Henrik’s paper was awaiting publication, Jacobi also was lead to new ideas of elliptic functions and integrations, and mailed two notes, dated June 25 and August 2, to Schumacher in Altona, to be printed in the Astronomische Nachrichten. Spurred on by his competitor, Niels Henrik’s production on the subject increased. His work laid the foundation of all future studies into the field.
Finally, there came the time that people began to notice him. Legendre, who had failed to read Niels Henrik’s masterpiece earlier, started corresponding to both Niels Henrik’s and Jacobi, describing them as two of “the foremost analysts of out times.” Hermite said that Abel had laid out five hundred years’ of work for future generation of mathematicians. Mathematicians all across Europe were calling Niels Henrik for a professorship.
Unfortunately, it was too late. His tuberculosis slowly jeopardized Niels Henrik’s health. In December of 1828, he left Oslo to spend Christmas with his fiancee hoping to come back to the university. He, however, lost his battle to the disease on April 6, 1829 by the side of his fiancee, aged twenty-six years, eight months. Two days later, Crelle sent him notice that he had finally been able to secure a position for him at the University of Berlin.
A year after his death, the masterpiece that had been “misplaced” by Cauchy was given the recognition it deserved and awarded the grand prize by the Academy. To Anne Marie Abel, the mother of Niels Henrik, the money was a gift from heaven. 1500 francs were to be paid to her. During the last few months of his life, Niels Henrik wrote to his friend Keilhau, imploring him to assist her in whatever way he could. Keilhau had a deep sense of obligation toward Niels Henrik. He had never seen Christine but he felt that the best way in which the request of his friend could be fulfilled would be for her to consent to become his wife. They later married.
Niels Henrik’s memoir was prepared for publication by the young Italian mathematician Guglielmo Icilio Bruto Timoleone Libri who studied under Legendre. It was not published until 1841. Libri had become interested in Niels Henrik through this project and he wrote the first fairly extensive account of Niels Henrik’s life in French.
Niels Henrik’s tragic life contributed many to the field of mathematics such as Abel’s proof of no solution to the quintic equation, the solutions to definite integrals. In terms of the ordinary differential equation, Abel’s theorem and formula were all valuable.
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