Engel was born and died in Parchim, in the Duchy of Mecklenburg-Schwerin.
He studied theology at Rostock and Bützow, and philosophy at Leipzig, where he took his doctors" degree.
In 1776 he was appointed professor of moral philosophy and belles-lettres in the Joachimstal gymnasium at Berlin, and a few years later he became tutor to the crown prince of Prussia, afterwards Frederick William III. The lessons which he gave his royal pupil in ethics and politics were published in 1798 under the title Fürstenspiegel ("Mirror for Princes"), and are a favourable specimen of his powers as a popular philosophical writer
In 1787 he was admitted a member of the Academy of Sciences of Berlin, and in the same year he became director of the royal theatre, an office he resigned in 1794.