**William Berwick**'s father, William Edward Berwick (born in Shipley, Yorkshire about 1843), was a wool merchant in Bradford. His mother was Mary Hodgson who was born in Wrose Windhill, Yorkshire and the pair were married on 31^{st} May 1881. William had had an older sister Mary (born about 1884).

William Berwick was educated at a small private school before entering Bradford Grammar School where he soon showed his mathematical potential. He completed his schooling in 1906 and was awarded a Brown Scholarship from the Grammar School to assist in supporting him during his university studies. He was also awarded an entrance scholarship by Clare College, Cambridge, and he studied there for the Mathematical Tripos.

In 1909 he took Part I of the Tripos and was placed as joint fourth Wrangler. This means that he was placed fourth in the ranked list of those who were awarded a first Class degree. It is interesting to look at Berwick's fellow students, particularly those who were ranked above him in the Tripos. In first place was P J Daniell, who had been coached by R A Herman, while E H Neville, who was also coached by Herman, was placed second. Neville went on to become a colleague of Hardy's at Trinity College and it was Neville who met with Ramanujan while lecturing in India and made his visit to England possible. Daniell went on to become professor of mathematics at the University of Sheffield, but the most famous of Berwick's fellow students was L J Mordell. It was during his undergraduate years at Cambridge that Berwick became interested in number theory, and in this he was particularly influenced by G B Mathews who lectured at Cambridge. Berwick sat Part II of the Mathematical Tripos in 1910, then submitted an essay entitled *An illustration of the theory of relative corpora* for the Smith's Prize in the following year. He was awarded the second Smith's Prize and, also in 1911, his first paper was published. This paper, *On the reduction of arithmetical binary cubics which have a negative determinant*, was written jointly with Mathews and it was in fact the only paper Berwick jointly authored throughout his career.

By the time that Berwick's first paper was published he had left Cambridge to take up an assistant lectureship at the University of Bristol. He held this post until 1913 when he moved to a lectureship at University College Bangor. World War I broke out in 1914 and lasted four years. For two of these four years Berwick undertook valuable war service on the Technical Staff of the Anti-Aircraft Experimental Section of the Munitions Inventions Department at Portsmouth. During session 1919-20 the head of the mathematics department at Bangor was absent and Berwick was acting Head of Department. Following this he moved to the University of Leeds as a lecturer but, in 1921, he was promoted to Reader in Mathematical Analysis there. In the same year he was elected to a fellowship at Clare College, Cambridge.

While Berwick was in Leeds he married Daisy May Thomas in 1923. She was from Sheffield and the daughter of Dr W R Thomas. Two years later Berwick received another recognition of his mathematical distinction when he was awarded a Sc.D. by the University of Cambridge in 1925. The Chair of Mathematics at Bangor fell vacant and in 1926 Berwick was appointed to the post. Berwick wrote only 13 research papers, a monograph, and a number of other articles on mathematical recreations and puzzles. Ill health prevented him from undertaking more extensive work and in fact he only published five papers after his appointment to Bangor. Davenport writes in [2]:-

The years immediately preceding and following[his appointment to Bangor]constitute his most productive period. It was not long, however, before his health began to deteriorate, and at the height of his powers he fell victim to a long and progressive illness, which he bore with great fortitude and patience. He continued to carry out his lecturing and administrative duties as long as possible, and it was with much difficulty that he could be persuaded to delegate any part of them. By1940it became impossible for him to carry on any longer, and he resigned his chair.

Berwick was an algebraist who worked on the problem of computing an integral basis for the algebraic integers in a simple algebraic extension of the rationals. He also studied ideals in rings of algebraic integers. His book *Integral Bases* published in 1927 is a significant contribution described by Davenport as:-

... quite an ambitious undertaking.

The existence of such a basis can be easily proved but practical methods to compute such a basis are much harder. The heavy numerical computations involved in Berwick's work kept it outside the mainstream of algebraic number theory. As Davenport writes:-

... a great deal of courage and patience must have been needed for such a complex and laborious analysis.

In fact the work gained an importance which saw it republished in the 1960s.

Berwick also gave, in 1915, necessary and sufficient conditions for a quintic equation to be soluble by radicals. This, like nine of his thirteen papers, was published in the *Proceedings of the London Mathematical Society.*

In fact Berwick was a staunch supporter of the London Mathematical Society and served on the Council from 1925 to 1929, being Vice-President of the Society in 1929. His wishes were that a sum of money being given to the Society on his death to establish two prizes. His widow Daisy May Berwick presented the Society with the money and the Council decided to name the two prizes the Senior Berwick Prize and the Junior Berwick Prize. These prestigious prizes continue to be awarded by the London Mathematical Society.

We have a fine picture of Berwick through Davenport's words in [2]:-

In physical appearance Berwick was tall and impressive, and he had a clear and distinctive voice. He had many highly individual traits of character, and his conversation, even on the most commonplace subjects, had a pungent forthrightness which often startled his hearers. The workings of his mind continually took the most unexpected and original turns, which were a source of stimulation and delight to his friends. All those who appreciated the valuable qualities which lay behind his somewhat abrupt and reserved manner held him in high esteem.

His interest in teaching can be seen in the articles he published on mathematical recreations and also in the addresses he gave to several meetings of the British Association. His main hobby was chess and he enjoyed playing in the chess clubs of the various universities in which he worked.

Davenport sums up Berwick's mathematical papers as follows:-

Berwick's work does not make easy reading, though his exposition is clear and logical. He presumes in the reader a considerable background of knowledge - usually more than is suggested by the introductory paragraphs of his papers - and he was himself of course a master of all the classical theory of algebraic numbers.

**Article by:** *J J O'Connor* and *E F Robertson*

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