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Agner Erlang was descended on his mother's side from Thomas Fincke. His father was a schoolmaster and Erlang was educated at his fathers' school when he was young. He took his examinations in Copenhagen at the age of 14 and passed with special distinction after having to obtain special permission to take the examinations because he was below the minimum age.
He returned to Lonberg and taught at his father's school for two years. In 1896 he passed the entrance examination to the University of Copenhagen with distinction and, since his parents were poor, he was given free board and lodgings in a College of the University of Copenhagen.
His studies at Copenhagen were in mathematics and natural science. He attended the mathematics lectures of Zeuthen and Juel and these gave him an interest in geometrical problems which were to remain with him all his life.
After graduating in 1901 with mathematics as his major subject and physics, astronomy and chemistry as secondary subjects, he taught in schools for several years. During this time he kept up his interest in mathematics, and he received an award for an essay on Huygens' solution of infinitesimal problems which he submitted to the University of Copenhagen.
His interests turned towards the theory of probability and he kept up his mathematical interests by joining the Mathematical Association. At meetings of the Mathematical Association he met Jensen who was then the chief engineer at the Copenhagen Telephone Company. He persuaded Erlang to apply his skills to the solution of problems which arose from a study of waiting times for telephone calls.
In 1908 Erlang joined the Copenhagen Telephone Company and began applying probability to various problems arising in the context of telephone calls. He published his first paper on these problems The theory of probability and telephone conversations in 1909. In 1917 he gave a formula for loss and waiting time which was soon used by telephone companies in many countries including the British Post Office.
In addition to his work on probability Erlang was also interested in mathematical tables. This interest is described in [1]:
A subject that interested Erlang very much was the calculation and arrangement of numerical tables of mathematical functions, and he had an uncommonly thorough knowledge of the history of mathematical tables from ancient times right up to the present. Erlang set forth a new principle for the calculation of certain forms of mathematical tables, especially tables of logarithms...References (2 books/articles)
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An article on Erlang and his work is in PASS magazine at Cambridge, UK