[1] Bugge had at the same time under Christen Hees' (1712–1782) instruction studied mathematics, a subject he had shown a remarkable talent for.

He continued his studies of mathematics and physics and began working as an assistant for Christian Horrebow (1718–1776) at the Rundetårn Observatory.

Following Kofoed's death, Bugge was hired as geographical surveyor by the Royal Danish Academy of Science.

In 1765, he was appointed as trigonometrical observer and head of the Office of Surveying, where in that capacity, he carried out a triangulation of most of Zealand and Øresund and determined the polar height of a number of trig points.

Bugge became a member of the Royal Danish Academy of Science in 1775, further increasing his influence on its work with surveying of the country, and from 1780 until his death headed it alone.

In 1774, he and professor C. C. Lous [da] created tables for the General Widows' Pension Fund (Den Almindelige Enkekasse) based on Leonhard Euler's formulas.

A description of the observatory and observations from the three first years can be found in Bugge's privately published Observationes astron.

On Bugge's initiative, a series of small observatories were also established in Norway, Iceland, Greenland and Tranquebar with Abraham Pihl, Erasmus Lievog, Andreas Ginge and Engelhart as the first observers.

He managed to save the original drawings and copper plates as well as the associated journals and calculations for the special maps of Denmark.

He was a foreign member of the scientific academies in Petersborg, Pisa, London, Stockholm, Mannheim, Haarlem and Trondheim.