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After winning the Rayleigh Prize in 1929, Semple was appointed to a lecturing post at the University of Edinburgh. After holding this post for one year (1929-30) he was awarded his doctorate by Cambridge for a thesis on Cremona transformations, was elected a Fellow of St John's College, Cambridge, and, still in the same year 1930, was appointed to the Chair of Pure Mathematics at Queen's University, Belfast.

The Department of Mathematics at Queen's University flourished under Semple's leadership. He was a very active researcher, publishing nine important papers during six years. The topics he studied included representations of Grassmann manifolds on linear spaces, invariants of composite surfaces in higher space, and studies of the singularities forced on a surface under the condition that it has contact with a prescribed order with a given curve. In addition to his research, he started courses for secondary school teachers to allow them to keep them up-to-date with new mathematical developments. Administrative duties can take up much time and be particularly frustrating to young mathematicians keen to push forward their research. Semple, however, did not try to avoid such duties, rather he took on more than a reasonable share of them being Dean of the Faculty of Arts for three years and serving on the University Senate. He was honoured by election to the Royal Irish Academy in 1932 when still only 28 years old.

He remained at Belfast for six years before taking up the chair of Pure Mathematics at King's College, London, where he was to remain for the rest of his life. Before leaving Belfast, however, he married Daphne Hummel, who was the daughter of one of Semple's older colleagues. They had two children, John Semple who joined the medical profession, and Jessie Semple who worked in the art world.

In London he quickly became close friends with the Head of the Mathematics Department, George Temple (to have a Semple and a Temple in the same department must have been very confusing!). The two worked closely on the running of mathematics in London but, within three years of Semple taking up his chair, World War II broke out. In the early days of the War, London came under heavy bombing raids and a decision was taken to move London University Colleges to safer environments. The Mathematics Department of King's College was moved to Bristol, and this translation Semple had to organise since Temple was seconded to war work. These were difficult years and it was Semple's leadership that kept the department functioning and having it well placed to reopen in London in 1943.

Soon after King's College reopened Semple took on two major tasks for the London Mathematical Society , namely Secretary of the Society and Editor of the Journal of the London Mathematical Society. He held these positions from 1944 to 1947. During this period he began a collaboration with Roth and together they wrote the first of three famous texts which Semple was to co-author. Introduction to algebraic geometry was published in 1949. Zariski , in reviewing the work, praises two chapters particularly highly:

... chapter VII (Special rational surfaces and plane Cremona transformations) ... and also chapter VIII (Linear systems of surfaces, rational manifolds, and higher Cremona transformations), have a rich geometric content and a generous supply of special but highly interesting examples. A careful reading of these two chapters cannot be too highly recommended.

For more general comments on the work by Zariski , see the biography of Roth .

Roth and Semple also worked together setting up and running the London Geometry Seminar which operated for 40 years and provided one of the major focal points for geometry research throughout the world. Semple also worked with Du Val who joined the London Geometry Seminar but they only wrote one joint paper.

Semple's work was on various aspects of geometry, in particular work on Cremona transformations and work extending results of Severi . He wrote two famous texts Algebraic projective geometry (1952) and Algebraic curves (1959) jointly with G T Kneebone. In the Preface of the first edition of Algebraic projective geometry the authors explain their approach to geometry:

Projective geometry is a subject that lends itself naturally to algebraic treatment, and we have had no hesitation in developing it in this way - both because to do so affords a simple means of giving mathematical precision to intuitive geometrical concepts and arguments, and also because the extent to which algebra is now used in almost all branches of mathematics makes it reasonable to assume that the reader already possesses a working knowledge of its methods. ...

In spite, however, of treating geometry algebraically, we have tried never to lose sight of the synthetic approach perfected by such geometers as von Staudt , Steiner , and Reye . If one consideration has been more prominent in our minds that any other it is that of giving precedence to the geometrical content of the system and the geometrical way of thinking about it. Nothing, in our opinion, could be more undesirable than that this traditionally elegant subject should be allowed to take on the appearance of being merely a dressing-room in which algebra is decked out in geometrical phraseology.

In 1953, between the publication of these two books, Temple moved to a chair at Oxford and Semple became Head of Mathematics at King's College. Around this time he seemed to become somewhat disillusioned with the direction that research in algebraic geometry was going and 1957 saw the publication of the last research paper that Semple would write for over ten years. Administration, which had always been a major part of his life, now expanded to fill the time left free when he almost stopped research. This administration took him into a wider role in the University, well outside the work of his own Department. He served on the Academic Council, The University Court, and on the Senate. It was a time when he became particularly interested in the overseas colleges which were linked to the University of London, and he visited such colleges in Rhodesia, the West Indies, and Hong Kong.

If Semple rather regretted the direction that research in algebraic geometry was taking, he certainly did not show it in his book Algebraic curves. In the Preface to the book the authors argue convincingly for the importance of algebraic geometry:

... the authors observe that the subject of algebraic geometry is one to which mathematicians have been powerfully drawn over a very long period of time. One of the reasons of this enduring influence of algebraic geometry is the fact that the subject makes a strong appeal to the imagination in that it not only illuminates properties of geometrical figures that we are all able to draw or visualize but also extends the range of geometrical thinking far beyond the bounds of intuition, and one can think about it in a much deeper sense than that of granting formal assent to its conclusions.

His research interest reawakened after ten years when he began to read over his early papers. He suddenly felt that he wanted to look again at the problems he had studied more than 30 years earlier. The result was a series of fascinating papers and one further book Generalized Clifford parallelism (1971). Todd , in reviewing this book writes:

This tract contains a mass of detailed information ... . Its style is reminiscent of the 1930s rather than the 1970s (the reviewer intends this remark to be a compliment!) and is elegant throughout. It provides impressive evidence of the power of strictly classical projective geometry when applied to the right sort of problem.

Semple had retired from his chair at King's College two years before the publication of this book. His revived interest in research, however, meant that four research papers appeared in the three years 1968-70 written while he worked on his book on Clifford parallelism.

Tyrrell, who collaborated with Semple on four of his later works, described him in :

Foremost among the qualities for which he is remembered were his uprightness and determination, and the strong leadership that he was able to bring to projects of widely different sorts. In his own subject, he had in a high degree the gift of organising research, both for groups and for individuals; as a lecturer, he was much in demand and gave inspiration to many hundreds of students and other listeners; and, in his writing, he was able to pass on his love of geometry with infectious enthusiasm that few other authors have managed to achieve. [He] will be remembered ... for his good humour, his wisdom and the innumerable acts of kindness that he performed.

George Temple writes about Semple in :

He was a wonderful companion, and a great friend. He showed great courage and even cheerfulness when ill health compelled him to give up golf and concentrate on gardening. His haunting Irish accent was always a great charm.

Source:School of Mathematics and Statistics University of St Andrews, Scotland