Rouet became Professor of Oriental Languages at Glasgow in 1751, and Professor of Church History in 1752.
In 1741, as a young clergyman, he became a travelling tutor to John Maxwell of Pollock. Later on, in 1759 he became tutor to Lord Hope, an appointment which eventually led to his resignation from the University in 1761 or 1762. The University refused to grant his request of a leave of absence while he travelled abroad with the young Lord - when he went anyway, he was deprived of his office by a majority in the Senate of the University. Rouet's dismissal, however, was not acknowledged by the crown, but in the end, he chose himself to resign.1
On his return from the Grand Tour he “purchased estate of Auchendeneran on the west shore of Loch Lomond, the name of which, from his Italian predilections, he changed to Bel Ritiro: a learned and highly accomplished man living much both in fashionable and literary society”.2
Rouet was even a cousin to Baron Mure and a friend of David Hume.
Boswell met Rouet and Lord Hope at Brunswick on June 27, 1764. On June 29 he “talked a good deal [with Rouet]. He was a sensible forward fellow.”.