The code-breakers were mainly classicists and mathematicians. There were, however, notable exceptions to the rule that said that classicists were better than mathematicians. One or two of the younger mathematicians, Barnes in particular, proved to be highly skilled code-breakers. In conversation in later years Eric recalled that he gained his commission as a Lieutenant because of his success in cracking a Japanese code that had baffled the British experts at Bletchley Park.

... he was fascinated by his war work and used to give talks on cryptography. He was also familiar with modern cryptography (it sounded diabolically sophisticated and is in great demand for computer security).

So in 1949 and 1950 I ate, drank and slept mathematics: reading and writing out notes on or translations of papers in the Cambridge Philosophical Society Library, working at problems on binary quadratic and bilinear forms and attending lecture courses and seminars.

I am indebted to Dr J H H Chalk, who suggested the original problem, and to Professor L J Mordell for valuable suggestions and discussions.

I should like here to express my indebtedness to Dr J H H Chalk, who has proved (in an unpublished paper) that ... and who suggested to me the possibility of extending this result. My thanks are due also to Prof L J Mordell for his guidance in the writing of this paper.

It was suggested to me by Professor L J Mordell that the methods of this paper might be extended to deal with more general classes of forms. Such an extension is in fact possible, and I hope shortly to publish some results on the minimum of a general bilinear form in four variables.

Chapter I contains a summary of classical results in the theory of the reduction of indefinite binary quadratic forms. The following three chapters are concerned with the function

          $P = P(x, y, u, v) = Q(x, y ) Q(u, v)$,

where Q(x, y) is an indefinite binary quadratic form with real coefficients and discriminant D > 0, and the variables are integers subject to the condition

         $| xv - uy | = 1$.

Complete results are given for M(P), the lower bound of |P|, and for $M_{+}(P)$ and M- (P), the lower bounds of the positive and the negative values of P respectively. In each case it is shown that there exists an enumerably infinite sequence of isolated minima tending to a limiting value, and that the set of forms with minimum arbitrarily close to this limiting value has the cardinal number of the continuum.

A generalization of these results is given, in Chapter VII, to indefinite ternary forms Q(x, y, z) of determinant d ≠ 0. Here

          $P = Q(x_{1}, y_{1}, z_{1}) Q(x_{2}, y_{2}, z_{2} ) Q(x_{3}, y_{3}, z_{3})$,

and the variables are integers with determinant ±1. It is shown that $M(P) < \Large\frac{|d|}{2.2}\normalsize$ except when Q is equivalent to a multiple of one of three special forms, where $M(P)$ is either $\large\frac{2}{3}\normalsize |d|$ or $\large\frac{2}{25}\normalsize |d|$.

The remaining chapters, V and VI, are devoted to a study of the general bilinear form

          $B = B (x, y, z, t ) = \alpha xz + \beta xt + \gamma yz + \delta yt$

in integral variables satisfying $| xt - yz | = 1$. The form has two independent invariants under integral unimodular transformation, which may be taken as

          $D = (\beta + \gamma)^{2} - 4\alpha\delta$,

and $\omega = \Large\frac{|\beta-\gamma|}{√|D|}\normalsize$ (for $D ≠ 0$).

I was fortunate to attend a one-term course of lectures on linear algebra by Eric in Part 2 of the Cambridge Mathematical Tripos. His style was lucid and unhurried, his blackboard work always impeccable. He rarely consulted his notes during a lecture, giving the impression that he was not working from a planned script. Each lecture, however, ended precisely on time at a stage where there was a natural break in the mathematics; never was he part way through a proof when time ran out, which showed the whole had been meticulously planned. Eric had a dry sense of humour, far removed from any conscious joke. After introducing the concept of homomorphism he remarked: 'One of the Morph brothers'. The characteristic which I most appreciated was his ability to make even quite difficult concepts easy to grasp.

... is awarded for the most meritorious contributions to knowledge and society in Australia or its territories, conducted mainly in New South Wales by an individual who is from 5-15 years post-PhD or equivalent on 1 January of the year of the award, together with signs of leadership. The recipient may be resident in Australia or elsewhere.

He has made many fruitful contributions to the Geometry of Numbers, showing high skill in the use of simple methods to solve problems which lie quite deep in the theory of numbers. For earlier work in this field he was awarded a Smith's Prize and was elected to a Fellowship of Trinity College, Cambridge.

During his first decade in Adelaide, Eric played a leading role in connection with school mathematics and began his involvement in university entrance matters. He served terms as Chief Examiner in Mathematics for the Public Examinations Board and as Chairman of the Board.

Perhaps to the detriment of his mathematical research, he became increasingly involved in administration. A partial list of the responsible positions he held in the University is impressive: Head of Department, First Dean of the Faculty of Mathematical Sciences, Chair of Education Committee, Member of University Council, Deputy Vice Chancellor.

Eric Barnes was an exceptionally quick and incisive thinker with an excellent memory. He devoted time and hard work to any matter which he took up and had a remarkable ability to master complex detail and identify the essentials. He was also a gifted expositor whose presentations were clear, logical and appropriate to their audiences. These qualities underlay his mathematical research and teaching and contributed greatly to his work in administration and academic management.

Eric's intellectual abilities could at times be daunting, and by nature he had a low tolerance for inaccuracy. In debate on large University committees, these characteristics sometimes led to an abrasive style of argument.

Eric gained and retained the respect and affection of his colleagues in pure mathematics. While he could argue cogently for his own point of view at departmental meetings, he was not dominating and was very much the opposite of a 'God-Professor'. He had a genuine concern for students at all levels and took a special interest in students from overseas, mathematically gifted students, and those who did not quite fit the system. His colleagues found him approachable, supportive and encouraging, particularly to new arrivals, those with less experience, and new researchers.

Among Eric's recreational interests were music, bridge, chess, reading and a love of language and words. He could lighten the atmosphere with a witty turn of phrase and he remained fluent in idiomatic French. He was a competent pianist and those who were there remember a happy Mathematics Department party at the home of the Potts family, around 1963, when musical entertainment was provided by a trio (Eric Barnes, piano, George Szekeres, viola, Ren Potts, clarinet) and novice pianist Maurice Brearley.