He was born in the
Tettenhall district, on the outskirts of
Wolverhampton on 31 August 1885, the eldest of five sons of William Peveril Turnbull, HM Inspector of Schools. He was educated at
Sheffield Grammar School then studied Mathematics at
Cambridge University graduating MA. After serving as lecturer at St. Catharine's College, Cambridge (1909), the University of Liverpool (1910), and the University of Hong Kong (1912), Turnbull became master at St. Stephen's College in Hong Kong (1911–15), and warden of the University Hostel (1913–15). He was a Fellow at
St John's College, Oxford (1919–26), and from 1921 held a chair of mathematics at the
University of St Andrews. In 1922, he was elected a Fellow of the
Royal Society of Edinburgh. His proposers were
Arthur Crichton Mitchell, Sir
Edmund Taylor Whittaker,
Cargill Gilston Knott, and
Herbert Stanley Allen. He won the Society's
Keith Prize for 1923-25 and the
Gunning Victoria Jubilee Prize for 1940–1944. In 1932, he was elected a
Fellow of the Royal Society. He was a keen mountain climber and served as President of the
Scottish Mountaineering Club from 1948 to 1950. He retired in 1950 and died at
Grasmere in the
Lake District on 4 May 1961. ==Family==