Georg Klügel's father was a businessman. Klügel attended the Johanneum Grammar School, the renowned humanistic school in Hamburg. From that school he progressed to the Hamburg Gymnasium Academicum where he received a solid mathematical education. He did not decide to follow a mathematical career at this stage, however, and he entered Göttingen University in 1760 with the intention of reading for a degree in theology.
At Göttingen University Klügel took a mathematics course as part of his degree and so he met Kästner who quickly saw his talents in mathematics. Klügel was fascinated by the topic and was soon ready to follow Kästner's advice and change his course to read for a degree in mathematics. Again following Kästner's advice he wrote a thesis on the parallel postulate. In this work he listed nearly 30 attempts to prove the fifth axiom and correctly concluded that the 'proofs' were all false. His work is cited by almost all later contributors to non-euclidean geometry. He defended his thesis on 20 August 1763 and after this he continued to undertake mathematical research in Göttingen.
Klügel remained in Göttingen until 1765 when he moved to Hannover to take up the appointment as editor of the Intelligenzblatt. In 1767 he was appointed professor of mathematics at Helmstedt then he moved to the chair of mathematics and physics at the University of Halle. He remained in this post for the rest of his career.
It was in Helmstedt and Halle that Klügel made his most important contributions to mathematics. These were somewhat of a mixture between encyclopaedic style accumulation of facts together with some real innovative ideas.
Klügel made an exceptional contribution to trigonometry, unifying formulae and introducing the concept of trigonometric function, in his Analytische Trigonometrie. Euler, who studied similar problems 9 years later, in some respects achieved less than Klügel in this area. Folta writes in :-
Klügel's trigonometry was very modern for its time and was exceptional among the contemporary textbooks.
It was his mathematical dictionary, however, which led to his fame. This was a three volume work which appeared between 1803 and 1808. In 1808 Klügel became seriously ill and could do no further work on the project. Another three volumes were added between 1823 and 1836 by Mollweide and Grunert and the dictionary was widely used for several generations making Klügel's name widely known.
Among the honours which Klügel received for his contributions to mathematics was election to the Berlin Academy which took place on 27 January 1803.
Article by: J J O'Connor and E F Robertson