**Joseph Wedderburn**'s father, Alexander Stormonth Maclagen Wedderburn, was a medical doctor. Alexander Wedderburn came from a family of Ministers of the Church with his father being the Parish Minister of Kinfauns and his grandfather (Joseph's great-grandfather) being Parish Minister of Blair Atholl. Joseph's mother was Anne Ogilvie and she came from a family of lawyers; Anne's father had been a lawyer in Dundee. Anne and Alexander Wedderburn had a large family, Joseph being one of fourteen children, eight boys and six girls. In fact Joseph was the tenth child of the family.

Joseph was brought up in Forfar, north of Dundee, and he attended Forfar Academy from the age of five until he was thirteen. He then went to George Watson's College, an independent school in Edinburgh, for three years. In 1898 he completed his school education and won a scholarship to study at the University of Edinburgh. He entered Edinburgh University in 1898, at the age of sixteen and a half.

It was a time when Wedderburn made remarkable progress with his mathematics and in addition during 1902-03 he worked as an assistant in the Physical Laboratory of the University. He began mathematical research while still an undergraduate and his first paper, *On the isoclinal lines of a differential equation of the first order* was published in the *Proceedings of The Royal Society of Edinburgh* in 1903. Two other papers which he published in the same year in publications of the Royal Society of Edinburgh were on the scalar functions of a vector and on an application of quaternions to differential equations. He obtained an M.A. degree with First Class Honours in mathematics from the University of Edinburgh in 1903.

Wedderburn then pursued postgraduate studies in Germany spending session 1903-1904 at the University of Leipzig and then the summer semester of 1904 at the University of Berlin. Already Wedderburn's mathematical interests were in algebra and his German trip allowed him to interact with Frobenius and Schur. He was awarded a Carnegie Scholarship to study in the United States and he spent 1904-1905 at the University of Chicago where he did joint work with Veblen. Chicago was, of course, an excellent place to continue his deepening interest in algebra for, in addition to Veblen, Eliakim Moore and L E Dickson were there at this time. The determination of finite division algebras was a very natural problem in the light of the work being undertaken in Chicago, and as soon as he arrived at Chicago, Wedderburn started to work on it, in close contact with Dickson.

Returning to Scotland in 1905, Wedderburn worked for four years at the University of Edinburgh as an assistant to George Chrystal. The depth of Wedderburn's contribution to algebra during these years in Edinburgh was remarkable. In 1905 he showed that a non-commutative finite field could not exist. In the paper he published in that year he gave three proofs of this theorem which were all based on a clever use of the interplay between the additive group of a finite division algebra *A*, and the multiplicative group *A** = *A*-{0}.

In [10] Parshall discusses this theorem. She notes that the first of the three proofs has a gap in it which was not noticed at the time. This is in fact significant since Dickson also found a proof of this result but, since Wedderburn had already found his first "proof" (which Dickson believed to be correct), Dickson acknowledged Wedderburn's priority in a paper he wrote on the topic. Dickson noted in the paper that it is only after having seen his proof that Wedderburn constructed his second and third proofs. Parshall's work here shows that really Dickson should be credited with having found the first correct proof.

This theorem gave, as a corollary, the complete structure of all finite projective geometries. These geometries consisted of a set of "points", a set of "lines" and an "incidence relation" between points and lines, subject only to the conditions that two distinct points are on a single line, two distinct lines have a single common point and a line contains at least three points. Wedderburn and Veblen showed that in all these geometries Pascal's theorem is a consequence of Desargues' theorem. They published the paper *Non-Desarguesian and non-Pascalian geometries* in the *Transactions of the American Mathematical Society* in 1907 in which they constructed finite projective geometries which are neither "Desarguesian" nor "Pascalian" (this is Hilbert's terminology).

In 1907 Wedderburn published what is perhaps his most famous paper on the classification of semisimple algebras. In this paper *On hypercomplex numbers* which appeared in the *Proceedings of the London Mathematical Society,* he showed that every semisimple algebra is a direct sum of simple algebras and that a simple algebra was a matrix algebra over a division ring. From 1906 to 1908 he served as editor of the *Proceedings of the Edinburgh Mathematical Society.*

In 1909 Wedderburn returned to the United States being appointed a Preceptor in Mathematics at Princeton where he joined Veblen. We should say a word about the Preceptors at Princeton. They were the idea of Woodrow Wilson (who was to become the 28^{th} President of the United States in 1913). Woodrow Wilson had been Professor of Political Science at Princeton and, in 1902, he was appointed President of Princeton. He set out to change the nature of Princeton by making it a leading research active university. To do this, Wilson said:-

... 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.

Fifty Preceptors were to be appointed to achieve this in the whole university and Henry Fine, Dean of Mathematics, was put in charge of finding young mathematicians to fill the mathematics posts. Between 1905 and 1909 Eisenhart, Veblen, Bliss, George Birkhoff, and Wedderburn were appointed. The next five years were especially happy ones for Wedderburn and his fellow Preceptors described him during this time:-

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. ... His taste in literature ran to books of travel and he accumulated a large library of travel.

However the five happy years came to an end with the outbreak of the First World War. Immediately Wedderburn volunteered for the British Army but, being an exceptionally modest man, he enlisted only in the role of private. Records show that he was the first person at Princeton to volunteer for war service and that he had the longest war service of anyone on the staff. He served in France between January 1918 and March 1919, making use of his scientific skills. In France, as a Captain in the Fourth Field Survey Battalion, he devised sound-ranging equipment to pinpoint the positions of enemy guns.

On his return to Princeton he took up his post as Preceptor in Mathematics but he was soon promoted to Assistant Professor in 1920, obtaining permanent tenure as Associate Professor in 1921. He served as Editor of the *Annals of Mathematics* from 1912 to 1928. From about the end of this period Wedderburn seemed to suffer a mild nervous breakdown and became an increasingly solitary figure. It looks as if from this time on he suffered from depression. Certainly he stopped seeing his friends and although he seemed to recognise that his problems came from loneliness, rather than seek to be with people he deliberately cut himself off. Some of his friends made a strenuous effort to penetrate the barrier he was erecting and found that underneath was still the friendly, deep thinking, brilliant mathematician.

A comment on his teaching by Robert Hooke [5]

Wedderburn's lecturing style was unique, to say the least. He was apparently a very shy man and much preferred looking at the blackboard to looking at the students. He had the galley proofs from his book "Lectures on Matrices" pasted to cardboard for durability, and his "lecturing" consisted of reading this out loud while simultaneously copying it onto the blackboard. Ernst Snapper, who claimed to be only the fourth person ever with the courage to write a dissertation under Wedderburn(and one of the other three had lost his mind)told me this story explaining why Wedderburn was a bachelor. It seems that an old Scottish tradition required that a man, before marrying, accumulate savings equal to a certain percentage of his annual income. In Wedderburn's case his income had gone up so rapidly that he had never been able to accomplish this.

By 1945 Princeton gave him early retirement in his own best interests. From this time on his isolation became almost total. Although we have given 9 October 1948 as the date of his death, in fact he probably died a few days earlier than this. The people who looked after the house and grounds in Princeton were he lived found him on that day but the subsequent medical examination revealed that he had died of a heart attack several days earlier.

Parshall writes in [9]:-

According to officials at the bank which settled Wedderburn's estate ..., the papers remaining at his death were subsequently destroyed, thereby limiting historical study of Wedderburn's life and work almost exclusively to published sources.

Wedderburn made important advances in the theory of rings, algebras and matrix theory. His best mathematical work was done before his war service and we have referred to some of it above. In total he published around 40 works mostly on rings and matrices. His famous book is *Lectures on Matrices* (1934). This work was described by Jacobson, who was a student of Wedderburn's, in [11]. Jacobson writes:-

That this was the result of a number of years of painstaking labour is evidenced by the Bibliography of661items(in the revised printing)covering the period1853to1936. 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 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 ...

Among the honours which Wedderburn received were the MacDougall-Brisbane Gold Medal and Prize from the Royal Society of Edinburgh in 1921, and election to the Royal Society of London in 1933.

**Article by:** *J J O'Connor* and *E F Robertson*

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