Mathematically-inclined readers of this book should probably read it alongside Tom Hawkins' excellent book The Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869-1926 (New York: Springer-Verlag, 2000). That book is a masterly and thorough account of Lie's ideas, their original formulation, their immediate reception, and the way it became transformed into the theory of Lie groups and Lie algebras that plays such a central role in modern mathematics. This book is an equally thorough and masterly account of Lie's life. Like Stubhaug's previous book, on Abel, this one is handsomely printed, well illustrated, and exhaustively researched. Stubhaug has taken the bold decision to say almost nothing technical about Lie's mathematics. While this may well be right for his original Norwegian readership, because short accounts of what Lie did are notoriously obscure, this does mean that he never even explains that what Lie called a group is not a group in the modern sense at all.
That said, this is a curiously involving story of the life of one of the major creators of modern mathematics. We learn how he came to prominence in Norway, and how he matured, rather late, as an inventive mathematician. He had to push to get enough funds to study abroad, and once there he was lucky. He met, and impressed, Felix Klein, and on a journey to France in 1870-71 also impressed several French mathematicians. This was just as well, because Darboux's intervention seems to have rescued him from prison, where he had been put on surface in suspicion of being a German spy (this being the time of the Franco-German war). We then follow his career back in Norway, his time as a professor in Leipzig, and his eventual return to Norway, cut short by an early death from pernicious anaemia at the age of 57. For much of his career he was assisted by Friedrich Engel, and also, later by Georg Scheffers and we get a picture of how these collaborations worked. We also get a picture of his family background, his marriage to Anna, who was 12 years younger than him, and of his family life (they had three children).
Lie belonged to three over-lapping groups: he was an increasingly eminent Norwegian man of Science, he belonged to the mathematical communities of Norway and of Leipzig, and he belonged to the international community of mathematicians. In each of these worlds he cut quite a figure. He was, we learn, a rather exciting lecturer, happier when he could improvise than when he was constrained to read from notes, and, for the better students at least, full of stimulating insights. This contrasts rather oddly with his books and papers, which are often very difficult to read, and it would have been interesting to learn more. The oft-told tale that Lie swung his style over from geometry to analysis in order to be understood by the heavyweights in Berlin is neither confirmed nor denied here, but left unexplored. We also get a picture of Lie as someone increasingly confident of his own merits and achievements, with sharp opinions about the work of others, and active in supporting journals for Scandinavian mathematicians (although he did not get on well with Mittag-Leffler). The book is enlivened with letters and quotes that give a vivid picture of many of the people he came across in all these areas of his life. The competitive spirit of Lie's colleague C.A. Neumann at Leipzig stands out as having been particularly difficult for him.
Lie is famous for having some sort of nervous breakdown in 1889. What this was, how it affected him, the extent to which it changed him afterwards are delicate matters, and it is not easy, perhaps not possible, to sort the matter out over a century later. Stubhaug presents a wealth of information. We learn that Lie was eventually discharged from hospital without being regarded as cured, and that he resumed his research. Stubhaug thinks it unlikely that the mental illness was connected to the pernicious anaemia which in those days was usually fatal within six months of diagnosis (as it was in Lie's case). There is a wealth of German testimony that Lie became a much more difficult person, prone to make accusations that people were stealing his ideas (for example, Killing) and willing to make hurtful comparisons in print (as he did with Klein). Stubhaug holds all these German stories somewhat at arms' length, although he notes that non-mathematicians such as Wilhelm Ostwald also thought that Lie had become a different, more ill-tempered person. It would seem that Stubhaug believes that a definitive verdict is not possible now, and maybe is not possible for anyone in Lie's situation.
The book is written in sections that introduce various themes, themselves presented chronologically, which gives it a slightly repetitive feel that also has its virtues. The balance of the Norwegian matter, which is done is rich detail, to French and German matter, necessarily covered in less detail, is very well struck. The translation reads very well, and when German or French has been translated it is usually done well (I did wonder about the use of capitals, and some mathematical terms come across uncertainly). But it is a fine example of a biography of a mathematician. There are not enough of those, and there are many mathematicians to go round, so we can hope that others will come along with Stubhaug's energy and skill. For in these pages you may indeed meet, and enjoy the company of, the mathematician Sophus Lie.
Jeremy Gray ( firstname.lastname@example.org) has worked at the Open University since 1974. He is also an Affiliated Research Scholar at the Department of History and Philosophy of Science of the University of Cambridge, England. He works on the history of mathematics in the 19th and 20th Centuries, with a particular interest in complex function theory and geometry.