Despite writing a thesis on astronomy, Bäcklund published a paper on geometry Några satser om plana algebraiska kurvor, som gå genom samma skärningspunkter in 1868. After being appointed as an assistant at the astronomical observatory in Lund on 19 January 1869, he was appointed as a docent in geometry later in the same year. Hill retired from the chair of mathematics at Lund in 1870 and a complex appointing process was set up to fill the chair. Bäcklund, although relatively inexperienced at this stage in his career, applied for the position as did Edvard von Zeipel, the assistant professor, Carl Fabian Emanuel Björling, a lecturer in Halmstad, Göran Dillner, an assistant professor in Uppsala, and Sophus Lie. Lie soon withdrew his application and the others were evaluated by Axel Möller in 1871. All the candidates gave lectures and, because he was not already on the staff of a university, Björling was asked to defend one of his recently published papers on roots of algebraic equations. Perhaps surprisingly, since he was also a candidate for the chair, Bäcklund was asked to act as Björling's opponent for this defence. Bäcklund pointed out that many of the results claimed by Björling had been proved many years before by Euler. Björling did not dispute this but said that to discover theorems previously discovered by Euler only showed his exceptional quality.
Bäcklund was the youngest of the candidates and did not stand much chance of being appointed despite being able to present five papers in algebraic geometry. The first of these we have already mentioned above but the most significant was Några satser om plana algebraiska kurvors normaler (1869) concerning normals of real algebraic curves. Möller rated Edvard von Zeipel as the leading candidate, Bäcklund as third, and Björling in last place. However, Björling disputed Möller's assessment and, after furious letters to the government and to the King, he was appointed to the chair of mathematics in 1873. After the tensions caused by this competition for the chair, the mathematics department must have been a rather unhappy place over the following period. Bäcklund, however, was able to spend time away from Lund after he received funding to make a trip to Germany to further his studies. He spent most of the months from February to July 1874 at Leipzig and Erlangen working with Felix Klein and Ferdinand von Lindemann. Having failed to win the chair of mathematics with his contributions to algebraic geometry, he now changed topic feeling that this would give him a better chance of promotion. It was in this new area of differential equations that Bäcklund produced his more notable results, namely on what are today called Bäcklund transformations.
All Bäcklund's papers on algebraic geometry are in Swedish and published in Swedish journals. His first paper in an international journal, written in German in Mathematischen Annalen, was Über Flächentransformationen (1875). At the start of the paper Bäcklund explains the background to the work:-
Recently, I became concerned with the question if there are surface transformations of a three-dimensional space that leave invariant the second-order tangency rather than the first-order tangency. I discussed this question in Volume 10 of the 'Annual Reports of Lund University' (September 1874) and came to the conclusion that the transformations that leave invariant the first-order tangency, i.e. Lie's contact transformations, are the only ones for which the higher-order tangency conditions are invariant. Simultaneously, Lie's paper appeared in Volume 8 of 'Mathematischen Annalen' (1874) where the similar question on osculating transformations was raised. Therefore I would like to undertake a detailed presentation of the previous investigation and to dwell on certain points that were only hinted at before.After this paper on surface transformations which are bijections, he went on to study general surface transformations, preparing the way in the two-part paper Über partielle Differentialgleichungen höherer Ordnung, die intermediäre erste Integrale besitzen (1877, 1878). He continued his investigations with Zur Theorie der partiellen Differentialgleichungen zweiter Ordnung (1879) and Zur Theorie der partiellen Differentialgleichung erster Ordnung (1880), returning to study surface transformations in Zur Theorie der Flächentransformationen (1881). For more information about the content of these important papers, see .
He was unsuccessful in his application for a professorship in mechanics at Uppsala University, but it was not too long before he was appointed as an extraordinary professor of mechanics and mathematical physics at Lund on 8 February 1878. Carl Wilhelm Olseen, who studied at Lund University in the 1890s, describes Bäcklund's teaching at that time in the obituary he wrote :-
Bäcklund devoted himself to his teaching duties with ardour and delight and he often talked about the pleasure his lectures had given him. During the time that the author (C W Olseen) heard him (the 1890's) Bäcklund carried a double burden. Outside of the regular four lectures a week he gave a beginner's course in mechanics with exercises, also four hours a week. His interest in his audience was manifested by, among other things, invitations to a yearly party where lobster was a standing ingredient.Bäcklund continued in the position of extraordinary professor until 1897 when he was made full professor of mechanics and mathematical physics. He was appointed professor of physics on 21 September 1900. During his years as professor of physics, he served as rector of Lund University from 1 June 1907 to 31 May 1909. Olseen, who was by this time a leading Swedish physicist, also describes the contributions that Bäcklund made to physics in the obituary he wrote :-
There are many kinds of physical theories. For experimental research most useful, but in the long run least tenable, are those which, on the basis of accepted knowledge, examine recent experiments and their consequences for a systematic theory. For theoretical physics most important is the theoretical work, which, from contradictions between reality and established theory, carves out a new basic hypothesis for the theory. As an example we may mention Planck's quantum hypothesis. Yet another kind is the theoretical work which, by a deeper mathematical treatment of an existing theory, attempts to eliminate a contradiction between theory and experiment or be useful to experimental research. Bäcklund's physical theory falls into none of these categories. With contemporary experimental work it had no connection. Neither had it any ties with the theoretical work which at the same time was done in England or on the continent. The aim of his theory was neither to be useful to experimental work nor to eliminate some contradiction between theory and experiment. Then, what was the aim of this theory? To reduce physics to mechanics! To prove that all physical phenomena can be deduced from various forms of motion! This is the object, more philosophical than physical, which Bäcklund in his activity as a mathematical physicist tried to realize. In his youth this was the established aim of theoretical physics. He remained true to this conception while around him physics went through a development which for the majority of physicists radically changed the circle of problems. For this reason his theory was never tested in experiment. At the time when it was so developed that experimental tests were possible, the centre of gravity in physics had moved from mechanics to electromagnetism. The foundations of his theory had failed before the theory could be tested. This was the tragic element in Bäcklund's life as a theoretical physicist. Or perhaps, this was just one side of his tragedy. There was another one. In the extensive world he had created he was the only living being. During his twenty-five years of work in theoretical physics he was completely alone. There are things he has said where it is clear that he felt this loneliness as a tragedy.An interesting long letter by Bäcklund to Poincaré concerning his work in physics is published in .
Lars Garding, in , sums up Bäcklund's career with regrets that he moved to physics from mathematics:-
Bäcklund's mathematical work was more successful. His discovery and characterization of generalized contact transformations is a permanent piece of mathematics which remains of interest one hundred years after it was done. But also in the mathematical papers it is possible to see the combination of power and insensitivity which Oseen describes. It is perhaps regrettable that Bäcklund was not able to devote himself to mathematics all his life.
Article by: J J O'Connor and E F Robertson