In the present month Professor G H Hardy, who for eleven years has been resident in Oxford, will return to Cambridge and occupy the Sadleirian chair of mathematics, vacant by the resignation of Professor Hobson. The occasion seems a favourable one to recall the history of the Sadleirian chair and of the three distinguished living mathematicians who have occupied it. All of them have been Presidents of the Mathematical Association, and have exercised a great influence upon the teaching of our subject. We shall deal at some length with this aspect of their work. Some account will also be given of their researches, but although from a higher standpoint this is the most important of their activities, an adequate estimate of their original work will not be attempted on this occasion.
The foundation of the Sadleirian (or Sadlerian) chair can be traced back [This account of the history of the Sadleirian chair is derived, from Professor Forsyth's obituary notice of its first occupant, Professor Cayley (Proc. Royal Soc., Vol. 58,1895; reprinted in Cayley's Collected Mathematical Papers).] to a benefaction of Lady Mary Sadleir, who by her will of 1701 left the University of Cambridge an estate, the income from which was to be used to maintain lecturers in algebra at nine colleges. The benefaction became available in 1710 and the lectureships were duly established. However, with the ever-growing range of studies, the restriction to a single branch of mathematics gradually deprived the lectures of much of their value. In fact, after a time, it was difficult to persuade anyone to attend, and the benefaction ceased to fulfil the wishes of the founder. At last this state of affairs was so obviously bad that a proposal was made to suppress the lectureships and use the money for the endowment of a professorship, to be called the Sadlerian Professorship of Pure Mathematics. This was sanctioned in 1857 and came into operation in 1863. The spelling Sadlerian, which occurs in the statute establishing the chair, was always used by its first two occupants, but since their time the form Sadleirian has been preferred, as conforming more closely to the name of the founder. The duties of the Professor were to give one course of lectures in one term of the year, and "to explain the principles of pure mathematics." After 1886 the stipend of the post, at first modest, was increased, and two courses of lectures were required. It was hoped that the Sadleirian and other professors of mathematics would come into touch with undergraduate students, but the cast-iron regime of the Mathematical Tripos prevented this. It was impossible for undergraduates, whose future career depended upon their positions in the order of merit in a highly competitive examination, rigidly confined to a stereotyped syllabus, to "waste their time" with professors who were eagerly extending the bounds of knowledge, and seeking after new truths generally too complicated to be dealt with in a three hours' examination. Thus arose the strange paradox that Cambridge possessed a number of eminent professors whose lectures had little (if any) influence on even the best students, and with whom most of the undergraduates were wholly unacquainted.
The first Sadleirian Professor was Arthur Cayley (1821-1895), one of the greatest mathematicians of the nineteenth century. There is no need to go into details of his life and work. An exhaustive account will be found in the article by Professor Forsyth mentioned in the note above.
The Second Sadleirian Professor was Andrew Russell Forsyth. Born in Glasgow on 18th June, 1858, he was educated at Liverpool College and at Trinity College, Cambridge. He graduated as Senior Wrangler and First Smith's Prizeman in 1881 and was elected a Fellow of Trinity in the same year. For a short time (1882-3) he was professor of mathematics at University College, Liverpool (now Liverpool University). He returned to Cambridge as a College and University lecturer and assistant tutor in 1884. In 1895 he succeeded Cayley, and his first task was to edit the unpublished portion of his predecessor's collected works. He remained as Sadleirian Professor until his resignation in 1910. A brief period spent in India led to the publication in 1913 of his lectures on Functions of two or more Complex Variables, delivered, by special invitation, to the professors and doctors of the University of Calcutta. In 1913 he became Chief Professor of Mathematics in the Imperial College of Science and Technology. His retirement from this position in 1923 has not meant a relaxation of his activities, but rather an opportunity for a widening of interests.
Professor Forsyth has always been a prolific writer. He soon acquired a great reputation by his research work and books, which at first were chiefly related to differential equations. His Treatise on Differential Equations, first published in 1885 and now in its sixth edition, was described by the Mathematical Gazette (II, p. 295, May 1903) as the most lucid, accurate and exhaustive exposition of the subject in our language. It has been translated into both German and Italian. This was followed by his monumental Theory of Differential Equations, whose six volumes appeared at intervals from 1890-1906. It is doubtful if even the Germans have produced any treatment of the subject on so extensive a scale. Among Professor Forsyth's own researches Cajori's History of Mathematics specially mentions Differential Invariants and Reciprocants, and Singular Solutions. He has applied the methods of differential equations to find systems of invariants and covariants that are algebraically complete, and given a complete discussion of certain differential equations that had been rather cursorily dealt with by writers on relativity. A valuable summary of the present state of knowledge in partial differential equations, indicating opportunities for further research, was given to the London Mathematical Society (Presidential Address) in 1906. But Professor Forsyth's interests have never been restricted to a single subject. In 1893 appeared his Treatise on the Theory of Functions of a Complex Variable, now in its third edition. Among the topics on which he lectured during his tenure of the Sadleirian chair may be mentioned Differential Geometry and the Calculus of Variations. His lectures on Differential Geometry appeared in a volume published in 1912. His lectures on the Calculus of Variations were the earliest in Cambridge to expound the Weierstrass theory: these were embodied in a treatise published in 1927 which extended the whole range of the subject and included much new research. In 1928 he edited the late Professor Burnside's Theory of Probability, and in 1930 he published his own two-volume Geometry of Four Dimensions. Among his minor writings may be mentioned Mathematics in Life and Thought (1929) and several biographical notices; his wide and detailed knowledge enables him to deal with the life and work of eminent mathematicians in a manner impossible to those of more restricted range. Naturally Professor Forsyth has been the recipient of numerous honours. He was elected a Fellow of the Royal Society in 1886, served on the Council 1893-5, and was awarded the Royal Medal in 1897. He was President of the Mathematical Association 1903-5 and of the London Mathematical Society 1904-6. Honorary degrees have been conferred on him by Aberdeen, Calcutta, Christiania, Dublin, Glasgow, Liverpool, Oxford, and Victoria. He is an honorary member of several learned societies, including some in Italy, Russia, and the United States.
The aspect of Professor Forsyth's work that will appeal most to us is the part he played in the improvement of geometrical teaching. For over thirty years the mathematical teachers represented by our Association endeavoured to free the schools from the tyranny of Euclid, but their efforts were in vain until they found allies in the British Association, in engineers like Professor Perry, and in Cambridge mathematicians like Professor Forsyth. Following a discussion at Glasgow in 1901, in which Professor Perry took the leading part, the British Association appointed a committee to report on improvements that might be effected in the teaching of mathematics. The report of this committee, drawn up by its chairman, Professor Forsyth, will be found in the Mathematical Gazette (II, pp. 197-201, October 1902). It was cautiously worded, avoiding the exaggerations that had weakened the arguments of some of the more enthusiastic reformers, and it undoubtedly paved the way for the decisive step, namely the adoption by Cambridge of the recommendations of a Special Syndicate (of which, among others, Professor Forsyth, Messrs Barnard, Godfrey, and Siddons were members) of a new syllabus in Geometry that for the first time did not make Euclid compulsory. These recommendations related to the Previous Examination: shortly afterwards, they were adopted by the Cambridge Local Examinations Syndicate for their examinations open to the whole country. Thus we may look upon Professor Forsyth as the Moses of our Association, bringing us at last to the promised land of geometrical reform after many weary years in the wilderness. Other contributions to the work of the Association are the opening of the discussion on the Coordination of the Teaching of Mathematics and Science (V, pp. 244-252, March 1910), the presidential address to the London branch on Differential Equations in Mechanics and Physics (XI, pp. 73-81, May 1922), and the article on Dimensions in Geometry (XV, pp. 325-338, March 1931).
The third Sadleirian Professor was Ernest William Hobson. Born at Derby on 27th October, 1856, he was educated at Derby School and Christ's College, Cambridge. He graduated as Senior Wrangler in 1878. He was elected to a Fellowship at Christ's College and became a tutor. In 1903 he became Stokes Lecturer, and he held this post until his election to the Sadleirian chair in 1910. He retained this chair until 30th September of the present year.
In 1891 he published the first edition of his Treatise on Plane Trigonometry. The later portions of this book were for many years the only place (with the exception of Chrystal's Algebra) where could be found an accurate account in English of complex numbers and of infinite series. In 1907 the fame of his Trigonometry was eclipsed by that of his Treatise on the Functions of a Real Variable and the Theory of Fourier's Series. Of this Professor W H Young remarks (Mathematical Gazette, XI, p. 428, December 1923) that it "was at the time the only systematic account of theories so novel in their character, even to the ordinary professional mathematician, that author and publishers alike may well have had doubts as to the success of the venture." Later on the book was doubled in size and divided into two volumes, which appeared in 1921 and 1926 respectively. A third edition of the first part, still further enlarged, appeared in 1927. The complete work constitutes the most exhaustive account of the subject that has yet appeared in any language. A comprehensive treatise on The Theory of Spherical and Ellipsoidal Harmonics is on the point of publication. Professor Hobson's smaller books are Squaring the Circle (1913) and The Domain of Natural Science (1923; a series of Gifford lectures delivered at Aberdeen).
Most of Professor Hobson's researches have been connected with the theory of functions of real variables, but he has also dealt with Legendre's and Bessel's functions, integral equations, potential theory, the conduction of heat, and the calculus of variations. His presidential address to the London Mathematical Society in 1902 was entitled The Infinite and Infinitesimal in Mathematical Analysis. The London Mathematical Society's Proceedings contain thirty-nine of his papers.
Professor Hobson was elected a Fellow of the Royal Society in 1893, served on the Council 1903-5 and 1928-30, and was awarded the Royal Medal in 1907. The London Mathematical Society chose him as President in 1900-2, and gave him their De Morgan Medal in 1920. Honorary degrees have been conferred upon him by Aberdeen, Dublin, Manchester, Oxford, St Andrews, and Sheffield, and he is a member of learned societies in Ireland, Germany, and Italy.
Professor Hobson acted as our President in 1911-3. His presidential addresses were entitled The Democratization of Mathematical Education, and On Geometrical Constructions by Means of the Compass. These will be found in the Mathematical Gazette, VI, pp. 234-242, March 1912, and VII, pp. 49-54, March 1913. Perhaps the greatest service rendered by Professor Hobson to the cause of reform in mathematical teaching was the prominent part he took (in conjunction with Professors Forsyth, Baker, and Hardy) in advocating the abolition of the order of merit in the Mathematical Tripos. The wonder is that there could endure for so long a system which classified William Thomson (afterwards Lord Kelvin) as second to one who was utterly lacking in originality, but this happened in 1845, and it was not until 1909 that the system came to an end, thanks to a group of determined reformers among whom the three Sadleirian Professors were prominent.
The fourth and present Sadleirian Professor is Godfrey Harold Hardy. Born on 7th February, 1877, he was educated at Winchester and at Trinity College, Cambridge. He was Fourth Wrangler in 1898. In 1900 he took the second part of the Tripos and was placed in the First Division of the First Class. In the same year he was elected to a Fellowship at Trinity. For a time he took pupils in conjunction with the other Smith's Prizeman of 1901, Mr J H (now Sir James) Jeans, one teaching pure mathematics and the other applied. He became a lecturer for Trinity College in 1906 and succeeded Dr H F Baker as Cayley lecturer in 1914. These posts were held until 1919, when he was appointed to the Savilian chair of Geometry in the University of Oxford. He resigned this to take up his duties in Cambridge as Sadleirian Professor in October.
Professor Hardy's output of research has been very great. In the London Mathematical Society's Proceedings alone there have appeared over sixty papers, and there are a great many more in other English and foreign journals. Most of these deal with convergence of series or the analytic theory of numbers. Several have been written in collaboration with Professor Littlewood. Landau's Vorlesungen über Zahlentheorie (1927) gives great prominence to a set of theorems which he calls the first, second, third and fourth Hardy-Littlewood theorems. He also mentions the Hardy identity, the Hardy-Landau identity, and Hardy's theorem on the roots of the Zeta-functions. Hardy's convergence theorem is now standard; it will be found in Whittaker and Watson's Modern Analysis, Chap. VIII. It is interesting to notice that some foreign writers (e.g. in Acta Mathematica) make the Hardy-Littlewood methods the starting-point of their own work. Many eminent Cambridge mathematicians remain almost unknown to the rest of the mathematical world, but Professor Hardy has never been isolated.
It was one of his writings (the tract on Orders of Infinity, 1910) that helped to fin the imagination of the Indian genius Ramanujan. The subsequent correspondence led ultimately to Ramanujan settling in Cambridge. However, he was peculiarly weak in the power of expressing himself, and his papers would probably never have been published if it had not been for the self sacrificing labours of Professor Hardy in putting them into intelligible form.
Professor Hardy has written three of the Cambridge Tracts in Mathematics and Mathematical Physics. One has already been mentioned; the others are The Integration of Functions of a Single Variable (1905) and The General Theory of Dirichlet's Series (1915, in collaboration with M Riesz). His only text-book, A Course of Pure Mathematics, first appeared in 1908. The late Mr A Berry, in reviewing the fifth edition (Mathematical Gazette, XIV, pp. 428-9, April 1929), said "he has shown in this book and elsewhere a power of being interesting, which is to my mind unequalled by any of the eminent men (with the possible exception of M Picard) whom I have just mentioned. I suggest to Professor Hardy that he would probably be increasing his service to English mathematics if he were to divert to this purpose" (the writing of a substantial treatise on analysis) "some of the mental energy and time that he would otherwise devote to drawing somewhat closer the cordon that surrounds the unknown zeroes of Riemann's Zeta-function, and to similar problems."
Professor Hardy was elected a Fellow of the Royal Society in 1910 and awarded the Royal Medal in 1920. He was President of the London Mathematical Society in 1926-8 and of the Mathematical Association in 1924-6. Honorary degrees have been conferred upon him by Birmingham, Manchester, Marburg, and Oslo, and he is a member of learned societies in Austria, Czechoslovakia, Denmark, Germany, India, Poland, Russia, Sweden, and the United States.
Professor Hardy's contributions to the Mathematical Gazette have one dominant theme running throughout. All his life he has been fighting against the tendency for English mathematics to become stereotyped out of touch with current tendencies abroad. It will be remembered that the generation that followed Newton adhered to his methods exclusively to the neglect of the more powerful methods that had developed upon the continent. This brought Cambridge and English mathematics into an isolated state, from which they were rescued at the beginning of the nineteenth century by the labours of Woodhouse and of the Analytical Society (Peacock, Babbage, and Herschel). Apparently this state of affairs tends to recur. Professor Hardy's earlier contributions to the Gazette included several reviews, in which he severely attacked text-books which reproduced the errors which have unfortunately become traditional among English authors. The best statement of his views are contained in his two presidential addresses What is Geometry? (XII, pp. 309-316, March 1925) and The Case against the Mathematical Tripos (XIII, pp. 61-71, March 1926). He declared it broadly true that "Tripos mathematics was a collection of elaborate futilities," and quoted the opinion of a foreign friend that the peculiar characteristics of English mathematics had been "occasional flashes of insight, isolated achievements sufficient to show that the ability is really there, but, for the most part, amateurism, ignorance, incompetence, and triviality." These evils he ascribes to the Mathematical Tripos. "The system is vicious in principle, and ... the vice is too radical for what is usually called reform. I do not want to reform the Tripos, but to destroy it." This address was one of the most striking ever delivered to our Association, and it made a deep impression upon all who listened to it. The majority agreed with the denunciation of the present system, but there was some fear that the drastic remedies proposed might bring forth even worse results.
The Mathematical Association will wish Professor Hardy every success in his tenure of the Sadleirian chair.
H T H Piaggio.
University College, Nottingham
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