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.

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.

... 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.

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.

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.

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.

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.

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.

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.

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.

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.