# Dorothy Mary Elizabeth Foster

### Quick Info

Darlington, County Durham, England

St Andrews, Fife, Scotland

**Dorothy Foster**studied at Bedford College and was awarded a Ph.D. by the University of London in 1960. Except for a few years as an Assistant Lecturer in Mathematics at Royal Holloway College, London, she spent her whole career at the University of St Andrews. She was an expert on geometric number theory.

### Biography

**Dorothy Foster**was the only child of Thomas Sidney Foster, a bank clerk, and his wife Elizabeth Flintoff, a bank cashier. Sydney Foster (born 1902) had married Elizabeth Flintoff (born 1896) in 1932 in Darlington, County Durham. The family came from Darlington, County Durham and, with her cousin Joy Tabrum, was the direct descendent of the Dixon family of Raby, Cockfield and Newcastle-upon-Tyne. This came through Dorothy's maternal grandparents who were William Flintoff (1851-1903) and Mary Amy Dixon (1849-1914). The Dixon fortunes originated in the coal seams under Cockfield Fell, but the family included Jeremiah Dixon (1733-1779), an astronomer who surveyed part of the Mason-Dixon Line, and his brother George Dixon (1731-1796), who first lit a house with coal gas and produced coal tar. For more information about Jeremiah Dixon and George Dixon, see THIS LINK.

Other Cockfield family members were engineers who worked with George and Robert Stephenson on the Stockton to Darlington and Liverpool to Manchester Railways.

The Newcastle branch of the family also produced three noted engineers. John Dixon (1835-1891) and his brother Waynman designed a special ship, the Cleopatra, to transport the obelisk known as Cleopatra's Needle to London from Alexandria. Their brother, Raylton, owned and ran the Cleveland Dockyard, the largest boatyard on the Tees; he was Mayor of Middlesbrough and was knighted for services to shipbuilding. Of these three engineers, Waynman was a genial man and worked with both brothers, but John and Raylton were rather more combative and competitive. Joy Tabrum [1] suspects this was a common Dixon family trait.

Dorothy Foster grew up in Darlington, living at 15 Dale Road. She attended Polam Hall School from 1941 to 1952. This school had been founded by the Quakers in 1848 as a small finishing school for girls. The main School building was once the private mansion of Jonathan Backhouse, the founder of Barclays Bank. Dorothy excelled at Polam Hall School, consistently ranked 1st or 2nd in her class. The reports for her mathematics studies in the Autumn Term of 1951 are as follows [1]:-

**Pure Mathematics**: Dorothy is working very hard and her work is always of a high standard. Dorothey works independently and shows powers of selection when tackling problems. She should do well!

**Applied Mathematics**: Dorothy continues to do work of a high standard. She deserves to do very well.

*Scripture:**Always interested and most attentive.*

*English:**Style imaginative, but Dorothy worked extremely well in her coaching periods, and overcame many difficulties she had previously experienced.*

*Latin:**Dorothy is a keen and steady worker and deserves to do well in her examination.*

*Games:**Dorothy could be a really good tennis player if she gave more time to practice.*

*Form Mistress's Report:**Dorothy has devoted herself so wholeheartedly to her work that it has been a real pleasure to teach her; I hope she meets with the success she so well deserves.*

*General Conduct:**Dorothy has always maintained a high standard of conduct, and had set a good example to the school.*

Since you have passed in the required papers in Mathematics and Further Mathematics and in Latin at the Advanced Level, you are qualified to begin reading for a B.A. Honours degree in Mathematics on entering the College in October. A place has been reserved for you in Reid Hall.She began her studies at Bedford College on Tuesday, 7 October 1952, graduating with a B.Sc. in Mathematics in 1956. She received a letter dated 13 July 1955 congratulating her on being placed First Class in the examinations. It also informed her that she had been awarded a postgraduate studentship.

She described the Mathematics Department at Bedford College as being:-

... quite small in the 1950s and staff-student relations were informal and friendly. All our lecturers had a heavy teaching load, not uncommon at that time, so there was no provision for weekly tutorials.One of her lecturers at Bedford College was Geoffrey Kneebone. Dorothy wrote:-

Dr Kneebone, as well as giving excellent lectures, always managed to discuss our weekly written work with us individually. In our first year both he and Mrs Kneebone invited us to their home. On revisiting him on the occasion of the sesquicentenary celebrations of Bedford College in 1999 he was as friendly and welcoming as ever.Her first paper

*Indefinite quadratic polynomials in n variables*, published in 1956, gives her address as Bedford College. The research for this work must have been done soon after she sat her degree examinations. She was appointed as an Assistant Lecturer in Mathematics at Royal Holloway College, London in 1957. Her next paper was

*On a class of quadratic polynomials in n variables*(1958) written soon after she began working for a Ph.D. advised by John Henry H Chalk (1922-2004). She writes in this paper [4]:-

I am very grateful to Dr J H H Chalk for all his help and encouragement during my work on this problem.Dorothy's thesis was

*Some arithmetical properties of quadratic polynomials*(May 1960). She writes in the Preface [3]:-

The problems discussed in this thesis arise in the geometry of numbers, a part of the theory of numbers which originated under the inspiration of Minkowski at the turn of the last century.Wilfred Norman Bailey (1893-1961) was Head of Mathematics at Bedford College when Dorothy studied there, retiring in 1958 before she completed her research. In fact there had been a difficulty with Dorothy's Ph.D. studies for, in the middle of her work, her Ph.D. supervisor J H H Chalk had left Bedford College and moved to the University of Toronto in Canada. Supervision had been by letter over the final stages but she had overcome these difficulties seemingly with ease. The Vice-Chancellor, acting on behalf of the Senate, conferred the degree of Ph.D. in the Faculty of Arts (Theory of Numbers) on her 'in absentia' on 30 August 1960.

The first chapter is introductory and provides the background to the main part of the thesis. The well known theorems which are needed later are stated without proofs, reference being given in the appendix. In each of the remaining three chapters a different problem is considered and is preceded by a short summary. The theorems are numbered according to the chapter in which they occur.

The subject matter of this thesis has been suggested by my supervisor, D J H H Chalk. I am very deeply indebted to him for all his invaluable help and encouragement throughout its preparation.

... Finally I should like to thank Professor W N Bailey for suggesting the theory of numbers as a subject for research, and the University of London for their award of a postgraduate studentship for fifteen months.

In February 1960, even before she had submitted her thesis to Royal Holloway College, Dorothy Foster was appointed to the Department of Mathematics at the University of St Andrews. She received a letter from David M Devine, Joint Clerk to the University Court, written on 8 February 1960 [1]:-

I am writing to confirm the offer which was made to you verbally on Saturday of appointment as Lecturer in Mathematics in St Salvator's College in this University.The head of the Department of Mathematics was Edward Copson who was the Regius Professor of Mathematics. At this time St Andrews had two mathematics departments, the other being the Department of Applied Mathematics. In the second half of the 1960s the Department of Mathematics changed its name to the Department of Pure Mathematics. Dorothy continued as a member of Pure Mathematics throughout her career, although the Department later became the Division of Pure Mathematics in the School of Mathematics and Statistics.

She wrote 12 papers between 1956 and 1992, five of them co-authored with George Phillips, namely

*The arithmetic-harmonic mean*(1984);

*A generalization of the Archimedean double sequence*(1984);

*The approximation of certain functions by compound means*(1984),

*General compound means*(1985); and

*Double mean processes*(1986). Her final three single authored papers were:

*Estimates for a remainder term associated with the sum of digits function*(1987);

*A lower bound for a remainder term associated with the sum of digits function*(1991); and

*Averaging the sum of digits function to an even base*(1992).

As a lecturer Dorothy was always exceptionally well prepared, with well thought out lectures. When I [EFR] was a student attending her lectures in the first half of the 1960s it was still a chalk and blackboard time and Dorothy would write extremely fast, with seldom a pause. I attended a complex variable course she taught but the other one of her courses I attended was tragic. It began as a course on linear algebra given by Arthur Hinton Read who taught the material in a "linear transformations of vector spaces" style. In the middle of the course, one December Monday morning, Professor Copson came into the classroom and gave us the tragic news that Arthur Read had died in an avalanche while mountain climbing that weekend. Dorothy Foster then took over the linear algebra course and taught it in her style, namely matrices as boxes of numbers. Dorothy gave an outstanding advanced course on number theory. One student, Nigel Watt, studied Dorothy's number theory course in the 1980s and took her advice to carry on to study for a Ph.D. in number theory with Martin Neil Huxley at Cardiff University. Nigel Watt has had a highly successful university careeer with around 30 number theory publications.

Dorothy was the local organiser of the 1984 Edinburgh Mathematical Society Colloquium held in St Andrews. Werner Georg Nowak attended this conference and in a letter sent from Vienna to Dorothy Foster on 28 June 1988, he writes:-

I have to apologise deeply for being so terribly late in fulfilling my promise to send you some of my material (4 years is in fact an enormous delay!); all I can say as a sort of excuse is that, for some time, I was waiting till my work on the problems we discussed in St Andrews (divisors in arithmetic progressions and similar topics) would have reached a certain final stage. This, however, did not happen up to the moment; in fact, I am going to talk on anIn 1992 Dorothy retired from her lectureship at the University of St Andrews. I [EFR] remember the day when she walked past my office carrying her electric fire. My office door was always open and she called to me from the corridor, "I'm off now. Goodbye." I replied, "But we'll see you back in the Mathematical Institute from time to time, won't we, Dorothy." She replied simply, "No. Goodbye." Of course I saw her frequently over the years of her retirement but I do not remember seeing her in the Mathematical Institute again.-estimate concerning the divisor problem in arithmetic progressions in a splinter group of this year's St Andrews Colloquium.

Anyway, I look forward with great pleasure to meet you again ... Needless to say that I am still very grateful to you for all you did for me during my stay at St Andrews, in particular for the nice excursion in your car and the tea at your home. ...

In 2003 Geoffrey Kneebone, who had taught Dorothy at Bedford College, died and Dorothy, together with fellow student Glenys Ingram, founded the Geoffrey Kneebone Scholarship which is available for a full-time postgraduate student studying MSc Mathematics for Applications or MSc Mathematics of Cryptography and Communications at Royal Holloway.

John Tuck, the Director of Library Services, Royal Holloway College, wrote to Dorothy on 14 September 2015:-

Royal Holloway Library Services is pleased to announce that over the coming months 2,000 Royal Holloway and Bedford New College research theses will be made available online, and yours is one of the theses that has been selected.In 2018 the London Mathematical Society recorded that Dorothy had been a member of the Society for 60 years.

We wish to celebrate Royal Holloway and Bedford New College's outstanding record of scholarship through this project, and promote the research output of Royal Holloway and Bedford New Colleges as widely as possible. We see this collection as being of interest for the theses themselves, and also to throw light on the kinds of research that were undertaken in the earlier years of the College.

Dorothy had many interests in addition to mathematics. She enjoyed sports, playing tennis on the lawn behind Lower College Hall in the 1960s before the North Haugh tennis courts were built. In the late 1960s and early 1970's the University of St Andrews had small honours classes studying Pure Mathematics and some of the lecturers would go putting on the Himalayas on Wednesday afternoons with the Senior Honours students. Dorothy was an expert putter and we were all surprised to see just how competitive she was. But Dorothy was multi-talented - she was an accomplished pianist and had an enthusiastic interest in literature. She was, for many years, a strong supporter of the Jane Austen Society. She was also an excellent baker and I enjoyed several delightful afternoon teas at her home in Irvine Crescent, St Andrews. Dorothy never lost her interest in mathematics, however, and one of her near neighbours, a retired professor of German, told me that one of the last times he spoke to her she was explaining the 'Seven Bridges of Königsberg problem' to him.

She died [5]:-

... peacefully, in her sleep, in St Andrews, aged 87 years, on 2nd July, 2021. Loved by her family, friends and will be sadly missed.

### References (show)

- B Featherstone,
*Personal communication*. - D M E Foster,
*Papers*. - D M E Foster,
*Some arithmetical properties of quadratic polynomials*, Ph.D. Thesis, Bedford College, University of London (1960).

https://www.proquest.com/docview/1792281286?pq-origsite=gscholar&fromopenview=true - D M E Foster,
*The Quarterly Journal of Mathematics*(2)**9**(1958), 241-256. - Dorothy Foster,
*Notices, Fife Today*.

https://www.legacy.com/obituaries/fifetoday-uk/obituary.aspx?n=dorothy-foster&pid=199471160&fhid=41139

### Additional Resources (show)

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Written by J J O'Connor and E F Robertson

Last Update September 2021

Last Update September 2021