There was only one obvious next step for research - the excellent http://www.scotlandspeople.gov.uk/ - the genealogical website of the General Register Office of Scotland. Searching the pre-1855 Old Parish Records enabled me to trace the Donaldson line back to Samuel Donaldson, merchant who married in Leith in 1759.
Research of the Register of Sasines for Edinburghshire in the National Archives of Scotland brought to light property transactions of Samuel Donaldson of Leith, 1770-1785. [An instrument of sasine is a Scottish legal document recording the transfer of ownership of a piece of land or property]
Samuel's descendants included his youngest son Robert, grandson Robert who went from South Leith to South Shields, and his son another Robert who moved to Portsmouth - the linking factor the sea, with family occupations as a merchant, master mariner, seaman, roper, ship's carpenter, caulker and river policeman.
A story of mariners and miners was revealed in sidelines of the family:
The White family of mariner origins with the name Matthew Iley White recurring down the generations.
The Hibbert family of coal miners from Derbyshire, Yorkshire and South Shields.
The Hawkyard family with a lodging house in Alnwick, Northumberland.