Oswald Veblen, who was elected an Honorary Fellow of the Royal Society of Edinburgh in 1943, was born in Decorah, Iowa, U.S.A., on June 24, 1880. He died on August 10, 1960, in Brooklin, Maine, while he was on holiday with his wife from their home in Princeton, New Jersey. He was of Norwegian descent, his paternal grandfather and his mother having both been born in Norway. His parents were Andrew A Veblen and Kirsti Veblen (née Hougan), of whose eight children he was the eldest. At the time of his death one of his three brothers and all four of his sisters were still living. He was a nephew of the economist Thorstein Veblen. In 1908 he married Elizabeth M D Richardson of Dewsbury, Yorkshire, sister of the physicist 0 W (later Sir Owen) Richardson. He was related by marriage also to another distinguished physicist, Clinton Joseph Davisson, who was the husband of his wife's sister Charlotte.

In 1898 Veblen graduated Bachelor of Arts (A. B.) at the University of Iowa, where his father had become Professor of Physics. He stayed at the University for a further year as an associate in Physics, during which time he took over his father's classes while the latter was ill with typhoid fever. He then went to Harvard, where he graduated A.B. in 1900. He spent the next three years as a graduate student at Chicago (where his uncle Thorstein was Professor of Political Economy), working under the supervision of E H Moore, and gained his Ph.D. in 1903. He then became an Associate in Mathematics at Chicago, where he remained until he was appointed Preceptor in Mathematics at Princeton University in 1905. In accepting the latter post he became, in the phrase of the then President of the University, Woodrow Wilson, one of "the fifty new preceptor guys" whose appointment helped to raise the academic status of a university that already had 259 years of distinguished history behind it. In 1910 Veblen was promoted to a full Professorship at Princeton, and in the middle 1920s he was appointed to the newly-established Henry Burchard Fine Professorship of Mathematics. He remained in this post until the Institute for Advanced Study was founded in Princeton in 1932-33, when he, along with James Wendell Alexander, Albert Einstein, John von Neumann and Hermann Weyl, became one of its first Professors. On his retirement in 1950 he received the title of Professor Emeritus of the Institute, and was elected President of the International Congress of Mathematicians for the meeting held in that year at Harvard.

He gave distinguished service in the two World Wars. From 1917 to 1919 he was a captain, and later major, in the Ordnance Department of the U.S. Army, and in the Second War he helped to build up a high-grade research team for the Ballistic Research Laboratories. At the time of his death the U.S.A. Chief of Ordnance wrote of "the key role Dr Veblen played in the expansion of activities of the Ballistic Research Laboratories ..., where he pioneered the development of ballistics research .... Dr Veblen's work typified the patriot-scientist and was in the finest tradition of American citizenship".

If all of Veblen's services to mathematics are taken into account he is certainly to be ranked among the world's greatest mathematicians of the first half of the twentieth century. His own research work and scholarship were of the highest distinction: he was a source of inspiration to many others and particularly to young mathematicians: and he played a leading part in making Princeton a world centre of mathematics. His work was widely recognized: he received honorary doctorates from the Universities of Oslo, Oxford, Hamburg and Chicago, and honorary membership or fellowship of learned societies in Denmark, England, France, Ireland, Italy, Peru, Poland and Scotland.

Most of the results of his research, published initially in the usual way in periodicals, were collected together and further extended in books that helped to lay the foundations of much of the abstract mathematics of the 1940s and 1950s. It is interesting to read, for example, in the preface to Volume II (1918) of Veblen and Young's

The use made by Einstein in 1916 of tensor analysis and Riemannian geometry in the formulation of general relativity brought these subjects to the forefront of mathematical research and led in the 1920s and early 1930s to a rapid development of generalized differential geometry. In this development Veblen played a large part and was able to use his deep knowledge and understanding of geometry and topology. It was characteristic of him that his work in this field came to be represented more by books written by himself alone, or in collaboration with others, than by research papers. In 1927 there appeared his Cambridge tract Invariants of quadratic differential forms, which gave an account, typical in its conciseness and clarity, of the theory of differential invariants treated from a geometrical and essentially elementary standpoint. A specially useful feature of this tract was the historical section that appeared at or near the end of each chapter.

What may in the future come to be regarded as his most important work was done in collaboration with a young English Commonwealth Fund Fellow, J H C Whitehead, who later became Waynflete Professor of Pure Mathematics at Oxford, and whose tragically sudden death preceded that of Veblen by a few months. This work also appeared in a Cambridge tract, namely

Veblen's next major publication was

In his later years, Veblen's main mathematical interest was in the theory of spin-tensors or spinors, which arose out of the need to bring Dirac's equations (1928) into conformity with the relativistic principle of covariance. In 1936 there appeared a mimeographed volume

In appearance Veblen was tall, well-built, slender and good looking. His voice was quiet, his speech often containing a very slight hesitancy that added to his general charm. He showed a special interest in younger mathematicians, to whom he and his wife showed great kindness. His home life was completely happy, though to the great regret of him and his wife they had no children. He was liberal in his social outlook and took as a compliment a remark made to him by a candid friend that his views were those of a late-Victorian radical. Shortly after his death his colleagues of the Institute for Advanced Study wrote of him that:

During the last two or three years of his life he suffered from a strained heart due, perhaps, to the continuance into old age of his favourite exercise of chopping logs. He was also afflicted with blindness; though when he realized that he had retained some peripheral vision, he set about inventing, with characteristic enthusiasm and fair success, an instrument designed to assist the sight of people similarly afflicted.

I am grateful to Mrs Veblen for supplying me with many of the personal details included in this notice, and to Mr R Fletcher, research student, for help in compiling the following list of Veblen's publications.

In 1898 Veblen graduated Bachelor of Arts (A. B.) at the University of Iowa, where his father had become Professor of Physics. He stayed at the University for a further year as an associate in Physics, during which time he took over his father's classes while the latter was ill with typhoid fever. He then went to Harvard, where he graduated A.B. in 1900. He spent the next three years as a graduate student at Chicago (where his uncle Thorstein was Professor of Political Economy), working under the supervision of E H Moore, and gained his Ph.D. in 1903. He then became an Associate in Mathematics at Chicago, where he remained until he was appointed Preceptor in Mathematics at Princeton University in 1905. In accepting the latter post he became, in the phrase of the then President of the University, Woodrow Wilson, one of "the fifty new preceptor guys" whose appointment helped to raise the academic status of a university that already had 259 years of distinguished history behind it. In 1910 Veblen was promoted to a full Professorship at Princeton, and in the middle 1920s he was appointed to the newly-established Henry Burchard Fine Professorship of Mathematics. He remained in this post until the Institute for Advanced Study was founded in Princeton in 1932-33, when he, along with James Wendell Alexander, Albert Einstein, John von Neumann and Hermann Weyl, became one of its first Professors. On his retirement in 1950 he received the title of Professor Emeritus of the Institute, and was elected President of the International Congress of Mathematicians for the meeting held in that year at Harvard.

He gave distinguished service in the two World Wars. From 1917 to 1919 he was a captain, and later major, in the Ordnance Department of the U.S. Army, and in the Second War he helped to build up a high-grade research team for the Ballistic Research Laboratories. At the time of his death the U.S.A. Chief of Ordnance wrote of "the key role Dr Veblen played in the expansion of activities of the Ballistic Research Laboratories ..., where he pioneered the development of ballistics research .... Dr Veblen's work typified the patriot-scientist and was in the finest tradition of American citizenship".

If all of Veblen's services to mathematics are taken into account he is certainly to be ranked among the world's greatest mathematicians of the first half of the twentieth century. His own research work and scholarship were of the highest distinction: he was a source of inspiration to many others and particularly to young mathematicians: and he played a leading part in making Princeton a world centre of mathematics. His work was widely recognized: he received honorary doctorates from the Universities of Oslo, Oxford, Hamburg and Chicago, and honorary membership or fellowship of learned societies in Denmark, England, France, Ireland, Italy, Peru, Poland and Scotland.

Most of the results of his research, published initially in the usual way in periodicals, were collected together and further extended in books that helped to lay the foundations of much of the abstract mathematics of the 1940s and 1950s. It is interesting to read, for example, in the preface to Volume II (1918) of Veblen and Young's

*Projective Geometry*(which was in fact written wholly by Veblen himself, because Young was too busy to be able to contribute to the second volume) the dry remark that "I shall pass by the opportunity to discuss any of the pedagogical questions which have been raised in connection with the first volume and which may easily be foreseen for the second. It is to be expected that there will continue to be general agreement among those who have not made the experiment that an abstract method of treatment of geometry is unsuited to beginning students. In this book, however, we are committed to the abstract point of view". Four years after the publication of this volume there appeared Veblen's book on analysis situs, a subject which, now usually called topology, has played a dominant part in the revolutionary developments of mathematics of recent decades. In the preface to his book*Topology*(1930), S Lefschetz pays tribute to the fundamental work on topology done by Poincaré, but says that the foundations of the subject were left by him "in rather unstable equilibrium. It is largely to Veblen and Alexander that we owe the remedy for this state of affairs, and the present improved situation. A date marks the transition: 1922, when there appeared Veblen's excellent Cambridge Colloquium Le*ctures: Analysis Situs,*which has deservedly become the standard work on the subject. The ground being thus well prepared, new developments came rapidly ...".The use made by Einstein in 1916 of tensor analysis and Riemannian geometry in the formulation of general relativity brought these subjects to the forefront of mathematical research and led in the 1920s and early 1930s to a rapid development of generalized differential geometry. In this development Veblen played a large part and was able to use his deep knowledge and understanding of geometry and topology. It was characteristic of him that his work in this field came to be represented more by books written by himself alone, or in collaboration with others, than by research papers. In 1927 there appeared his Cambridge tract Invariants of quadratic differential forms, which gave an account, typical in its conciseness and clarity, of the theory of differential invariants treated from a geometrical and essentially elementary standpoint. A specially useful feature of this tract was the historical section that appeared at or near the end of each chapter.

What may in the future come to be regarded as his most important work was done in collaboration with a young English Commonwealth Fund Fellow, J H C Whitehead, who later became Waynflete Professor of Pure Mathematics at Oxford, and whose tragically sudden death preceded that of Veblen by a few months. This work also appeared in a Cambridge tract, namely

*The foundations of differential geometry*(1932), and was intended as a companion to Veblen's earlier tract. It gave a detailed analysis of the assumptions made in generalized differential geometry and placed the whole subject upon an axiomatic basis. To an important extent it helped to pave the way for what has come to be known as "differential geometry in the large", the main development of which has taken place since the end of the Second World War. In 1934, after Whitehead had returned to England, Veblen was heard to make the joking remark, "Well, if I've done nothing else, I think I can claim to have made a mathematician out of Henry Whitehead " - a remark intended to reflect neither upon Whitehead's ability nor upon other senior mathematicians under whose guidance he had worked: it was rather an amused comment upon his young collaborator's vitality and*joie de vivre.*Whitehead's death, coming within a few hours of a visit paid to Veblen in Princeton, was a source of great grief to the latter.Veblen's next major publication was

*Projektive Relativitätstheorie*(1933). It was written in German not because the publishers would have been unwilling to publish it in English but because writing it helped him, Veblen, to improve his knowledge of the German language or so he said. It was a unified theory of gravitation and electromagnetism which, if it shared with other unified theories the quality of cutting little or no physical ice, was a construct of great elegance. Its main value was perhaps pure mathematical in the account it gave of projective differential geometry as formulated by Veblen.In his later years, Veblen's main mathematical interest was in the theory of spin-tensors or spinors, which arose out of the need to bring Dirac's equations (1928) into conformity with the relativistic principle of covariance. In 1936 there appeared a mimeographed volume

*The geometry of complex domains*(revised edition, 1955), which was a record of a seminar conducted by Veblen and J von Neumann and contained notes on lectures given by Veblen and his then pupil J Wallace Givens. This gave a full account of the theory of linear and antilinear transformations in complex projective n-space and of its application to spinor theory. An interest in spinors remained with Veblen for a long time, well on into his years of retirement, during which he developed various approaches to the theory and analytical techniques for dealing with them. It is regrettable that this work was never published.In appearance Veblen was tall, well-built, slender and good looking. His voice was quiet, his speech often containing a very slight hesitancy that added to his general charm. He showed a special interest in younger mathematicians, to whom he and his wife showed great kindness. His home life was completely happy, though to the great regret of him and his wife they had no children. He was liberal in his social outlook and took as a compliment a remark made to him by a candid friend that his views were those of a late-Victorian radical. Shortly after his death his colleagues of the Institute for Advanced Study wrote of him that:

His effect on mathematics, transcending the Princeton community and the country as a whole, will be felt for decades to come ..., and he was a powerful force in establishing the highest academic standards in general.

*He loved simplicity and disliked sham .... He possessed the art of friendship .... We are grateful for his unusual wisdom, for his unflinching integrity and honesty, for his uncompromising ideals, and, not least, for his generous friendship."*During the last two or three years of his life he suffered from a strained heart due, perhaps, to the continuance into old age of his favourite exercise of chopping logs. He was also afflicted with blindness; though when he realized that he had retained some peripheral vision, he set about inventing, with characteristic enthusiasm and fair success, an instrument designed to assist the sight of people similarly afflicted.

I am grateful to Mrs Veblen for supplying me with many of the personal details included in this notice, and to Mr R Fletcher, research student, for help in compiling the following list of Veblen's publications.

Oswald Veblen's RSE obituary by H S Ruse appeared in

*Royal Society of Edinburgh Year Book 1962,*52-59.