I was a student in Cambridge from late 1933 to mid-1938, and attended lecture courses given by Hardy in that time. The subject I chiefly remember was Fourier Series. Soon after that time those lectures appeared as a booklet with that title, in the series Cambridge Tracts in Mathematics. The booklet was jointly authored by Hardy and W.W. Rogosinski, who later became Professor of Pure Mathematics in Newcastle upon Tyne. At the time of the lectures Lebesgue's theory of integration was only thirty years old, and many members of the audience were unfamiliar with it. Its impact on Fourier Series theory was not slight even then, but Hardy was considerate of his audience and usually tempered the wind to the shorn lambs, even though he did not habitually suffer fools gladly.
The audience consisted of perhaps fifteen undergraduates in their third year, intending to sit the Tripos, both Part II and Part III (formerly known as Part II Schedule B); there were also a few graduates, and several mathematicians from Europe. In these lectures I became friendly with Harry Pitt, with whom I used to go cross-country running, and with whom I shared the Smith's Prize for 1937. He has a Tauberian theorem to his name, and later became Vice-Chancellor of Reading University. Some of the mathematicians from Europe may have been fleeing from Hitler's Third Reich. They were more advanced than the rest of us, and some had doctorates. Among these last was Richard Rado, who later became Professor of Pure Mathematics at Reading. Occasionally there was a guest visiting Hardy; in particular I remember the mighty E. Landau. Many years later I learned that Landau found the best possible constant in what came to be known as Hardy's Inequality. Even though Hardy disliked travelling he kept in personal contact with leading European mathematicians, a practice which was none too widespread in Britain at that time; Landau may well have been making a return visit.
Among other topics on which Hardy gave lectures in those years were Dirichlet Series and Special Functions. The latter course was largely about Legendre Polynomials, a subject which had previously fired my imagination under the influence of Professor Tom Cherry's lectures, in my studies for B.A. in The University of Melbourne. Those early studies may have had an important influence on my much more recent researches on Legendre Functions (a major generalization of Legendre polynomials). Although E.W. Hobson had written the treatise `` Spherical and Ellipsoidal Harmonics" only a few years before, Hardy's lectures added one or two important new things to the subject, notably a Rieman-Lebesgue-type theorem for Legendre polynomials. My generalization of this has played an important part in my recent work. In such ways Hardy brought his original mind to bear on various aspects of the subject suggesting new ways of approaching some of the classical results. One of his ideas was for deriving the asymptotic expansion of P_n(\cos \theta) as n \to \infty , from Laplace's First Integral. I thought this particularly attractive but, sad to relate, I have not been able to make it work in the intervening sixty years!
Perhaps Hardy's most noteworthy book is `` Inequalities", written jointly with Littlewood and Polya and published in 1934. Its authors collected material from far and wide, a major task since inequalities tend to occur in almost every branch of mathematics. The book is still, and will remain, a mine of information. In 1987, more than one hundred mathematicians gathered in Birmingham under the leadership of W.N. Everitt to discuss the progress made since the book was published and the prospects for the future. Everitt had hoped to mount this conference in 1984, the fiftieth anniversary of the publication of `` Inequalities", but practical difficulties prevented this.
An almost more encyclopaedic book, by Hardy alone, is his `` Divergent Series"; it was only completed during his final illness. He dedicated it to L.S. Bosanquet, and wrote `` without whose help this book would never have been finished". While its appeal is to a much more specialized readership than that of `` Inequalities", I believe that it still remains the undisputed authority on its subject. Summability of series, as presented in it by Hardy contributed materially to the `` rigorization" of mathematics, as it emphasised the need for explicit definitions and opened the way to exploring the consequences of definitions, in other branches of mathematics as well as in this one. When outlining a concept to students, before the idea was completely crystallized, Hardy would sometimes avoid too much complication by saying that an expression was to be understood `` in some Pickwickian sense".
Undergraduates mostly know of only one of Hardy's books, his `` Pure Mathematics". First published in 1908, the demand for it was such that it went through nine editions in less than forty years. There was a fairly major revision at the seventh edition, in 1938. To do the proof- reading for this edition Hardy employed me for one British pound an hour. I have read in two places lately that what he employed me for was to extract suitable exercises and problems from old examination papers. This may be so, but I do not remember doing it, whereas I have clear recollections of doing proof reading. He may have had other proof-readers as well, but if so I did not know about them. I don't know why he chose me; I was not one of his research students, and one would expect that he would know his research students, such as Harry Pitt or Frank Smithies, a good deal better than he knew me.
One day Hardy announced that he would fine me a shilling, because I had not noticed a mistake in the proof of the Heine-Borel covering theorem. I don't remember whether he extracted the shilling, but I do know that I didn't like that theorem at that time. The version of it in his book was concerned only with the real line; it was not until years later, when I was teaching the version for plane sets, that I came to terms with the theorem. I visualized it as concerning a family of mats on the floor. Incidentally, on one occasion when I went to Hardy's rooms in Whewell's Court, the floor was covered with reprints; presumably he was sorting them. I did not have the presence of mind to mention covering theorems then; in fact I was not at that time aware of any covering theorems for plane sets.
One of Hardy's research students at that time was Frank Smithies, a Scotsman who has remained in Cambridge to this day. He recounts that their association began with Smithies announcing to Hardy that he would like to work on integral equations. Hardy's response was: `` Good; now I shall learn something about integral equations". So even sixty years ago mathematics was too large for even one so eminent as Hardy to know it all. Eventually Smithies went on to write a Tract in the Cambridge series, on integral equations. Bôcher had written one many years before; but Smithies's tract must have superseded it, otherwise the editors of the series would not have printed the new one. Still further down the line Smithies was giving lectures on functional analysis, the offspring of integral equations. I attended these lectures in a year when I was in Cambridge on leave, and I well remember their superb clarity.
No account of Hardy can fail to mention his legendary collaboration with Littlewood. It extended over half a lifetime, and there has probably been no comparable collaboration in the history of mathematics, either in respect of its importance for the subject or in respect of its marathon longevity. These two great mathematicians were world leaders in the early part of the twentieth century, even though there were other giants, such as Lebesque, F. Riesz and perhaps Fredholm in those years. Hardy and Littlewood both lived in Trinity College, Cambridge, within three minutes' walk of one another, yet it is said that they communicated with one another mostly by notes, rather than face to face. This may be understandable when one considers the complexity of their work. It is also said that they agreed that neither was obliged to answer notes from the other.
Another remarkable collaboration was that of Hardy and Ramanujan. The latter had been a clerk in the Madras post office. From there he sent Hardy a collection of strange summations of series, without proofs, and Hardy found it no trivial matter to prove some of them. When Hardy succeeded, against many difficulties, in bringing Ramanujan to Cambridge, it seemed that Ramanujan did not even know what a proof was. It was clear that Ramanujan had arrived at his remarkable results by unknown and unconventional methods. What those methods were is still a mystery, locked in the mind of a genius who survived for only a few years in the damp and misty climate of Cambridge. In later years Hardy was to say that Ramanujan was the only romance in his life. This is surprising, as more than one writer has described Hardy as `` a charming man". On the other hand, this is not the description I myself would use; he struck me as austere.
Like most people, Hardy had his idiosyncracies. One of these was evident in a toast which he is said to have proposed on more than one festive occasion. It was: `` Here's to Pure Mathematics, may it never be of any use." Again, it was said of the electrical engineer Oliver Heariside that he and Hardy kept one another at arm's length because their outlooks on mathematics were in such extreme disagreement. Heariside proceeded without clear foundations, saying in defence that the definitions would make themselves in time; whereas Hardy insisted on precise definitions. I think mathematics in general has benefitted from both these outlooks. When Hardy talked of using a method in a `` Pickwickian sense" he was really getting close to Heaviside's style of thinking.
As relaxation, Hardy would go on a summer afternoon to watch cricket at Fenner's. He took much more than a casual interest in the game; in fact one of his moderately sophisticated inequalities had its genesis in a consideration of players' batting averages and their satisfaction therewith. I have forgotten the details, but he expressed that satisfaction by a well defined measure. What is more, he actually played cricket, and there is a photograph of him leading his team on to the field. The members of the team were all mathematicians, and prominent among them was Bosanquet.
In 1916 Bertrand Russell was expelled from Trinity and from his lectureship, for allegedly treasonable propaganda against Britain, stemming of course from his pacifism. Many of the Fellows supported the expulsion, and many opposed it. Hardy did not usually concern himself with political matter either collegial or national, although it is said that he had a large picture of Lenin in his rooms. However he took the expulsion of Russell extremely seriously. In due course a ground swell against it gathered strength, and eventually Hardy played a leading part against the dismissal by writing and circulating among the Fellows of Trinity a full account of the discussions which led to the dismissal. This pointed strongly to the probability that a serious miscarriage of justice had occurred. It carried the day, and Russell was reinstated. Looking back on these events, perhaps Russell's chief misdemeanour was to be ahead of his time.
In this note I have given a few sidelights on Hardy's character, mostly gleaned from events which I saw or experienced myself and from stories told me by friends. I have certainly not done justice to the man's towering intellect and achievements; these will no doubt show up in the contributions of others. But I do stress that he was so different from most people, perhaps even from most academics. His values were more like those of the prophets of old than those of the general public. He despised the widespread bourgeois, consumerist and sometimes hypocritical attitudes of the majority. He followed his lights wherever they led; no one else could lead him. Few others have his clear view of the world or his insight into the things that really matter.
The University of Melbourne