# William Leonard Edge

### Quick Info

Born
8 November 1904
Stockport, England
Died
27 September 1997
Bonnyrigg, Scotland

Summary
William Edge graduated from Cambridge and lectured at Edinburgh University. He wrote many papers on Geometry.

### Biography

William Edge's parents were both schoolteachers. He was educated at his local school, Stockport Grammar School, and from there he went to Cambridge where he studied mathematics at Trinity College. After graduating, he continued working for his doctorate at Trinity on projective geometry. Cambridge was at that time a centre for geometry research with Baker's school flourishing there. Edge's fellow students included P du Val and J G Semple but other famous geometers joined the group while Edge was at Cambridge including the slightly younger men H S M Coxeter and J A Todd.

After holding a fellowship at Trinity he was offered a lectureship at the University of Edinburgh by E T Whittaker which he accepted and took up the post in 1932. Edge was to spend the rest of his career at Edinburgh and David Monk, writing in [2], suggests that the reason that Edge never moved to a chair in another university was because:-
... the Scottish hills and mountains, which he loved, kept him in Edinburgh.
Edge played a major role in the success of the Mathematics Department at Edinburgh, first under Whittaker's and then under Aitken's leadership. He formed a close friendship with both these men and supported their work with his high international reputation for research and his lecture courses packed with gems. He did not, however, find administration to his liking and he preferred to avoid this whenever possible.

After studying classical geometry, Edge moved towards the topic which is most associated with him, namely finite geometry. He had an amazing geometrical feel for complex situations as well as a skill at handling intricate combinatorial arguments which were characteristic of his work.

Edge wrote nearly 100 papers and his mastery of the area ranks him with Coxeter as one of the leading geometers of the 20th century. His work was a continuation of work started by the great geometers of the late 19th and early 20th centuries, in particular Castelnuovo, Cayley, Clebsch, Cremona, Fano, Fricke, Humbert, Klein, Plücker and Schläfli.

Georges Humbert discovered a plane sextic curve of genus 5 having five cusps for its singular points. These have interesting geometrical properties and Edge investigated them in a series of papers spanning 40 years. In 1890 Castelnuovo studied and classified algebraic surfaces with hyperelliptic prime sections. Edge continued and completed Castelnuovo's investigations. Castelnuovo proved that a non-ruled surface whose prime sections have genus 2 is the projection of a non-singular rational surface of order 12 in projective 11-space. Edge explicitly examined one such projection in a paper on Castelnuovo's normal surface.

The equation of the scroll of tangents of the common curve of two quadrics is due to Cayley in 1850. Salmon, in his famous text, gave an equation in covariant form. Edge gave a procedure for finding this equation in 1979. Bring's curve was first studied in Klein's 1884 book in connection with the transformation to reduce the general quintic equation to the form $x^{5} + ex + f = 0$. Some of Edge's work on Bring's curve extends work due to Clebsch.

Edge investigated a pencil of canonical curves of genus 6 on a del Pezzo quintic surface in a 5-dimensional projective space. He investigated the group of self-projectivities of the space, which is isomorphic to the symmetric group $S_{5}$. He also used geometrical configurations to investigate groups and, although his work was out of fashion at a time when group theorists were moving towards the classification of finite simple groups, his work did provide a deeper understanding of some of these groups, for example Conway's simple groups. Edge was not someone uninterested in modern techniques, however, and it may come as a surprise to some that in a 1991 paper he included computer-drawn pictures.

Other topics Edge worked on, all of which exhibit his mastery of the subject, include nets of quadric surfaces, the geometry of the Veronese surface, Klein's quartic, Maschke's quartic surfaces, Kummer's quartic, the Kummer surface, Weddle surfaces, Fricke's octavic curve, the geometry of certain groups, finite planes and permutation representations of groups arising from geometry.

His papers are almost all written as single author papers but he did collaborate with his friends Coxeter and Du Val. In fact when he attended the celebrations for Coxeter in Toronto in 1979 it was the first time Edge had crossed the Atlantic and he said only his great friendship with Coxeter had made him overcome his reluctance to travel.

I [EFR] asked Edge a few years ago if he would come to St Andrews and give a talk on the history of mathematics. He said he knew nothing of the history of mathematics. I did not give up that easily and asked him if he would not speak on Cayley's mathematics. "Never met Cayley" replied Edge. He paused for a second before adding "Knew his landlady though".

For many years Edge was someone I [EFR] expected to see whenever I went to the Edinburgh University Staff Club. For some reason I never quite understood, there was frequently a note on the blackboard at the entrance to the club saying there was a message for W L Edge. He was a tall straight man with an imposing figure, certainly someone who one noticed. Typically he wore a green corduroy jacket and his hair blew about in an uncontrolled manner.

A colleague now at St Andrews, C M Campbell, attended Edge's courses in the 1960s. He described them as difficult lectures which required a lot of work to appreciate their content but, once this work had been put in, the quality and insight in Edge's lectures became apparent. Edge taught the algebra courses at Edinburgh at this time but he taught algebra with a strong geometric flavour reflecting his deep knowledge, feel and love for geometry.

Edge had a deep concern for his students, both while they were studying at Edinburgh and after they had graduated. He kept in touch with these students in many different ways including sending his best wishes when he saw a notice of marriage in the press.

Monk [2] describes Edge's lifestyle and interests outside mathematics as follows:-
Edge never married. He lived in a succession of lodgings, carefully chosen for the quality of the cooking and space for a piano. Music was an abiding interest and he had a fine singing as well as a sonorous speaking voice.
More details of his music are given in [1] (and have been described to us in similar terms by Ledermann):-
Apart from mathematics his great loves were walking and music, and his lodgings had always to accommodate a grand piano. Together with Aitken (violin), Walter Ledermann (viola) and Robin Schlapp (cello), he formed the "mathematical quartet".

They performed in particular on the first Friday of each month that the Edinburgh Mathematical Society met. There was always a dinner for the speaker at Whittaker's house, and Whittaker, who hated small-talk, would say after dinner, "Edge, would you care to perform?"

The quartet alternated between Mozart's G minor and his E flat (his only two piano quartets), and played nothing else on these occasions. Edge was also a capable singer, and performed the solo in a Bach cantata for participants at one of the St Andrews colloquia ...

### References (show)

1. Obituary in The Times
2. D Monk, Professor W L Edge, The Scotsman (Edinburgh, 4 Oct 1997).