[2] He attended the High School of Dundee before reading classics at the University of Edinburgh; studying under Arthur Berriedale Keith, he took papers in Sanskrit and graduated with a first-class MA degree in 1939.
[3] In 1946, Brough left his post at the British Museum and was appointed to a lectureship in Sanskrit at the School of Oriental and African Studies in the University of London.
[6] Brough published Selections from Classical Sanskrit Literature (1951), The Early Brahmanical System of Gotra and Pravara (1953), The Gāndhārī Dharmapada (1962) and Il Regno di Shan-Shan: Una Tappa nel Viaggio del Buddhismo dall'India alla Cina (1965).
In 1967, he left his chair in London to take up the Professorship of Sanskrit at the University of Cambridge, which he held with a professorial fellowship at St John's College.
[9] During this time, the advent of new courses on modern Indian languages led to fewer students taking Sanskrit; the university took the decision that Brough's chair would lapse after his appointment ended, and so he became its last holder.