Herbert Westren Turnbull: Fourteen Ways Up the Maiden Rock
In 1932, rummaging through the library at St Andrews, a professor of mathematics found letters that John Collins had written to James Gregory two and a half centuries earlier. What they revealed was that Gregory had made remarkable discoveries in the calculus and the theory of numbers and had never published a word of them. Herbert Westren Turnbull spent much of the rest of his life on the reading of such letters — and he taught his students to climb sea cliffs while he was at it.
The Inspector's Son
Turnbull was born on 31 August 1885 in Tettenhall, Wolverhampton, the eldest of six. His father, William Peveril Turnbull, was one of Her Majesty's Inspectors of Schools and a man with genuine mathematical interests, and the enthusiasm passed straight down. Herbert went to Sheffield Grammar School and then to Trinity College, Cambridge, where he came out Second Wrangler in the Mathematical Tripos and took the Smith's Prize in 1909.
Second Wrangler was, in the Cambridge of that era, a national result — and also, forever after, a reminder that someone had been first.
A Career That Went the Long Way Round
What followed was not the smooth ascent the Tripos usually purchased. Turnbull lectured at St Catharine's College, Cambridge in 1909, moved to Liverpool for 1910–11, and then went to Hong Kong. From 1911 to 1915 he was master at St Stephen's College and warden of the university hostel there, lecturing in mathematics on the side. He married Ella Drummond Williamson in 1911; they had a daughter.
Back in England, he taught mathematics at Repton School from 1915 to 1918 and then served as a schools inspector, following his father's trade, until 1921. He held the Fereday Fellowship at St John's College, Oxford from 1919.
Then, in 1921, he was appointed Regius Professor of Mathematics at the University of St Andrews — a chair he would hold for twenty-nine years.
The Last of the Classical Algebraists
Turnbull was, in the London Mathematical Society's phrase, "a mathematician of the classical school." His subject was invariant theory, and specifically the symbolical calculus of Clebsch and Aronhold, which he loved with an enthusiasm that never dimmed.
The trouble was timing. His approach, as an assessor put it, was "concrete and formal in the sense that he sought to solve problems by an effective formalism rather than by a conceptual analysis" — and during the 1920s the fashion in algebra swung hard the other way, towards exactly the conceptual, structural methods his formalism was not. The verdict of his colleagues was blunt and a little sad: "his original work on invariants did not receive the recognition which it would have found two decades earlier."
He had better luck as an expositor. *The Theory of Determinants, Matrices, and Invariants* (1928) was judged one of the most readable textbooks on its subjects, and Turnbull deserves specific credit for introducing matrix theory into the British undergraduate curriculum at a time when it was almost never taught. *Theory of Equations* followed in 1939, and in 1945 *An Introduction to the Theory of Canonical Matrices*, written with A. C. Aitken.
Newton, Gregory, and the Historian's Turn
Turnbull's other gift was for the history of his subject, and here he was ahead of his time rather than behind it. *The Great Mathematicians* (1929) laid out his purpose exactly: "I have tried to show how a mathematician thinks, how his imagination, as well as his reason, leads him to new aspects of the truth." The Gregory discovery of 1932 fed the *James Gregory Tercentenary Volume* of 1938. *The Mathematical Discoveries of Newton* appeared in 1945.
After retiring in 1950 he took on the largest job of his life: editing *The Correspondence of Isaac Newton* for the Royal Society. He completed the first volumes of what became a seven-volume edition before his death — the foundation of everything modern scholarship knows about how Newton actually argued with his contemporaries.
The Rope and the Piano
Turnbull was an accomplished pianist who played in chamber orchestras; he and Ella performed two-piano duets described as "highly musical and exquisitely blended," and their home was open to a constant stream of students and friends.
He was also a serious mountaineer — an Alpine Club member who made unguided ascents, president of the Scottish Mountaineering Club from 1948 to 1950, and the man who worked out fourteen distinct routes up the Maiden Rock near St Andrews. He taught climbing on those local cliffs to the same students he taught matrices to, which tells you something about the kind of professor he was.
Why Herbert Is Called a Genius
The case, made honestly, is not about raw power. Turnbull was Second Wrangler and a Smith's Prizeman — formidable, but not singular. His research programme was overtaken; the LMS obituary says so directly.
What he had was a rarer and more specific gift: he could read mathematics as a human act. Where most mathematicians see a finished proof, Turnbull saw the mind that had been groping towards it, and he could reconstruct that groping from fragments — which is what let him look at three-hundred-year-old letters in a Scottish library and recognise unpublished calculus. The stated aim of *The Great Mathematicians*, to show how imagination as well as reason leads to new truth, is a genuine intellectual thesis, and it made him one of the people who turned the history of mathematics from anecdote into scholarship. The Newton edition is the proof: it required simultaneous command of seventeenth-century mathematics, seventeenth-century handwriting, and seventeenth-century personality.
The honest counter-case is that no theorem bears his name and no result of his changed the direction of algebra. He was a great teacher, a great editor and a great historian who happened to be a merely good original researcher — and much of what we admire is craft and stamina rather than lightning. Walter Ledermann, who knew him, reached for other words entirely: "inexhaustible kindness and patience." Another colleague simply said "he was a lovely man." That is what people who worked beside him actually testify to, and it would be dishonest to overwrite it.
Legacy
Turnbull was elected to the Royal Society of Edinburgh in 1922 and the Royal Society in 1932, and won the RSE's Keith Medal and Gunning Victoria Jubilee Prize. He was three times president of the Edinburgh Mathematical Society — in 1926, 1933 and 1950 — and was made an honorary fellow in 1954.
He died on 4 May 1961 at Grasmere, in the Lake District, among the hills. In his lectures he had insisted, as the LMS obituary records, "that no mathematical jargon was permitted to mar the purity of English prose." The Newton correspondence, completed by others, remains the standard edition. Both are the same legacy, viewed from different angles: a conviction that mathematics is written by people, for people, and ought to read that way.
Achievements
- Fellow of the Royal Society — 1932
- Keith Medal — 1923
- Fellow of the Royal Society of Edinburgh — 1922
- Smith's Prize — 1909
- Gunning Victoria Jubilee Prize — 1940
- Held posts at St Catharine's College, University of Liverpool and University of Hong Kong
