English mathematician who made contributions to algebraic invariant theory and to the history of mathematics.

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 1932 he was elected a fellow of the Royal Society.

Turnbull's work on invariant theory built on the symbolic methods of the German mathematicians Rudolf Clebsch (1833-1872) and Paul Gordan (1837-1912). His major works include The Theory of Determinants, Matrices, and Invariants (1928), The Great Mathematicians (1929), Theory of Equations (1939), The Mathematical Discoveries of Newton (1945), and An Introduction to the Theory of Canonical Matrices (1945), which was cowritten with A.C. Aitken. He edited the first three volumes of The Correspondence of Isaac Newton (1959–1961).