# Rosalind Cecilia Hildegard Tanner

### Quick Info

Born
6 February 1900
Göttingen, Germany
Died
24 November 1992
Croydon, England

Summary
Cecily Tanner was the daughter of the mathematicians William Henry Young and Grace Emily Chisholm. She published excellent papers on Riemann-Stieltjes integrals, then she became interested in the history of mathematics. Her efforts led to Thomas Harriot's work being much more widely known and understood.

### Biography

Rosalind Cecilia Hildegard Tanner only took the name Tanner after her marriage. She was the daughter of the English mathematicians William Henry Young and Grace Emily Chisholm who were studying in Göttingen, Germany when she was born. She was given the names Rosalind Cecilia Hildegard Young but the name on her birth certificate has German versions of her names, appearing as Rosalinde Cäcilie Hildegard Young. In fact she was known as Cecily and, to avoid confusion with other Youngs, we will call her Cecily throughout this biography. In fact her publications appear with the names R C Young, R Cecily Young or Rosalind Cecily Young until he marriage in 1953 after which she published under the name R C Tanner.

Cecily's mother Grace Chisholm undertook research at Göttingen, Germany, advised by Felix Klein and in 1895 she obtained a Ph.D. for her thesis which applied Klein's group theory to spherical trigonometry. She sent a copy of her thesis to William Henry Young who had been her tutor for a term at Cambridge. By this time her parents were elderly, her father being 86 years old, so she returned from Germany to England to help look after them. On 11 June 1896 she married William Henry Young who had studied at Peterhouse, Cambridge, had been fourth Wrangler in the Mathematical Tripos of 1895 and had been elected a fellow of Peterhouse in 1896. Felix Klein came to Cambridge in 1897 to receive an honorary degree and encouraged the Youngs to go the Göttingen to undertake research.

William and Grace's first child Francis (known as Frank) was born on 4 June 1897 in Cambridge and in September the Youngs left Cambridge for Göttingen. Grace wrote [10]:-
At Cambridge the pursuit of pure learning was impossible. ... Everything pointed to examination, everything was judged by examination standard. There was no interchange of ideas, there was no encouragement, there was no generosity.
After a short while in Germany, William, Grace and Frank Young left Göttingen and travelled in Italy for a year spending some time studying geometry in Turin, then in September 1899, they returned to Göttingen where their second child Rosalind Cecilia Hildegard, the subject of this biography, was born in February 1900. Felix Klein's daughter, Luise Klein (1879-1961), acted as Cecily's unofficial godmother (Luise could not be her official godmother because she was not a member of the Church of England). Four further children were born to the Youngs in Göttingen: Janet Dorothea Ernestine Young (born 14 December 1901), Helen Marian Kinnear Young (born 20 September 1903), Laurence Chisholm Young (born 14 July 1904), and Patrick Chisholm Young (born 18 March 1908).

Let us give a few details about Cecily Young's siblings. Frank Young, although showing considerable mathematical abilities, was keen to become a pilot. During the First World War he was 2nd Lieutenant, Royal Flying Corps. He was killed in active service on 14 February 1917, aged 20 years, when his plane was shot down. Janet Dorothea Ernestine Young became a physician and married Stephen Ernest Michael. She died in London in July 1997. Helen Marian Kinnear Young married Jean M Canu (1898-1989) in 1929. She died in Germany in 1947. Laurence Chisholm Young became a mathematician and has a biography in this Archive. Patrick Chisholm Young became a chemist and pursued a career in finance and business. He married Marjorie Sarjent in 1950 and died in 1996.

In 1905 William and Grace Young published First book of geometry. This book is interesting here because it shows how the Youngs used non-conventional ways of teaching mathematics to Cecily and their other children. G H Hardy writes [9]:-
The book is a genuine 'book for children' of a very interesting and original kind. The central idea is that children should be encouraged to think of geometrical objects in three dimensions, to think of a plane, for example, as a boundary of a solid, and of a line as an intersection of two planes, or as a fold in one. I am no authority on such a matter, but I should have thought that the idea was sound, and that a book based on it should be more concrete and more stimulating, for most learners, than those of the more conventional and abstract pattern. The authors, however, were asking too much of English teachers. It appeared that they could not, or would not, fold paper, and the book fell absolutely flat in England. It was much more successful abroad, has been translated into German, Italian, Magyar and Swedish, and used with success in German schools.
Two further children's books authored by Grace Young, written to introduce Cecily and her other children to science, were Bimbo (1906), and Bimbo and the Frogs (1907). We note that 'Bimbo' was the Young's nickname for their eldest child Frank.

In 1908, shortly after the birth of Patrick, the large Young family moved to Geneva, Switzerland where their children were educated. Cecily had begun her primary schooling in Göttingen before the family moved, but from 1908 continued her education in Geneva. Much of the time, however, William Young was away from home. He returned to Cambridge during term time where he both taught and examined. At various times, in addition to his role as examiner at Cambridge, he was also an examiner to the Central Welsh Board (1902 to 1905), the University of London, and the University of Wales. In 1913 he accepted two part-time chairs, one being the Hardinge Professorship of Pure Mathematics in Calcutta University which he held from 1913 to 1917, the other being the Professorship of Philosophy and the History of Mathematics at the University of Liverpool which he held from 1913 to 1919. Grace Young had, for the larger part of every year, the responsibility of bringing up the children on her own.

In 1915 the family moved to Lausanne where Cecily completed the two final years of her schooling and entered the University of Lausanne in 1917. In 1919 her father, William Young, [9]:-
... was appointed Professor of Pure Mathematics at Aberystwyth. Here he was as energetic as ever, but his residence abroad was sometimes a source of trouble, and disagreements about this, and about appointments to the staff, led to his resignation in 1923.
These were four years in which Cecily was studying for her licence at the University of Lausanne but in fact she was in Aberystwyth for much of these four years acting as an unofficial assistant to her father. Edward Collingwood was on the staff at Aberystwyth and between 1921 and 1924 she helped him translate Georges Valiron's French lectures into the book Lectures on the General Theory of Integral Functions. It is worth noting that the Preface to the book was written by William Young but nowhere in the book is Cecily Young's help acknowledged. After spending time in Aberystwyth, she returned to full-time study at Lausanne and was awarded her 'Licence ès sciences (mathématique et physique)' in 1925.

Cecily published her first paper in 1925, namely Les fonctions monotones et l'intégration dans l'espace à n dimensions which was published in L'Enseignement Mathématique. In the paper she proves certain facts which were stated in William Young's paper On multiple integrals (1917) without proof. The paper is written in French and she gives her address as Lausanne.

After completing her degree in Lausanne, Cecily matriculated at Girton College, Cambridge to undertake research for a Ph.D.; her mother, Grace Chisholm, had been an undergraduate at Girton. Her thesis advisor was Ernest Hobson and she developed some important ideas which, as well as making a very powerful Ph.D. thesis, she published over the following few years. At Cambridge she was close to her brother Laurence Chisholm Young who was studying at Trinity College, Cambridge. He was working on writing his first book The theory of integration and Cecily assisted him, typing out the text. The book is an introduction to Lebesgue-Stieltjes integration using techniques based on monotone sequences of functions and was published by Cambridge University Press in 1927. As well as translating French texts to English, Cecily translated texts from German to English, for example while studying at Cambridge she translated Konrad Knopp's Theorie und Anwendung der unendlichen Reihen and her translation was published as Theory and Application of Infinite Series (1928). She was awarded a Ph.D. in 1928 for her thesis Foundations for the generalisation of the theory of Stieltjes integration and of the theory of length, area and volume. An n-dimensional treatment.

During the years 1928-32, Cecily published a number of papers related to the work of her thesis. The main ones are: On the integral $\int_a^b \large\frac {F(x)dx}{x-t}$ (1928); On Riemann-Stieltjes integrals with respect to a Riemann integral in space of n dimensions (1928); Théorèmes sur les ensembles d'intervalles linéaires au sens général, avec application aux fonctions à limites unilatérales uniques et finies en tout point (1928); Functions of $\Sigma$ defined by addition or functions of intervals in n-dimensional formulation (1929); On Riemann integration with respect to a continuous increment (1929); On "Riemann" integration with respect to an additive function of sets (1929); On many-valued Riemann-Stieltjes integration (1931); On many-valued Riemann-Stieltjes integration, II. Integration of bounded functions with respect to functions of bounded variation (1931); Non-Uniform Convergence and Term-by-Term Riemann-Stieltjes Integration (1931); and The algebra of many-valued quantities (1931). Let us illustrate these papers by quoting from some of them.

Cecily begins the paper On Riemann-Stieltjes integrals with respect to a Riemann integral in space of n dimensions (1928) as follows:-
J M Whittaker and H J Ettlinger have lately discussed a very special point in the theory of Riemann-Stieltjes integration in one dimension, which can, in point of fact, be elucidated in a few lines, and in n-dimensional language, as a corollary to W H Young's necessary and sufficient condition for Riemann-Stieltjes integrability of bounded functions.
The following extract is from On Riemann integration with respect to a continuous increment (1929):-
In a previous communication to the Math. Zeitschrift (Functions of $\Sigma$ defined by addition or functions of intervals in n-dimensional formulation), I had the opportunity of immediately illustrating the use of the notions there introduced and of some of the results, in the simple considerations relating to "Riemann" integration of continuous functions with respect to continuous increments. This application has an independent interest and may be extended in a similar manner to that in which ordinary Riemann-Stieltjes integration of continuous functions is extended. With this extension the present paper is concerned.
...
In section I of the present paper, I discuss the definition of Riemann integration with respect to a continuous increment, obtaining at the same time the elementary propositions needed in the sequel, and I examine the connection between this definition and the current definitions of Riemann integration with respect to a function of bounded variation. In sections 2-5, integrability of bounded functions with respect to a non-negative increment is considered, and necessary and sufficient conditions deduced for this case. In sections 6-8, the restriction that the increment be non-negative is removed, and in section 9 the precise formulation of the criteria of integrability is obtained for the case of an unbounded integrand. The latter discussion reduces Riemann integration of an unbounded function to that of a bounded function equal to it where it lies between certain fixed bounds (depending on the integrator) and zero elsewhere; while the former reduces Riemann integration of a bounded function with respect to any continuous increment to-that with respect to non-negative increments. Both these questions I have failed to find explicitly treated in previous literature even in the simple case of ordinary one-dimensional Riemann-Stieltjes integration, the step taken in sections 6-8 having indeed been slurred over by previous writers.
Finally, we quote from On "Riemann" integration with respect to an additive function of sets (1929):-
In the present note I consider nominally seta of points of the field of Borel sets in space of n dimensions. The only idea, however, special to this type of set, actually involved is that of the distance between two elements (points), on which is based that of the span of a set, of the norm of a subdivision, as well as that of point of accumulation of a set and the consequent ideas of closed and open sets relatively to a containing set. The characteristic notions of product, sum, difference, limits of sets are not geometrical, and are common to all types of sets. The fields of sets considered may be restricted if desired, provided that we maintain the possibility of forming subdivisions of arbitrarily small norm of every set; of finding subsets of arbitrarily small span containing any element of any given set; and of proceeding to the limit of any sequence of sets, without leaving the field.
In the paper The algebra of many-valued quantities (1931), Cecily came up with ideas that were independently rediscovered over 20 years later [2]:-
The idea of considering whole sets of numbers instead of particular numbers and using them in computation is not new in mathematics and appears in a relatively modern form with studies of Young (1931) who used sets of limits of a given function in case the limit was not unique. Young considered intervals and other sets of numbers and defined generalisations of arithmetic operations and relations on them. For instance, she defined the sum $[a_1, a_2] + [b_1, b_2]$ as the set of all numbers $a+b$ where $a \in [a_{1}, a_{2}]$ and $b \in [b_{1}, b_{2}]$. These rules later reappear in the context of interval analysis, which emerged in the 1950s in connection with analysis of errors in computations on digital computers independently in the works of Dwyer, Warmus, Dunaga, and Moore, who referred to Dwyer's book. The basic arithmetic operations for intervals introduced by Young are reinvented in these works.
During the time Cecily was writing these papers she held a fellowship at Girton College. Her fellowship ended in 1932 and for a year she was unable to find another position. She had hopes of a position at the University of Michigan in the United States but, despite support from Norbert Wiener with whom she had written the joint paper The total variation of $g(x+h) - g(x$) (1933), she was unsuccessful. In 1933, however, she was appointed at the Imperial College of Science and Technology, University of London, where she spent the rest of her career. Her last publications on topics close to her thesis were two more papers in 1936. After this she became interested in the history of mathematics and in mathematical education.

Cecily attended several of the International Mathematical Congresses. She was at the 1932 Congress in Zurich, the 1950 Congress in Cambridge, Massachusetts, and the 1958 Congress in Edinburgh, Scotland. At the 1958 Congress she delivered the Short Address 'Equal and unequal'; none of the Short Addresses were published. By the time she attended the 1958 Congress, Cecily was R C H Tanner, for she had married Bernard William Tanner (1881-1954) in the Holy Trinity Church in Wallington, Surrey on 3 September 1953. Bernard Tanner, who was 72 years old, had been an electrical engineer and the chief maintenance engineer at Imperial College, London. Their marriage only lasted nine months since Bernard Tanner died on 12 June 1954 at The War Memorial Hospital Carshalton, Surrey.

We mentioned that Cecily became interested in the history of mathematics but she had a particular passion to make the work of Thomas Harriot better known. Her paper On the role of equality and inequality in the history of mathematics was published in 1962, see [12]. The paper begins:-
The following essay is adapted from one with the same title read to the British Society for the History of Science on 20 October 1945 - the anniversary, by a striking coincidence, of the birth of W H Young (1863-1942). To his memory I dedicated the talk, and now rededicate its publication, not only because I am his daughter and of all that means, but because he invented a method, the method of monotone sequences, which shows the powerfulness of inequalities as a mathematical tool supremely. In 1631, twenty-five years after Shakespeare died, eleven years before Newton was born, Walter Warner, mathematician and philosopher, gave out in print the Algebra of Thomas Harriot, his 'Artis Analyticae Praxis ad Aequationes resolvendas '. This is the work in which our modern inequality signs are used for the first time in their modern sense, the left-hand pointing arrow-head < to mean 'is less than', the right-hand pointing arrow-head > to mean 'is greater than'. Today these signs are indispensable to the mathematician. To us it appears obvious that here was one of the happiest conceits in the history of mathematical thinking.
It was the study of inequalities that made Cecily enthusiastic about making Thomas Harriot's contributions better known. She financed the Thomas Harriot Seminar in Oxford, Harriot's home town, which began in 1967. It ran until 1983 and contributed much to bringing Harriot to the fore and putting his work in context. She wrote a number of papers on Harriot, we mention in particular: The study of Thomas Harriot's manuscripts (1967); Thomas Harriot as mathematician: A legacy of hearsay (1967); Thomas Harriot (1560?-1621) (1969); Henry Stevens and the associates of Thomas Harriot ((1974); Nathaniel Torporley's 'Congestor analyticus' and Thomas Harriot's 'De triangulus laterum rationalum (1977); The ordered regiment of the minus sign: Off-beat mathematics in Harriot's manuscripts (1980); and The alien realm of the minus: Deviatory mathematics in Cardano's writings (1980).

Bernhard Neumann writes about her interests other than mathematics and her health problems in [11]:-
From childhood she suffered from impaired hearing, which became more severe over the years and restricted her contacts outside her family, though she learned to lip-read and had many friends, especially among fellow mathematicians. She was a talented violinist, wrote poetry, and was intermittently a churchgoing Christian. She donated generously to a number of organisations, especially those concerned with her historical interests, at Oxford, Durham, and Cambridge. She died in the Mayday Hospital, Croydon, from bronchopneumonia, acute myeloid leukaemia, and congestive heart failure on 24 November 1992, and was cremated in Croydon.
We now give two quotes by Cecily. The first is from a letter she wrote to her parents giving insight into the relation between them [13]:-
Another famous partnership, that of George Eliot and Lewes, can be taken as in many respects the counterpart of that of my parents. There it was the man who took the brunt of life off the woman's shoulders and spent his creative energies in fostering her genius. This my mother clearly appreciated.

When all is said, it remains that my father had ideas and a wide grasp of subjects, but was by nature undecided; his mind worked only when stimulated by the reactions of a sympathetic audience. My mother had decision and initiative and the stamina to carry an undertaking to its conclusion. Her skill in understanding and responding, and her pleasure in exercising this skill led her naturally to the position she filled so uniquely. If she had not that skill, my father's genius would probably have been abortive, and would not have eclipsed hers and the name she had already made for herself.
Finally, let us quote her views on mathematics and the British [12]:-
We are supposed to be a non-mathematical nation, just as we were once regarded as a non-musical nation. I have a theory that the opposite is true. Inequality is so deeply part of mathematics that it is taken for granted and one hardly thinks of pointing out that it had to be given a symbol in order to become safe and simple to use. Our mathematics also is so deeply assimilated that it is taken for granted. We don't see the fun of simplification and systematisation until the less fundamentally mathematical-minded insist on it. That is why the decimal system leaves the majority of us cold. It seems to me that this sort of antithesis has always stood between us and Continental mathematics, and is the chief obstacle Harriot encountered too, and of course Newton.

### References (show)

1. Archive of the University of Liverpool: Papers of Dr Rosalind Cecilia Hildegard Tanner; including papers of her parents W H Young and Grace C Young, University of Liverpool.
https://archiveshub.jisc.ac.uk/search/archives/f6f4ceae-2909-3a25-a079-7e252c19e566
2. R Belohlávek, J W Dauben and G J Klir, Fuzzy Logic and Mathematics: A Historical Perspective (Oxford University Press, 2017).
3. Dr Rosalind Cecilia Hildegard Tanner, Imperial College, London.
https://www.imperial.ac.uk/mathematics/for-staff/cecilia-tanner-research-funding-schemes/
4. L S G, Review: Cecily Tanner, British Society for the History of Mathematics Newsletter 23 (1993), 10-15, by I Grattan-Guinness, The College Mathematics Journal 25 (3) (1994), 251.
5. I Grattan-Guinness, Dr R C H Tanner (1900-92), British Journal for the History of Science 26 (1993), 229-231.
6. I Grattan-Guinness, Obituary: Cecily Tanner 1900-1992, British Society for the History of Mathematics Newsletter 23 (1993), 10-15.
7. I Grattan-Guinness, Obituary: Cecily Tanner 1900-1992, Association for Women in Mathematics Newsletter 23 (6) (1993), 21-24.
8. I Grattan-Guinness, Like Mother, Like Daughter, in B A Case and A M Leggett (eds.), Complexities: Women in Mathematics (Princeton University Press, 2016).
9. G H Hardy, William Henry Young, 1863-1942, Biographical Memoirs of Fellows of the Royal Society 4 (12 (1943), 306-323.
10. R Hersh and V John-Steiner, Loving and Hating Mathematics. Challenging the Myths of Mathematical Life (Princeton University Press, 2010).
11. B H Neumann, Tanner [née Young], Rosalind Cecilia Hildegard [Cecily] (1900-1992), mathematician and historian of mathematics, Oxford Dictionary of National Biography (2004).
12. R C H Tanner, On the role of equality and inequality in the history of mathematics, British Journal for the History of Science 1 (1962), 159-169.
13. S Wiegand, Grace Chisholm Young, Association for Women in Mathematics Newsletter 7 (3) (1977), 5-10.
14. R C Young, Foundations for the generalisation of the theory of Stieltjes integration and of the theory of length, area and volume. An n-dimensional treatment, Ph.D. Thesis (University of Cambridge, 1929).