The Biography of Niels Henrik Abel: The great journey abroad

After having written a personal letter to king Karl Johan of Sweden, Abel was able to advance the date of his journey abroad, and in September 1825 he departed from Christiania. The plan and the terms of the grant called for him to go first to Gauss in Göttingen and then on to Paris, but when Abel got to Copenhagen, he changed the route of his journey, and went to Berlin instead. This must be regarded as the greatest stroke of luck in his short life.

In Berlin he met the mathematically interested engineer, August Leopold Crelle (1780-1855), who found the courage after his meeting with Abel to pursue a goal that he had long wanted to achieve, namely to publish a mathematical journal in Berlin that could compete with the well-established journals in France. By the beginning of 1826, the first issue of Crelle’s Journal (Journal für die reine und angewandte Mathematik) was already published, and this was where Abel would publish most of the works that he managed to write. Largely thanks to Abel’s works, Crelle’s Journal quickly gained renown as one of Europe’s leading journals (and the journal is still published today, and continues to have good international repute).

Abel remained in Berlin for four months, and he had an inspiring time there together with Crelle and his circle of mathematicians.

Crelle’s Journal 
The first work that Abel published in Crelle’s Journal was an expanded version of the proof that the general fifth degree equation was not solvable by the extraction of roots. With acuity and virtuosity, he showed that a possible solution would have to possess properties that no radical can have. Many special equations of a higher degree than the fifth have solutions, and in his ongoing work on equation theory, Abel went a long way toward finding acceptable solutions to these equations. The first year of Crelle’s Journal contains seven other submissions from Abel – two minor problems with comments and five treatises – among other things a seminal work on binomial series, in which he fully discusses these series’ convergence criteria. In these works, Abel demonstrates the exactness and stringency that had often been lacking in mathematical proofs, and he becomes one of the pioneers of a more stringent modern type of proof.

Abel was accompanied on his travels by other young Norwegian scientists, most of whom were studying mineralogy and geology. For these young scientists, the most interesting study areas were in the south of Germany, in Austria, in Switzerland and in the north of Italy. Abel, who complained in many of his letters of how melancholy he could become when he was alone, followed his friends southward in Europe.

He did not get to Paris until July 1826, ten months after he had left Christiania. He reported, "At last, I have arrived at the focal point of all of my mathematical hopes," and he explained the long detour as follows, "My desire to see some of Europe was great, and should we confine our travels solely to the study of the strictly scientific?"

Milestone in the history of mathematics
Although Abel had now begun to publish in Crelle’s Journal in Berlin, he had saved what he thought were completely new insights for the venerable Paris Academy. Immediately after he had found a room in the city, Abel began the work that is called the Paris treatise. Scarcely any other mathematical treatise has garnered so much praise in posterity as Abel’s Paris treatise – an addition theorem for elliptic integrals. The impressive thing about it is Abel’s monumental generality in his statement of the problem. In addition, his scenario was greater than that of his predecessors, he demonstrated mathematical relationships that no one had previously dreamed of, and he opened up areas of research in which new discoveries are still being made. Abel’s Paris treatise continues to stand out as a milestone in the history of mathematics.

Abel submitted his Paris treatise to the scientific academy at the end of October 1826, signing it, «par N.H. Abel, Norvegien» (by N.H. Abel, Norwegian).

Tuberculosis – a death sentence
He continued to live in Paris for the rest of the year, and while he waited for an answer, he completed a couple of other works. However, Abel’s Paris treatise was laid aside and forgotten. As long as he lived, he was convinced that the work had been lost forever. His stay in Paris turned out to be a disappointment; he was unhappy, did not feel well, and had a fever and a cough. Someone or other, probably a medical student in the circle of scientists that Abel occasionally frequented, thought that Abel was suffering from tuberculosis – a death sentence at that time.

At the close of 1826, Abel left Paris, poor and weary, and returned to his friends in Berlin. He was offered the post of editor of Crelle’s Journal, but turned it down. He was homesick and wanted to put his scientific ability to the service of his "Fatherland". Crelle in turn continued his efforts to obtain a secure post for Abel in Berlin.