He was born in Bridgend, Glamorgan, Wales, son to physician William Morgan and Sarah (sister of Richard Price).
At eighteen he received medical training at Guy's Hospital, London, working also as an apothecary to pay his way.
He was not popular with his father's patients: they thought him inexperienced and they resented receiving treatment from someone with a deformity—Morgan suffered from a club foot.
After his father's death he left medicine and in 1774, on the recommendation of his mother's brother, the renowned Richard Price, he was appointed Assistant Actuary of the Equitable Life Assurance Society.
[5][6] Later in life, through his uncle, Richard Price, he became friends with noted radicals, including Tom Paine and Francis Burdett.