Congratulations to 16-year-old Aubrey William Ingleton, prefect at Tollington School Muswell Hill, who has been placed first out of 7,371 candidates in the recent Civil Service examination, general clerical class. Aubrey has had a brilliant school career and is exceptionally good at Mathematics. When he took the examination to enter Tollington from Tollington Preparatory School he gained 100 per cent for his arithmetic paper.

I am indebted to Dr F Smithies for suggesting this problem to me, and for assistance in the preparation of this paper. I also wish to thank Mr N A Routledge for his assistance in checking the manuscript, and Mr A W Ingleton for helpful advice.

George Temple had resumed as Head of the Mathematics Department after the war; other members of the Department with whom Ingleton interacted included Jack Semple, Richard Rado and Bernard Scott. Scott, who was a major source of exchange of ideas between colleagues, shared a room with Ruston. The King's Mathematics Department was known throughout the College and the University, not only for its academic excellence, but also for its informal and friendly atmosphere. There was one, short, departmental meeting a year, after the end of the summer term, when people would volunteer for the courses to be given the next year. The emphasis was very much on how to provide each member of staff with as much time as possible for research. There was also a strong tradition that anyone who had given a ten o'clock lecture (or who was to give one at eleven) would go down to the refectory to have coffee with the undergraduates.

The Hahn-Banach theorem on the extension of linear functionals holds in real and complex Banach spaces, but it is well known that it is not in general true in a normed linear space over a field with a non-Archimedean valuation. Sufficient conditions for its truth in such a space have been given, however, by A F Monna and by I S Cohen. In the present paper, we show that a necessary condition for the property is that the space be totally non-Archimedean in the sense of Monna, and establish a necessary and sufficient condition on the field for the theorem to hold in every totally non-Archimedean space over the field. This result is obtained as a special case of a more general theorem concerning linear operators, which is analogous to a theorem of Nachbin (A theorem of the Hahn-Banach type for linear transformations (1950)) concerning operators in real Banach spaces.

Let K be a field with a non-trivial non-Archimedean valuation, E a linear space over K having a real-valued norm with customary properties. It is proved that a necessary condition for the Hahn-Banach theorem to hold for linear functionals on subspaces of E is that E be non-Archimedean, that is, satisfy the strong triangle inequality: $|x + y| ≤ max(|x|,|y|$. For a given K it is shown that in order for the Hahn-Banach theorem to hold in every non-Archimedean space over K, it is necessary and sufficient that the intersection of any totally ordered collection of spheres in K be non-empty. This condition on K is equivalent to maximal completeness.

Ingleton was Offord's second appointment. Assistant lecturers at small UK institutions with well-connected heads-of-department were generally recruited through the grapevine, and Offord will have heard about Ingleton, probably from Ruston. As an analyst, Offord will have particularly appreciated a geometer (which the Department lacked) whose first research topic had been a generalised Hahn-Banach theorem. The Department was friendly and very pleasant to work in; though (except for the analysts, who organised a seminar) it was too small and heterogeneous to generate much productive interaction. However, the proximity of King's and University College largely compensated for this. The geometry in the London Honours degree, which (at least initially) took up most of Ingleton's teaching time, was predominantly the projective theory of quadrics, bits about twisted cubics and some classical differential geometry.

In this paper we are mainly interested in matrices of 0's and l's. The condition of being circulant imposes a restriction on the possible values for the rank of such a matrix and we can, in particular, define an integer $\rho(n)$ as the least possible rank of a non-recurrent circulant matrix of 0's and 1's of a given order n. We obtain the value of $\rho(n)$ for a certain class of values of n, but the results only yield upper and lower bounds for $\rho(n)$ in general. Further progress is possible, but the existence of a simple explicit expression for $\rho(n)$ valid for all n seems doubtful. The corresponding problem for matrices with arbitrary rational coefficients is, however, much more tractable and a complete solution is given.

This is a very valuable and interesting paper which surveys the representability problem for matroids and contains several new constructions and results.

Influential in this move was E C Titchmarsh, who was then Savilian Professor of Geometry at Oxford (a chair associated with New College), and who had known Ingleton from his lectures at University College London and thought very highly of him.

He was known to be a man of sound judgement, who was totally open, one who did not court disputes but who was not afraid to speak his mind, though never in such a way as to give offence.

... spent few words, but those words he spent were words well spent. You always knew that what he told you was true, and that what he promised you would get done. He had no great taste for college feasts or festivals, and ceased to attend them once he was no longer obliged by duty to do so. However, even as an emeritus fellow, he liked to come regularly into the common room to read the newspapers amid the companionship of the fellows. ... Most of us knew little of his hobbies: of his gift for composing chess problems, for instance, or his learned interest in the local history of Headington, or his encyclopaedic knowledge of vintage railway engines. But his passion for the Hebrides, the invariable scene of his annual holidays, was an open secret. ... I recall Aubrey as a paradigm of the type of devoted tutor and conscientious college officer that has been, throughout the century just ended, the backbone of the Oxford collegiate system.

... on a personal note, my memory of Aubrey is as a totally reliable, wonderful colleague. We met every week at the Tuesday combinatorics seminar. He was usually quiet, not one to venture wild conjectures or the like, but what he said was almost always correct. Indeed, I lost count of the times that shortly after the seminar, either that evening or the next day, I would get a phone call or precisely written note with a counterexample to some conjecture or problems that had been put forward at the tea conversation after the seminar.