features Westward Ho! Musing on Mathematics and Mechanics Alan Champneys CMath FIMA, University of Bristol In the latest feature from Bristol and the again. The hotel became a success and vil- West Country, Alan Champneys considers las were erected in the 75 acre estate sur- the life and work of the somewhat elusive rounding it. As further development took 20th century mathematician G.N. Watson place, the expanding settlement soon ac- (1886–1965), probably the most important quired the name of Westward Ho! figure to be born in the unusual resort of The Royal North Devon Golf club, Westward Ho! which opened the same year as the ho- Watson will be known to many gen- tel, is the oldest in England and Wales. erations of students through his influen- Then, in 1874, the United Services Col- tial textbook on mathematical analysis co- lege was opened nearby. It boasts Rudyard authored with E.T. Whittaker. A punctil- Kipling among its famous former pupils, ious man, a computer before the age of and his story collection Stalky & Co is modern computers, he was the first curator based on his experiences there. Staff of Ramanujan’s notebooks including the fi- for the school, as well as for hotel, golf nal ‘lost’ one. Watson’s legacy is most course and other attractions soon brought keenly felt at the University of Birming- permanent residents – they would have ham, where he spent most of his career. otherwise found it hard to commute via That legacy is viewed here through the | Dreamstime.com © Elliottphotos the steep track from Bideford. Holy lens of another, less famous, Birmingham Trinity Church was built in 1869, soon mathematician, David Wishart (1928–2003), a polymath, racon- followed by a Baptist church and several pubs. More growth in teur and amateur printer. the tourist business came with the coming of a railway line to We shall also consider Watson’s most famous result, a much- Bideford in 1901. Westward Ho! gained the status of a separate used lemma that bears his name, and in so doing ponder on the village with all the facilities one might expect. nature and importance of such ‘elementary’ mathematics. The exclamation mark is an intentional part of the village name. It is the only official place name in the British Isles to con- ‘Elementary my dear Watson’ tain such a punctuation mark. However it is not a world record. That goes to a small Canadian hamlet which has, not one, but he Devon coastal resort of Westward Ho! is noted for its two (!!) exclamation marks as part of its name. Saint-Louis- unusual name, a name that actually arises from a market- du-Ha! Ha!, just South of the St Lawrence estuary in Quebec is T ing ploy. It is named after the title of Charles Kingsley’s thought to get its name from the haha – from an archaic French 1855 novel [1] which is set in a fictional ocean facing quay near word for obstacle – formed by the nearby lake Témiscouata which the town of Bideford. The book was a blockbuster and the 5th Earl is a major obstacle to local travel. of Portsmouth, with the unlikely name of Isaac Newton Wallop, Westward Ho!’s most famous mathematical son, George saw the opportunity to cash in. In 1863 he formed the Northam Neville Watson was born on 31 January 1886 in the village. He Burrows Hotel and Villa Building Company and sought investors should not be confused with British number theorist G.L. Watson with the express purpose of opening a ‘family hotel’: (1909–1988) or Australian statistician G.S. Watson (1921–1998) The want of such accommodation has long been felt, both of whom made strong contributions to 20th century math- and as no attempt to supply it has hitherto been made ematics. At the time of his birth G.N.’s father, George Went- by individuals .... The salubrity and beauty of the worth Watson, was a teacher at the United Services School. North of Devon have long been known and appre- G.W. Watson would later go on to play a large role in the pub- ciated ... the recent publication of Professor Kings- lication of The Complete Peerage, a 13-volume encyclopedia of ley’s ‘Westward Ho’ [sic] has excited increased pub- the British peerage, generally accepted as the greatest British lic attention to the western part, more especially, of achievement in the field of genealogy from the pre-digital age. this romantic and beautiful coast. Nothing but a want Neville Watson, as the younger George Watson came to be of accommodation for visitors has hitherto prevented known, was educated at St Paul’s School from where he won a its being the resort of families seeking the advan- scholarship to study maths at Trinity College Cambridge in 1904. tages of sea bathing, combined with the invigorating He was Senior Wrangler in 1907, and gained a fellowship at Trin- breezes of the Atlantic ...[2]. ity in 1910. After a brief spell as an assistant professor at Uni- The foundation stone was laid in February 1864 and The versity College London for the duration of the First World War, Royal Hotel was completed in time for the 1865 season. Charles Watson was appointed in 1918 as the Mason Professor of Mathe- Kingsley himself was invited to come and open it. At first he matics at the University of Birmingham. He was elected a Fellow declined, believing that the development of the area would ruin of the Royal Society in 1919. In 1925 he married Elfrida Gwenfil its rugged beauty, so a quick change of name was undertaken. Lane, they set up home south of Birmingham and had one son, The hotel became The Westward Ho! Hotel. Charles Kingsley also called George. Watson remained at the University of Birm- attended, took his cheque and it is said never came near the place ingham for the rest of his career, taking retirement in 1951. Mathematics TODAY FEBRUARY 2018 18 The Francis Frith Collection Frith Francis The © Simon Greetham | Alamy Stock Photo © Figure 1: Modern day Westward Ho! (left) and in 1899 (right) with the hotel in the foreground. Generations of students, from the 1920s up to the beginning share mathematical interests, but it would seem they had similar of 21st century, will have become acquainted with G.N. Watson personalities. Both were meticulous – what we might call ‘details through the mathematical analysis textbook that bears his name, men’ in modern parlance – and humble, with a generous spirit, together with that of his collaborator and erstwhile teacher Ed- wishing to make a contribution to the world rather than seek glory mund Taylor Whittaker. Published in 1915, ‘A course on Math- for themselves. They were throwbacks to an earlier era, sharing ematical Analysis: an introduction to the theory of infinite pro- a dislike for telephones, a love of public transport, with a conse- cesses and analytic functions: with an account of the principal quent hatred of cars, and a strong interest in the history of their transcendental functions’ [3], to give it its full title, E.T. Whit- subject. Both remained devoted to Trinity throughout their lives, taker and G.N. Watson was actually a much expanded second edi- with the walls of the Watson family home being adorned with a tion of a 1902 first edition in Whittaker’s name alone. Third and collection of prints of the college and famous Trinity men. fourth editions were published in 1920 and 1927, with numerous Whittaker & Watson [3] was followed by a sole authored re-printings. In fact it remains in print today. book, considered by Watson himself to be his greatest achieve- The book is completely rigorous, in keeping with the tradi- ment, A Treatise on the Theory of Bessel Functions (1922) [5]. tions of Cambridge analysis. Yet, for its time, it takes a more The book has two stated aims; first, to introduce students to prac- modern pedagogical view, written primarily as a study aid for the tical applications of complex variable theory and, second, to com- diligent reader. It has numerous exercises – many taken from the pile everything that is known about the theory of Bessel func- world famous Cambridge Mathematical Tripos. The book’s in- tions and their application across mathematics and physics. A fluence also goes well beyond the English speaking world, with remarkable feature is that, after more than 600 pages of detailed evidence to suggest it was a prime study aid of the French Bour- mathematical argument, an appendix contains almost 90 pages baki movement [4]. of tabulated numerical values of various different Bessel func- tions, which Watson calculated by hand. Watson was a master calculator, his primary expertise being in the theory and practical application of asymptotic expansions. To the modern mathematician, it is easy to forget the impor- tance of the families of ‘special’ mathematical functions such as the Bessel functions. In the pre-computer age, the aim of the ap- plied mathematician was often to approximate and reduce prob- lems to integrals or differential equations that can be expressed in certain canonical forms. The solution of each different form is then expressed in terms of these special functions. For exam- Alamy Stock Photo Stock Alamy ple, the order-α Bessel function of the first kind Jα(x) can be | expressed as the solution of the differential equation d2y dy x2 + x +(x2 α2)y =0, dx2 dx − World History Archive World © or, for integer values α, via the integral π Figure 2: G.N. Watson and the front page of his textbook with 1 J (x)= cos(nτ x sin(τ)) dτ.