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William Cooke (d. 14 May 1589), who married Frances Grey, daughter of Lord John Grey of Pirgo, by whom he had four sons, including William Cooke of Highnam, Gloucestershire, who married Joyce Lucy, grand-daughter of Sir Thomas Lucy of Charlecote, and three daughters.
Gregory, born in London on 28 December 1776, was the fifth son of Benjamin Way (1740–1808), FRS, of Denham Place, Buckinghamshire, MP for Bridport in 1765, and of his wife Elizabeth Anne (1746–1825), eldest daughter of William Cooke, provost of King's College, Cambridge.