More than half of this book is a biography of Søren Georg Abel, and more than half is a biography of his son the famous mathematician Niels Henrik Abel — there is, of course, some overlap. There is also as much earlier material as the author, Arild Stubhaug, can find, some well-organised appendices with information about sources, a chronology, and a helpful summary of the main characters. Søren Georg Abel is an interesting if minor figure in the history of Norway, and his rise and fall accounts for the poverty that dogged Niels Henrik throughout his short life. This half of the book is much fuller than the comparable account in Ore's biography3540668349 of Abel (the standard English-language source) and the reader gains a vivid account of the circumstances of his life, his schooling and university education, and the issues that dominated the political and intellectual life of the region (Denmark, Sweden and Norway). The failure of his parents' marriage is hinted at as well as the sources seem to allow. The portraits of the important people in Abel's life are particularly well drawn, although his fiancée Christine Kemp remains slightly in the shade: it is easier to find out about energetic male explorers of remote Norway than most women of the period.

Niels Henrik Abel remains an elusive figure, although the pathos of his raging at his early death is well conveyed. He does not seem to have been talented except in mathematics. He could be charming but also shy. He was never going to be "a man of the world." He may have been hurt by the scandal around his father's political defeats, Stubhaug suggests as much, but he was not traumatised by them. However, while Stubhaug has been able to fill out Niels Henrik's family life admirably, he has little to add to the story of his mathematical life. Nor has he tried to delve deeply into the mathematics, aiming as he has at a general readership. He does give us a good and thorough account of his trip to Germany, Switzerland and France, in particular his successful meeting with Crelle that led to the publication of so much of Abel's work in the initial volumes of Crelle's *Journal für die reine und angewandte Mathematik*.

The translation by Richard Daly generally reads very well, but unfortunately stumbles at times with the mathematics. Some mistakes are trivial: "primary" numbers' for prime numbers throughout, "elliptical" functions for elliptic functions, "numbers" theory for number theory, "exchange" of variables, for change of variables. Some expressions are more annoying (whether they are due to the author or the translator). For example (p. 340): "In mathematics, a line of numbers with sizes that follow one another in a definite order are simply *a series*, and the individual sizes are called the series' *terms*." A little later (p. 342) we read of geometric series that "the sum of the *n* first terms *can* approach a finite limit when *n* increases beyond all limits, and when the series is said to be convergent, it is, on the contrary, divergent." Or this (on p. 468) "Crelle . . .wanted very much for Abel first to complete the solution of equations by means of square root" — presumably the solution of equations by means of radicals is what is meant. Lagrange studied the librations of the moon, which apparently (p. 202) "allow one to see the rotating lunar circumference, always with the same side towards the Earth." To give one last example, (p. 302 — Abel's approach to proving the insolubility of the quintic): "Abel posed the question: is it possible that the *form* the roots must necessarily have in a fifth degree equation *can* satisfy the equation?" The whole paragraph suffers from a confusion between the existence of roots and the possibility that they are not expressible as radicals (although Abel himself was capable of giving his paper on the subject a similarly misleading title, as Stubhaug points out on p. 306). The composer Hayden is Haydn. And in a confusion of Harolds and Haralds, the Danish King Harald III was defeated by the English King Harold II at Stamford Bridge on September 25 1066, but contrary to the footnote on p. 258 this is not the battle that William the Conqueror won. Harold's defeat and the last conquest of England began on October 14 1066 near Hastings.

The author set himself the task of writing a good book about Abel for a general audience, and in this he has succeeded. He did not analyse any of Abel's achievements in depth, and in the absence of new sources there would be little to add to Ore's account in any case. But if the mathematics is not always clear, his judgements are often sound, and he refrains from the habit of reading into every good idea of Abel's whole families of conclusions that later mathematicians were to reach. There are no grossly inflated priority claims here. There are some opinions that can be questioned, such as these three: Lagrange's father was closely related to Descartes (p. 201 and p. 529); Lagrange pushed through proposals for the decimal system by suggesting base 11 as a compromise with the old system of 12's and 20's (p. 210); Galois hurled a blackboard rubber at an examiner (p. 415). Quite certainly the correspondence between Legendre and Jacobi was begun by Jacobi, not Legendre (as claimed here on p. 453). Two oddities crop up on p. 204. Historians of mathematics doubt that Fermat's unproved theorems in number theory were considered most honourable in French mathematics in the 18th Century, and that Diderot lost a theological argument to Euler when Euler produced a nonsensical formula. There are several good historians of mathematics in Denmark who could have saved the author from the criticisms levelled here, and it is a pity they were not consulted. It is also regrettable that there are too few footnotes to allow the author's evidence to be assessed.

Rather than end this review with the customary small points reviewers trot out, however, it is more appropriate to observe that the book is exemplary in some respects. It is well organised, it is highly readable, it carries its learning lightly, it has no tendentious axes to grind. It has a good index, it has many fine illustrations, often in very good colour reproductions, and it is a physically lovely object, one of the most handsome on the shelves. Given the elegance of Abel's mathematics, that seems altogether right.

### References

Ore, Oystein, *Niels Henrik Abel, Mathematician Extraordinary*, University of Minnesota 1957, reprinted Chelsea New York 1974.

Jeremy Gray ( [email protected]) 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 19^{th} and 20^{th} Centuries, with a particular interest in complex function theory and geometry.