Edmondston's father for most of his long life was the only medical practitioner in the islands.
Arthur adopted his father's profession, entered the army, and served under Sir Ralph Abercromby in Egypt.
[2] Edmondston wrote two treatises on ophthalmia, published respectively in London, 1802, and Edinburgh, 1806.
His major work was his View of the Ancient and Present State of the Zetland Islands, published in 1809 in two volumes.
The book discusses the political and natural history of Shetland, its agriculture, fisheries, commerce, antiquities, and customs.