As nearly as I can determine, Thomas Clausen learned mathematics at home. It is mentioned that he was a trainee at the Munich Optical Institute in 1820. This was very important because by 1818 Joseph von Fraunhofer had become the director of the Optical Institute. Due to the fine optical instruments von Fraunhofer had developed, Bavaria overtook England as the centre of the optics industry. Even the famed Michael Faraday was unable to produce glass that could rival von Fraunhofer’s. It is not clear, so as to speak, that even today finer glass than von Fraunhofer’s can be produced. Unfortunately, von Fraunhofer was doomed to die young at age 39 in 1826 of heavy metal poisoning from the vapors used to dope the glass.
In 1824 Clausen was mentioned as being at the observatory at Altona near Hamburg. Heinrich Christian Schumacher (September 3, 1780 – December 28, 1850) was professor of astronomy at Copenhagen and head of the geodetic survey of Denmark. He was recorded as being very favorably impressed by Clausen’s paper on calculating longitude by the occultation of stars by the moon. I am not sure what went on, but Clausen’s biography speaks of him “eventually” returning to Munich and working there in 1842 when he took a position at the observatory at Tartu Observatory. He rose to become director of the Observatory from 1866-1872. I cannot tell if he retired (age 71), or merely continued but not as director.
In more than 4000 years since the Pyramids at Giza the efforts of thousands of mathematicians had, by 1400, only gotten pi correct to 10 decimal places. To be fair, it had taken almost 1000 years to get the 8th, 9th and tenth paces correct: the famed Chinese mathematician Zu Chongzhi (429–500 CE; traditional: 祖沖之; simplified: 祖冲之) had published pi to 7 places in around 480. A great leap forward of sorts happened when the English mathematician John Machin (1686? – June 9, 1751) calculated pi to 100 decimal places in 1706. In 1844 Johann Martin Zacharias Dase (June 23, 1824, Hamburg – September 11, 1861, Hamburg), the German calculating savant, took two months to calculate pi to 205 places – 200 of which were correct. Three years later Thomas Clausen published his own results which were correct to 248 places – he had been aiming for 250 places. As of this writing the current record is 13,300,000,000,000 (13.3 trillion) decimal places set in 2014 by houkouonchi (pseudonym).