His clerical prospects being very small, he went to Edinburgh to study medicine, and boarded with Thomas Blacklock.

In 1768 he published ‘The Land of the Muses; a poem in the manner of Spenser, by H. D.’ In 1769 he visited London for hospital practice, and in 1770, after proceeding M.A.

He also published ‘Poems to Thespia,’ Exeter, 1781, 8vo, and ‘The Death Song of Ragnar Lodbrach,’ translated from the Latin of Olaus Wormius, London, 1781, 4to.

In 1791 he published ‘Poems,’ second edition, London, 8vo, comprising the ‘Land of the Muses’ (with a second version) and ‘Ragnar Lodbrach.’ He was also contributor to Richard Polwhele's ‘Collections of the Poetry of Devon and Cornwall.’ Downman seems to have resumed medical practice at Exeter about 1790, and in 1796 he founded there a literary society of twelve members.

Downman wrote the opening address, and essays on ‘Serpent Worship,’ on the ‘Shields of Hercules and Achilles,’ and on ‘Pindar,’ with a translation of the 11th Pythian and 2nd Isthmian odes.

Two years before he died an anonymous editor collected and published the various critical opinions and complimentary verses on his poems, Isaac D'Israeli's (1792) being among them.