Now Schumacher began a project triangulating the district of Holstein in 1817 and a couple of years later extended it to a complete geodetic survey of Denmark. To aid this project, an observatory had been set up at Altona and Schumacher resided there as the director. He had got to know Holst since his surveying work had taken him to the district where Holst, the local astronomy expert, lived. In around 1819 Clausen began working for Schumacher at the observatory in Altona. At first he was employed as a clerk, but soon he was learning more advanced mathematics and helping Schumacher with the scientific work. Peter Andreas Hansen began working for Schumacher in 1821 and he played a role in increasing Clausen's mathematical knowledge. Clausen published his first paper Berechnung der Sternbedeckungen vom Monde zur Bestimmung der geographischen Länge in 1824. This paper, which presented a method of calculating longitude using stellar occultations by the moon, appeared in Volume 2 of Astronomische Nachrichten, a journal founded by Schumacher in 1821.
Clausen became an assistant at Altona Observatory in 1824 and in October of that year he met Carl Friedrich Gauss, who was conducting geodesic measurements nearby, for the first time. They became friendly and Gauss was impressed with the young man. Things were not going well between Schumacher and Clausen, however, and when Clausen broke an expensive barometer this was the last straw - Schumacher told Clausen to leave just before Christmas 1824. Now Clausen was in considerable difficulty but, remembering how well he had got on with Gauss, he travelled to Göttingen to see him. After explaining what had happened in Altona, he asked Gauss to write a letter to Schumacher supporting him. This Gauss did gladly and, with some reluctance, Schumacher reinstated him. The reluctance seems to have been on account of Clausen lacking breeding rather than on account of his work. After his first paper, mentioned above, Clausen published two further papers in Volume 2 of Astronomische Nachrichten in 1824: Mondssterne in Altona beobachtet , and Anzeige von Druckfehlern . Three further papers appeared in 1825: Längendifferenzen aus Mondsculminationen ; Auszug aus einem Briefe des Herrn Thomas Clausen an den Herausgeber ; and Resultate der Mondssterne beobachtungen . There were, however, distractions for Clausen since his father became ill, their farm was being sold, and three of his siblings were still too young to support themselves.
Relations between Schumacher and Clausen went from bad to worse when Clausen fell in love with Schumacher's niece who lived in the same house as her uncle (and Clausen). There was no way that Schumacher would allow his niece to marry someone of Clausen's lowly class so he tried to get him to leave Altona by arranging another position for him. In October 1826 Schumacher visited Munich where, following the death of Fraunhofer earlier that year, the Munich Optical Institute was in need of a good theoretical astronomer. Schumacher suggested Clausen for the position, and the offer was accepted. Clausen was delighted when informed of the job offer and wrote a letter of acceptance on 3 November. He signed a contract to begin work in Munich in May 1827. However, Wilhelm Bessel now suggested to the Munich Optical Institute that they employ his former student Carl August von Steinheil. Although von Steinheil was weaker than Clausen as a mathematician, he had the advantage of being rich and astronomers always need lots of money to carry out their research. Clausen's appointment was delayed, putting him in an extremely difficult position since he had resigned from Altona although he was given some money to compensate for the fact that he had a signed contract. Having resigned from Altona, he again pressed Schumacher's niece to marry him but was turned down again. The difficulty over his appointment was solved at last with both Clausen and von Steinheil being employed at the Munich Optical Institute. Clausen started work in December 1828. He did not carry out any observational duties in this post and was left on his own to undertake research into mathematics and astronomy. He began sending papers on pure mathematics to August Crelle's Journal für die reine und angewandte Mathematik which did not please the director of the Munich Institute. Towards the end of 1832 Clausen suffered a severe disappointment when he learnt that he would never succeed Fraunhofer.
Clausen's work was recognised by many of the top scientists of the day including Olbers, Gauss, Bessel, Hansen, Crelle, von Humboldt and Arago. However he began to suffer from mental illness in 1833. He suffered this illness for seven years, then in 1840 he left Munich, returning to Altona - he walked the whole way. Back in Altona he spent two years on his own doing some of the best science of his life. In 1840 he published a theorem on the Bernoulli numbers which had been proved around the same time by Karl von Staudt - today it is called the von Staudt-Clausen theorem. Also 1840 he published a paper on the quadrature of four cases of the lunes of Hippocrates of Chios. In 1842 he published a treatise on the comet discovered by Charles Messier in June 1770. In this treatise he computed the orbit of the comet and for this work was awarded a prize by the Royal Danish Academy of Sciences and Letters. Bessel described this work in the following terms :-
What a magnificent, or rather, masterful work! It is an achievement of our time which our descendants will not fail to credit him with.On 10 August 1842, Schumacher wrote to Gauss saying that Clausen had proved the nonexistence of orthogonal Latin squares of order 6 by dividing such Latin squares into 17 families. Sadly details of Clausen's proof have never been found although the remarkable combinatorial abilities he displayed leads one to believe that he did have a proof. He was also engaged in an argument with Jacobi. John McCleary writes :-
One of the prettiest results in the global theory of curves is a theorem of Jacobi (1842): The spherical image of the normal directions along a closed differentiable curve in space divides the unit sphere into regions of equal area. The statement of this theorem is an afterthought to a paper in which Jacobi responds to the published correction by Thomas Clausen (1842) of an earlier paper by Jacobi (1836).As soon as Clausen had arrived back in Altona at the beginning of 1840, Schumacher had tried to arrange a position for him at the Observatory in Tartu, at that time part of the Russian Empire and known as Derpt (similar to the earlier German name of Dorpat). Johann Heinrich Mädler, the director of the Observatory, would have preferred to employ Johann Gottfried Galle who, unlike Clausen, had an exceptional reputation as an observer. Galle did not want to go to Tartu and there followed a lengthy correspondence between Mädler, Schumacher and Gauss as to Clausen's suitability. In February 1842 he was appointed to the observatory in Tartu as Professor of Astronomy and began work there in the October of that year :-
... Schumacher and Bessel worried about how Clausen and Mädler would get along. But they worried in vain since these men made friends very quickly.Two years later Clausen received an honorary doctorate from the University of Königsberg. He had been put forward for this honour by Bessel, who thought very highly of Clausen's extraordinary talents. If Clausen had struggled to get positions earlier, this was no longer the case for he received an offer to go to the Observatory in Pulkowa, the leading Russian observatory. Despite being a tempting offer, he decided to stay in Tartu. In 1866 he was appointed to replace Mädler as director of the Tartu Observatory, a post he held until he retired in 1872. Clausen never married and, after moving to Tartu, appears to have had no further contacts with his family back in Denmark.
He received many honours during the years he spent in Tartu, in addition to the honorary doctorate just mentioned. He was elected to the Royal Astronomical Society of London in 1848, the Göttingen Academy of Sciences (at that time called Königliche Gesellschaft der Wissenschaften zu Göttingen) in 1854, and the St Petersburg Academy of Sciences in 1856. He received two prizes from the Göttingen Academy of Sciences for his outstanding work.
Clausen wrote over 150 papers on pure mathematics, applied mathematics, astronomy and geophysics. Among his work in pure mathematics, he factored the 6th Fermat number 2n + 1 where n = 26 in 1854 showing it was not prime. Since 264 + 1 = 67280421310721 × 274177 one can only marvel at how Clausen achieved this result. The first to show that not all the Fermat numbers were prime was Euler in 1732 when he showed that 2n + 1 where n = 25 was not prime. Clausen also gave a new method of factorising numbers. He computed π to 247 places which he published in 1847. He used the formula
He possessed an enormous facility for calculation, a critical eye, and perseverance and inventiveness in his methodology.Clausen's first love was mathematics, particularly number theory, although he had never received a formal mathematical training. He wrote in a letter in 1826 :-
... theoretical study of mathematics has always been the greatest interest to me.One can only guess what he might have achieved had the educational system in his time been able to give this remarkable man from a lowly background, the education he deserved.
Article by: J J O'Connor and E F Robertson