JOSEPH HENRY MACLAGEN WEDDERBURN

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2 V JOSEPH HENRY MACLAGEN WEDDERBURN Joseph H. M. W edderburn, Professor Emeritus in the Department of Mathematics of Princeton University, Princeton, New Jersey, U.S.A., was found dead in his residence at Princeton on 9 October Professor Wedderburn who was a bachelor had, for many years, lived alone. His body was found by the people who took care of the house and grounds. The medical authorities concluded, after examination, that death had occurred several days previously as a result of a heart attack. Wedderburn was born on 26 February 1882, in Forfar, Scotland, the tenth child in a family of fourteen children, which included seven brothers and six sisters. His father was Alexander Stormonth Maclagen Wedderburn, M.D., of Pearsie. His mother was Anne Ogilvie. On his father s side his grandfather was Parish Minister of Kinfauns and Professor of Exegesis in the Free Church College of Aberdeen. His paternal great-grandfather was Parish Minister of Blair Atholl and a Chaplain in the Black Watch. On his maternal side his grandfather was a lawyer in Dundee, as had been true of the family for several preceding generations. The detailed history of the Wedderburn family has been recorded in the Wedderburn Book. Of his many brothers and sisters only two survived him, a brother, Ernest Wedderburn of Edinburgh, Scotland, and a sister, Miss Elizabeth Wedderburn, of Paris, France. Wedderburn s early school years were spent at the Forfar Academy from 1887 to In the latter year he transferred for the last three years of his school education to George Watson s College in Edinburgh, From this college at the age of sixteen and a half years he obtained the leaving scholarship and entered Edinburgh University in the autumn of After five years as a student in this university he obtained, in 1903, the M.A. degree with First-Class Honours in Mathematics. His first research On the isoclinal lines of a differential equation of the first order was published in the Proceedings of the Royal Society of Edinburgh in the same year, when the author had just reached his twenty-first year. Two other communications to the Royal Society of Edinburgh date from the same year. During the year Wedderburn had served as Nichol Assistant in the Physical Laboratory of the University. In the following year Wedderburn pursued postgraduate studies in Germany. In the winter semester of he attended the University of Leipzig; in the summer semester of 1904 he was at work in the University of Berlin. Appointed Carnegie Fellow for the year he proceeded to the United States where the American phase of his scientific career began in postgraduate 6iq

3 620 Obituary Notices study at the University of Chicago. Two contributions to the Transactions of the American Mathematical Society stem from this period, one with S. Epsteen On the structure of hypercomplex number systems and a second entitled A theorem on finite algebras. Following the year in the United States was a period of four years, , once more in Edinburgh as Lecturer in Mathematics and assistant to Professor George Chrystal. Communications to the Royal Society of Edinburgh, to the Edinburgh and London Mathematical Societies, to the Annals of Mathematics and, in collaboration with Oswald Veblen, to the Transactions of the American Mathematical Society characterize these years as a creative scientist. His joint authorship with Veblen stemmed from the year in Chicago and was an advance notice of a future association of both in Princeton University in the years ahead. Meanwhile the researches already accomplished earned for Wedderburn the degree of Doctor of Science in Edinburgh University in From 1906 to 1909 he served as editor of the Proceedings of the Edinburgh Mathematical Society. Wedderburn entered on the Princeton phase of his life s work in the autumn of He was called to Princeton as Preceptor in Mathematics, joining a goodly company of distinguished members of the Princeton faculty. The preceptor guys as several generations of Princeton students have labelled them, not without deep affection and esteem, were the creation of Woodrow Wilson in his reorganization of the Princeton Faculty. That revolution required a large scale infusion of new blood, of scholars who would assume an intimate personal relation with small groups of undergraduates and impart to them something of their own enthusiasm for things of the mind. To Wilson, the preceptorial system seemed the only effectual means of making university instruction the helpful and efficient thing it should be. With a body of fifty preceptors which he brought to Princeton in 1905 he planned to transform the place where there are youngsters doing tasks to a place where men are doing thinking. President Wilson laid upon Dean H. B. Fine of the Mathematics Department of Princeton University the responsibility of shaping his policy in the area of the physical and natural sciences. It was thus that L. P. Eisenhart and Oswald Veblen, Gilbert A. Bliss, George D. Birkhoff and Joseph H. M. Wedderburn joined the preceptors of mathematics in Princeton in the years 1905 to Bliss left after three years to pursue his distinguished career in Chicago. Birkhoff stayed three years and proceeded to Harvard to become there eventually leader in mathematics and Dean of the Faculty of Arts and Sciences. Eisenhart, Veblen and Wedderburn stayed on at Princeton, building up its standards and traditions in mathematical research. Veblen left Princeton University in 1932 to establish, by a process of fission from the Department of Mathematics in the University, a Department of Mathematics in the newly organized Institute for Advanced Study also located in Princeton. Veblen, Alexander and von Neumann made the transition, Eisenhart and Wedderburn stayed behind. Five happy years as preceptor at Princeton were succeeded by the five grim

4 Joseph Henry Maclagen Wedderburn 621 years of World War I. Wedderburn responded immediately to the call to arms. In view of his already great scientific achievements it would not have been difficult at all for him to have obtained a commission in the British Army at the outset of hostilities. He had already seen service in the Territorial Forces before the war. It is altogether characteristic of the man and of his exceptional modesty that he volunteered at the beginning of hostilities as a private. The Princeton records show that he was the first resident of Princeton to leave for the war on the outbreak of hostilities, and had the longest war service of any member of the University or resident of the town. By November 1914, he had obtained his commission as Lieutenant in the 10th Battalion of the Seaforth Highlanders, receiving his Captaincy in January From 1915 to February 1918 he continued as Captain in the Seaforth Highlanders but transferred then to the Royal Engineers as Captain in the 4th Field Survey Battalion. In this capacity he saw service in France from January 1918 to March To this service he brought his exceptional gifts as mathematician and physicist taking an active part in the development of sound-ranging techniques for the location of enemy batteries and the practical employment of those techniques on the field of battle. His services were mentioned in despatches from the British Army in France. Wedderburn returned to Princeton after the conclusion of the war still in ihe capacity of preceptor. In he was Assistant Professor of Mathematics, and was promoted in the spring of 1921 to an Associate Professorship on permanent tenure. Seven years later he was promoted to Professor of Mathematics, a position which he held until his retirement in 1945, with a continuous faculty record of thirty-seven years of service. During those years he took an active part in all the activities of the Department of Mathematics. For a number of years he was in charge of the most important undergraduate course. That he was an exceptionally good teacher has been the testimony of a generation of undergraduates in a subject of study which, traditionally, earns such verdicts infrequently. When the Department of Mathematics at Princeton assumed responsibility for the publication of the Annals of, Wedderburn was appointed Editor-in-Chief and served for seventeen years, , in that capacity. In recognition of his eminence both as editor and scientist a volume of the Annals of Mathematics published in 1947 was dedicated to him. The frontispiece of the volume reveals a fine photograph of the man. His colleagues among the Princeton preceptors recall with enthusiasm the rich human qualities of Wedderburn. They recall his passion for play as well as for work, his desire for companionship and association with men. He loved the out-of-doors, found deep satisfaction in the wilderness, in the woods, canoeing along rivers and streams in the company of thoughtful men. As in his scientific work, he brought to the construction of the camp-site, the erection of the tent, the paddling of the canoe up- and down-stream the qualities of a complete perfectionist. In the wilds of Northern Canada, with congenial men, he found complete happiness. In the tasks of leisure hours he utilized the same resources of knowledge in mathematics, mechanics and physics that marked

5 622 Obituary Notices him out among his scientific colleagues in the class-room and the mathematics seminar. His taste in literature ran to books of travel and he accumulated a large library of travel. He admired John Buchan both for his native Scottish background and his appreciation of the open spaces of the earth. While Wedderburn had a broad and comprehensive knowledge of pure and applied mathematics, his research work was done exclusively in algebra. Indeed, algebra was his all-embracing scientific passion and it is in algebra that he made his eminent contributions, distinguished no less by their originality than by their depth. It was these characteristics which earned for him, from the Royal Society of Edinburgh, his first major distinction, the award to him in 1921 of the MacDougall-Brisbane Gold Medal and Prize for the period , in recognition of his investigations on hypercomplex numbers. The Royal Society of London, in 1933, elected him Fellow, also by reason of his distinction in the field of algebra. His mathematical discoveries include two famous theorems which bear his name, frequently quoted by mathematicians. One of his Princeton colleagues in mathematics considers him as the founder of a very important branch of algebra where many of his ideas still bear fruit in current research. Restricting his discussion of Wedderburn s work to the two theorems constituting his most outstanding discoveries, he writes: Before Wedderburn began his investigations, the classification of the semisimple algebras was done only if the ground field was the field of real or complex numbers. Interesting as this investigation was, it did not lead to a deeper insight into the nature of the hypercomplex numbers. Wedderburn attacked the problem in full generality and introduced new methods. This led to a complete understanding of the structure of semi-simple algebras over any given ground field. He showed that they are a direct sum of simple algebras and finally proved the celebrated theorem that a simple algebra consists of all matrices of a given degree with elements taken from a division algebra. These results were found in 1907 and stated in his famous paper entitled On hypercomplex numbers. This paper contains many other results and ideas and was the beginning of a new era in the theory. The second important contribution concerns the investigation of skew fields with a finite number of elements. The commutative case had been investigated before by E. H. Moore and had led to a complete classification of all commutative fields with a finite number of elements. Since a non-commutative finite field had never been found one could suspect that they do not exist. It was Wedderburn who showed in 1905 that this is indeed the case. This second theorem gives at once the complete classification of all semisimple algebras with a finite number of elements. But it also had numerous other applications in number theory and in projective geometry. It gave at once the complete structure of all projective geometries with a finite number of points and showed the interesting fact that in all these geometries the theorem of Pascal is a consequence of the theorem of Desargues.

6 Joseph Henry Maclagen 623 It is of the article On hypercomplex numbers communicated to the Proc. Lond. Math. Soc., 6, 77, that Hermann Weyl, in his book The classical, remarks of paramount importance for the modern development is Wedderburn s paper of the year 1908, where he investigates associative algebras in an arbitrary number field k\ After the war period and during the 1920 s came a group of papers dealing among others with The automorphic transformation of a bilinear form, a lengthy paper on Algebraic fields and another dealing with Algebras which do not possess a finite basis. During these years his contributions were communicated to the Transactions of the American Mathematical Society and to the Annals of Mathematics of which he was at that time Editor-in-Chief. Of the later period of Wedderburn s work after 1930, I am indebted to his distinguished pupil, Professor Nathan Jacobson of the Department of Mathematics in Yale University for the following comments: The most important publication by Professor Wedderburn after 1930 was undoubtedly his book Lectures on Matrices, New York, T result of a number of years of painstaking labor is evidenced by the Bibliography of 661 items (in the revised printing) covering the period 1853 to The work is, however, not a compilation of the literature, but a synthesis that is Wedderburn s own. It contains a number of original contributions to the subject. Though he did not follow the abstract point of view that had just become dominant, neither did he commit the error made by others of treating matrix theory as an art of juggling elements in an array. The important ideas of linear transformations, vector spaces, bilinear forms, though not set off, as is common in the most modern treatments, do appear in Wedderburn s book. Also, as in his best work, one finds here some neat and suggestive algebraic devices that make the book a very valuable reference book, even at the present time. Of the papers that Wedderburn published after 1930, I believe the best one was Non-commutative domains of integrity (Crelle 1931). In this paper the author gave an improvement and extension of known results on reducing a matrix with elements in a Euclidean domain to diagonal form. This stimulated further work on this problem and led to a complete solution a few years later. Also Wedderburn considered in this paper the important problem of imbedding a non-commutative domain in a division ring. He showed that this can be done if common multiples exist. This result was found independently by Ore. The paper The canonical form of a matrix contains a useful canonical form that displays the inseparability of the prime factors of the minimum polynomial. Boolean linear associative algebra presents a problem concerning which very little is known. In conclusion, I believe that Wedderburn s best papers appeared in the period However, all of his work indicated originality and great independence of ideas. The pace in algebra greatly increased in an important measure due to his stimulating discoveries and it is regrettable that for various

7 624 Obituary Notices reasons he did not keep abreast of the latest developments in the later years of his life. Towards the close of the 1920 s Wedderburn suffered what some of his friends have considered a mild nervous breakdown. More and more he withdrew from normal companionships, turned away from such leisurely pursuits and distractions as he had hitherto enjoyed. He followed an increasingly solitary life, dining alone at his club, making no effort to seek out his friends. Even on holidays in the Berkshire Mountains he would spend long intervals in solitude and silence. Those who deliberately penetrated beyond this self-imposed barrier found the same friendly human person that his earlier preceptor colleagues had known, with the same intellectual vigour, originality, depth of understanding and quality of effort that characterized all his scientific work. To one such visitor he occasionally remarked that he lived too much alone. To reverse the trend seems, however, to have been beyond his capacities if not his desires. In the last years his solitariness increased so that a new generation of undergraduates and research scholars saw less and less of him. By the year 1945 it became evident to the university authorities that the best interests of Wedderburn and the university would be served if he was placed on the retirement list some years ahead of the normal time of withdrawal. He was made Professor Emeritus in 1945 and spent the remaining years, generally alone, seeing little of colleagues and friends. It was possible occasionally to break down the barriers of his reserve as, for example, on the occasion of the Princeton Bicentennial Celebrations; at the conference on the physical sciences, when, as it happened, a dozen or so of the Fellows of the Royal Society were at one time in town he accepted an invitation to meet with them at tea. But these were rare and exceptional occasions, less and less frequent as the years passed on until, as the end proved and as he himself had said, he lived too much alone. The writer is deeply indebted to many of his friends and colleagues of Princeton University, of the Institute for Advanced Study and of the town of Princeton for much of the present record. We all knew Wedderburn from varied points of view, from a diversity of interests. We all share a recollection of a gifted scientist and scholar, who contributed richly to the standards of mathematical education in Princeton and to the elevation of scholarship and research in pure mathematics in a research centre more fortunate than most in the wealth of its mathematical talent. H. S. T aylor

8 Joseph Henry Maclagen Wedderburn 625 BIBLIOGRAPHY On the isoclinial lines of a differential equation of the joint order. Proc. Roy. Soc. Edinb. 24, O n the general scalar function of a vector. Proc. Roy. Soc. Edinb. 24, O n the application of quaternions in the theory of differential equations. Trans. Roy. Soc. Edinb. 40, N ote on the linear m atrix equation. Proc. Edinb. M ath. Soc. 22, (W ith S. Epsteen.) On the structure of hypercomplex num ber systems. Trans. Amer. M ath. Soc. 6, A theorem on finite algebras. Trans. Amer. M ath. Soc. 6, O n a theorem in hypercom plex num bers. Proc. Roy. Soc. Edinb. 26, N ote on hypercom plex num bers. Proc. Edinb. M ath. Soc. 25, (W ith O. Veblen.) N on-d esarguesian and non-pascalian geometries. Trans. Amer. M ath. Soc. 8, O n hypercom plex num bers. Proc. Lond. M ath. Soc. (2) 6, On the direct product in the theory of finite groups. Ann. M ath. Princeton (2), 10, On certain theorems in determ inants. Proc. Edinb. M ath. Soc. 27, N ote on the rank of a symmetrical matrix. Ann. M ath. Princeton (2), 15, O n continued fractions in non-com m utative quantities. Ann. M ath. Princeton (2), 15, N ote on the simple difference equation. Ann. M ath. Princeton (2), 16, N ote on the rank of a symmetrical matrix II. Ann. M ath. Princeton (2), 16, A type of prim itive algebra. Trans. Amer. M ath. Soc. 15, On long waves. A m er.j. M ath. 36, On m atrices whose coefficients are functions of a single variable. Trans. Amer. M ath. Soc. 16, On the equations of m otion of a single particle. Proc. Roy. Soc. Edinb. 41, T h e autom orphic transform ation of a bilinear form. Ann. M ath. Princeton (2), 23, On division algebras. Trans. Amer. M ath. Soc. 22, T h e functional equation j( x 2) = 2 x * + [g(jc)]2. Ann. M ath. Princeton (2), 24, Algebraic fields. Ann. M ath. Princeton (2), 24, Algebras which do not possess a finite basis. Trans. Amer. M ath. Soc. 26, A theorem on simple algebras. Bull. Amer. M ath. Soc. 31, T he absolute value of the product of two matrices. Bidl. Amer. M ath. Soc. 31, Note on matrices in a given field. Ann. M ath. Princeton (2), 27, N on-com m utative domains of in teg rity.^, reine. angew. M ath. 167, Lectures on matrices. Amer. M ath. Soc. Colloq. Pub. vii pp Boolean linear associative algebra. Ann. M ath. Princeton (2), 35, Note on algebras. Ann. M ath. Princeton (2), 38, A special linear associative algebra. Proc. Edinb. M ath. Soc. (2), 5, T h e canonical form of a matrix. Ann. M ath. Princeton (2), 39, T he division transformation. M ath. Gaz 23, Hom om orphism of groups. Ann. M ath. Princeton (2), 42, Desargues theorem. Edinb. M ath. Notes, 34, Pascal s theorem. Amer. M ath. Mon. 52, Pascal s theorem. Amer. M ath. Mon. 53, 86.