As was the standard practice of students in Germany in the second half of the 19th century, Lindemann moved from one university to another. He began his studies in Göttingen in 1870 and there he was much influenced by Clebsch. He was fortunate to be taught by Clebsch for he had only been appointed to Göttingen in 1868 and sadly he died in 1872. Later Lindemann was able to make use of the lecture notes he had taken attending Clebsch's geometry lectures when he edited and revised these note for publication in 1876.
Lindemann also studied at Erlangen and at Munich. At Erlangen he studied for his doctorate and, under Klein's direction, he wrote a dissertation on non-Euclidean line geometry and its connection with non-Euclidean kinematics and statics. The degree was awarded in 1873 for the dissertation Über unendlich kleine Bewegungen und über Kraftsysteme bei allgemeiner projektivischer Massbestimmung Ⓣ.
After the award of his doctorate Lindemann set off to visit important mathematical centres in England and France. In England he made visits to Oxford, Cambridge and London, while in France he spent time at Paris where he was influenced by Chasles, Bertrand, Jordan and Hermite. Returning to Germany Lindemann worked for his habilitation. This was awarded by the University of Würzburg in 1877 and later that year he was appointed as extraordinary professor at the University of Freiburg. He was promoted to ordinary professor at Freiburg in 1879.
Lindemann became professor at the University of Königsberg in 1883. Hurwitz and Hilbert both joined the staff at Königsberg while he was there. While in Königsberg he married Elizabeth Küssner, an actress, and daughter of a local school teacher. In 1893 Lindemann accepted a chair at the University of Munich where he was to remain for the rest of his career.
Lindemann's main work was in geometry and analysis. He is famed for his proof that π is transcendental, that is, π is not the root of any algebraic equation with rational coefficients. The problem of squaring the circle, namely constructing a square with the same area as a given circle using ruler and compasses alone, had been one of the classical problems of Greek mathematics. In 1873, the year in which Lindemann was awarded his doctorate, Hermite published his proof that e is transcendental. Shortly after this Lindemann visited Hermite in Paris and discussed the methods which he had used in his proof. Using methods similar to those of Hermite, Lindemann established in 1882 that π was also transcendental.
In fact his proof is based on the proof that e is transcendental together with the fact that eπi = -1. Many historians of science regret that Hermite, despite doing most of the hard work, failed to make the final step to prove the result concerning which would have brought him fame outside the world of mathematics. This fame was instead heaped on Lindemann but many feel that he was a mathematician clearly inferior to Hermite who, by good luck, stumbled on a famous result. Although there is some truth in this, it is still true that many people make their own luck and in Lindemann's case one has to give him much credit for spotting the trick which Hermite had failed to see.
Lambert had proved in 1761 that π was irrational but this was not enough to prove the impossibility of squaring the circle with ruler and compass since certain algebraic numbers can be constructed with ruler and compass. Lindemann's proof that π is transcendental finally established that squaring the circle with ruler and compasses is insoluble. He published his proof in the paper Über die Zahl π Ⓣ in 1882.
Physics was also an area of interest for Lindemann. He worked on the theory of the electron, and came into conflict with Arnold Sommerfeld on this subject. Eckert, in , looks at Lindemann's contributions to physics, using manuscript materials, including correspondence with Sommerfeld.
Another research interest of Lindemann was the history of mathematics. He also undertook, in collaboration with his wife, translating work. In particular they translated and revised some of Poincaré's writings.
Lindemann was elected to the Bavarian Academy of Sciences in 1894 as an associate member, becoming a full member in the following year. He given an honorary degree by the University of St Andrews in 1912.
Wussing writes in :-
Lindemann was one of the founders of the modern German educational system. He emphasised the development of the seminar and in his lectures communicated the latest research results. He also supervised more than sixty doctoral students, including David Hilbert.Hilbert was Lindemann's doctoral student in Königsberg. Another of his doctoral students was Oskar Perron who studied under him in Munich.
Article by: J J O'Connor and E F Robertson