Winifred Sargent was born into a Quaker family. John Grant Sargent and his wife Catherine Doubell, settled in Fritchley, Derbyshire in 1864 where they founded a very strict Quaker Community providing an alternative to the London Meeting. The Fritchley General Meeting met in 1870 and, although always small, it slowly increased up to about 1900. Winifred's father, Henry C Sargent, was a descendent of John Grant Sargent and Winifred was the only child of Henry Sargent's second marriage to Edith M Sargent. She was brought up in the Quaker Community at Fritchley and her primary education was provided partly by her father and partly in a small school run for children of the Fritchley Society of Friends (Quakers).
In September 1915 Sargent entered Ackworth School which had been founded by the Quakers in 1779 in the abandoned buildings of a Foundling Hospital in Ackworth near Pontefract and still occupied these building when Sargent became a boarder there. Although it was a mixed school, the boys and girls occupied separate wings in Ackworth School at that time and they were taught separately. Her mathematics teachers probably included Lydia S Graham M.A. and Victoria M Jevons B.Sc. both of whom were on the mathematics staff during her time at the school  and it seems that here Sargent first became enthusiatic about mathematics. She spent three years at Ackworth School, leaving on 28 July 1919, and, having won a Joseph Rowntree Entrance Scholarship, she began to study at The Mount School in York. This was a girls school founded in 1831, again run by the Society of Friends (Quakers). She was a boarder at this school for a while but did not find the mathematics challenging enough.
Sargent left The Mount School in York and became a day pupil at The Herbert Strutt School in Derby Road, Belper, Derbyshire. Unlike her two previous secondary schools, The Herbert Strutt School was comparatively new, having only opened in 1909 and gained full Secondary School status in 1913. Also unlike her previous secondary schools, this was not a Quaker school but it was an attractive prospect because of the high academic reputation which it had quickly attained.
In 1923, while studying at The Herbert Strutt School, Sargent gained a Derby County Scholarship, a State Scholarship, and a Mary Ewart Scholarship to study mathematics at Newnham College, Cambridge. The main emphasis there was on analysis and it was certainly a topic which attracted her. She gained further honours when she was awarded a Arthur Hugh Clough Scholarship in 1927, a Mary Ewart Travelling Scholarship in 1928 and, in the same year, a Goldsmiths Company Senior Studentship.
After graduating Sargent began to undertake research in analysis but after working for a while she felt that she was not producing results of the exceptionally high standard which she had set herself. She did produce good results, despite any feelings that she may have had about them, for she published On Young's criteria for the convergence of Fourier series and their conjugates in the Proceedings of the Cambridge Philosophical Society based on this work which appeared in print in 1929. This was a fine piece of work for the young mathematician who nevertheless gave up her research to become a mathematics teacher at Bolton High School.
Fortunately, even if Sargent did not believe strongly enough in her own mathematical abilities, others knew better. In 1931 she was offered the position of Assistant Lecturer in Mathematics at Westfield College, London. This College had been founded in 1882 as a pioneering college for the higher education of women, and was granted its Royal Charter in 1932. Although Sargent was somewhat unsure about the move, she was persuaded to make it and she began teaching at Westfield College in 1931. In 1936 she moved to Royal Holloway College. This was also a college for women, which had been founded in 1879 by Thomas Holloway, opened by Queen Victoria in 1886 and became a College of London University in 1900. Sargent's first appointment in Royal Holloway College was as an Assistant Lecturer in Mathematics, for although by the time of her appointment she had published a further two papers, she had never completed a doctorate.
In 1939 Sargent registered as a doctoral student of Bosanquet who was at that time a Reader in the University of London having interests in analysis which were close to those of Sargent. Of course World War II broke out not long after Bosanquet had begun to supervise her research and formal supervision came to an end; however he continued to influence and encourage her. She was promoted to Lecturer in Mathematics in 1941, then moved from Royal Holloway College to Bedford College in 1948. This was another London college for women situated after 1913 in Regent's Park, and becoming part of the University of London in 1900 at the same time as Royal Holloway College.
In 1947 Bosanquet inaugurated a weekly seminar and Sargent attended this from its beginnings until she retired, without missing a single week in 20 years. However, she did not attend conferences and seldom could be persuaded to address Bosanquet's seminar, although when she did she showed herself to be an outstanding speaker. In 1954 Sargent was awarded a Sc.D. by the University of Cambridge, and promoted to Reader at Bedford College. In 1965 Bedford College admitted male undergraduates for the first time. Two years later, in 1967, Sargent retired from her Readership and, unlike many mathematicians, seems to have given up research at this time.
Sargent's mathematical work involved the study of different types of integral. The Riemann integral is well known and had long been studied, but much effort had also been put into the study of the Lebesgue integral. Sargent concentrated on integrals such as the Perron integral, the Denjoy integral, the Cesàro-Perron integral, and the Cesàro-Denjoy integral. For example in 1941 Sargent published A descriptive definition of Cesàro-Perron integrals in the Proceedings of the London Mathematical Society which gives an inductive definition, using Cesàro derivatives, of the Cesàro-Denjoy integral which is equivalent to Burkill's integral. In the same year she published On sufficient conditions for a function integrable in the Cesàro-Perron sense to be monotonic in the Quarterly Journal of Mathematics. R L Jeffery, reviewing the second of these papers writes:-
There is ... provided a simple and direct proof for a theorem which is fundamental in the development of the Cesàro-Perron scale of integration.
Her 1948 paper On the integrability of a product extends results of Lebesgue from 1910 to the general theory of Denjoy-Perron integrals. Another paper published in the same year On the summability (C) of allied series and the existence of (CP) extends the conditions for the Cesàro summability of Fourier-Lebesgue series and of their conjugates given by Bosanquet in 1937 to the case of functions integrable in the Cesàro-Perron sense.
This is only a few examples of the work which Sargent carried out and published in 24 papers but it gives an indication of the type of problems which interested her. Eggleston, in , writes:-
Sargent's work is marked by its exceptional lucidity, its exactness of expression and by the decisiveness of her results. She made important contributions to a field in which the complexity of the structure can only be revealed by subtle arguments. Her work is in many ways an expression of her character. Although she was self-effacing and totally lacking in any ambition for self-advancement, yet she was independent and would not tolerate anything which she thought to be second-best. She never attempted to publicise her work ...
As to her interests outside mathematics :-
... she had a passion for walking and enjoyed tennis in her younger days.
Eggleston pays her this tribute:-
She will be remembered by her friends for her dedication to the search for mathematical truth, her absolute integrity and above all for the determination with which she imposed on herself and on those she taught the highest and most meticulous standards of accuracy and precision.
Article by: J J O'Connor and E F Robertson