Book Review

Called Too Soon by Flames Afar: Niels Henrik Abel and His Times Reviewed by Jesper Lützen

Called Too Soon by Flames Afar: Niels Henrik ideas were analyzed. Abel and His Times One could have tried Arild Stubhaug, translated by Richard H. Daly to reconstruct how Springer-Verlag, 2000, Berlin, Heidelberg, Abel successively de- New York veloped his ingenious $44.95, x + 580 pages, with 51 figures algebraic theorems ISBN 3-540-66834-9 and his revolutionary new approach to el- August 5, 2002, marks the bicentennial of the liptic integrals and birth of the Norwegian mathematical genius Niels functions, in particu- Henrik Abel. Many mathematicians will celebrate lar his idea of dealing this occasion at a meeting in his native country.1 with the problems in a But one can also celebrate the occasion at home by complex rather than a reading the new biography by Arild Stubhaug. It is real setting and his an extremely well-written account of Abel’s short idea of considering life (1802–29). It compares favorably with Oystein the inverse functions Ore’s biography Niels Henrik Abel: Mathematician rather than the elliptic integrals themselves. One Extraordinary, translated in 1957 from the Nor- could have contrasted these ideas with those of Abel’s predecessors, such as Lagrange and Le- wegian original Niels Henrik Abel; Et geni og hans gendre, with those of his contemporaries, such as samtid (1954), which is highly recommendable and Gauss, Jacobi, and Galois, and one could have an- was until now the most complete source on Abel’s alyzed the importance of Abel’s work for the later life. development of mathematics. One could have tried Stubhaug’s book is almost twice as long as Ore’s. to analyze the unpublished drafts that Abel left un- Does this reflect that much new Abel material has finished at his untimely death, and one could have been uncovered since Ore wrote his book? Not tried to reconstruct the ideas about integration in really. To be sure, Stubhaug has found some new finite terms that he wrote down in a paper that was traces of Abel’s life, but the primary material later lost, as well as his ideas on different algebraic dealing directly with the mathematical genius is still subjects that he hinted at on various occasions rather scant: his published mathematical papers, but did not have the time to develop. a few unpublished drafts, letters (in particular from Such an internal mathematical analysis—which his travel abroad), some official records, his friends’ would, so to speak, take the mathematics as the letters and diaries, obituaries, and a few other rec- main actor—is, however, not what Stubhaug pre- ollections from people who knew him. How does sents. His book is a real biography, in which the per- this material support a biography of 580 pages? son Abel is the central object of study. The book It could have served as a basis for an even longer is so long because the author wants to reflect Abel’s “scientific biography” in which Abel’s mathematical life in the world in which he lived—not so much the mathematical world he was a part of, but rather Jesper Lützen is professor of the history of exact sciences, the Norwegian society he grew up in and inhabited Department of Mathematics, Copenhagen University, Den- for most of his short life. It is the rich description mark, and at the California Institute of Technology. His e-mail address is [email protected]. of this local and regional society and its interac- tion with Abel that is the main novelty of this bi- 1The Abel Bicentennial Conference took place June 3–8, ography and that makes it so interesting to read. 2002, at the University of Oslo.

AUGUST 2002 NOTICES OF THE AMS 795 Stubhaug vividly de- Although mathematics is not the main issue in scribes Abel’s family Stubhaug’s book, it is also not marginalized com- background, the liv- pletely, as it has been in recent Hollywood pro- ing conditions in ductions about mathematicians and scientists. southern Norway, and Stubhaug clearly conveys the message that Abel’s the very exciting na- greatness is to be found in his mathematics. He tional Norwegian and briefly explains the mathematical problems Abel Scandinavian political addressed, the methods he used, and the results and cultural events he arrived at. These explanations will be sugges- that directly or indi- tive to people with some college knowledge of rectly influenced mathematics, and they are brief enough that they The church in Gjerstad, Norway. Abel’s life. will not put off a mathematically untrained reader. When I, as a Scan- Abel’s mathematics is also presented in its histor- dinavian reader, got hold of the 1996 Norwegian ical context. However, rather than describe the original of Stubhaug’s book, Et foranskutt lyn: Niels technical mathematical developments that pre- Henrik Abel og hans tid, I was extremely interested ceded Abel’s discoveries, Stubhaug tends to tell sto- to learn how the knowledge I had about Abel’s life ries about the mathematicians behind the devel- and work was connected in a meaningful way to the opments. For example, rather than try to explain cultural, social, and political history of the time, the problems concerning the solution of polyno- which I knew only superficially from school. The mial equations by radicals and the content of La- book is written in an elegant and captivating liter- grange’s papers that Abel read and used, Stubhaug ary style that made it almost impossible for me to tells stories about the person Lagrange (whom Abel put down. Stubhaug, who has studied mathemat- never met) and the exciting story, involving ics, history, and literature and has published poems Tartaglia and Cardano, about the discovery of the as well as papers on literary and cultural history, solution of the third- and fourth-degree equations. deservedly won the most prestigious Norwegian lit- And instead of trying to explain Galois theory, ex- erary prize (Brageprisen) for his biography of Abel. cept in the most general terms, Stubhaug tells the When I heard that the book had been translated dramatic story of Galois’s life. He also recounts the into English, I was a bit worried. Would it be pos- history of Fermat and his last (translated literally sible to render the captivating literary style in a as “great”) theorem and other good stories from the translation? It turned out that my worries were history of mathematics. Some of them have the unfounded. As far as I can tell, the translation by character of anecdotes, but generally the histori- Richard H. Daly is also a very good read. The style cal facts are accurate except for a few details, such of the translation is also literary and pleasant. as the claim that Pascal was the first to determine There are some, admittedly somewhat trivial, prob- the area under a cycloid. Stubhaug also succeeds lems with the translation. For example, on page 147 in conveying the importance and ingenuity of Abel’s the word Domkirke, meaning “cathedral”, is not mathematics by quoting contemporary and later translated, and this is unfortunate because the evaluations of it. connection to the Cathedral School being discussed But the main purpose of the biography is to at this point in the book becomes unclear. The place Abel in his local environment. The first third most glaring problem, however, is the translation of the book introduces Abel’s family and the early of mathematical terms and phrases. Here are some nineteenth-century southern Norway he grew up in. examples: solve algebraic equations with root in- At that time Norway and Denmark were ruled by dicators (page 204), equations of the fifth magni- the same king. Abel’s family had come to Norway tude (page 205), great and small infinities in the seventeenth century from Abild (whence the (page 210), Newton’s concept of marginal value name Abel) in Northern Schleswig, which also was (page 210), the Radon Transformer (page 285), the ruled by the Danish-Norwegian king. Abel’s grand- reverse function (page 301), Abelian theorem of lim- father was a parish vicar at Gjerstad near Risør in itation values (page 344), the Sylowian proposi- the coastal region of southern Norway. Abel’s father, tions (page 552). A mathematically trained reader Søren Georg, attended the Cathedral School in Hels- can easily interpret these somewhat peculiar ingør, Denmark, and studied theology at Copen- phrases, and since the mathematics is not the main hagen University before he married the daughter of issue in this book, the bad translations should not the richest merchant in Risør and became a vicar detract from its merits. It is remarkable, however, at Finnøy near Stavanger. Here the young couple had that such a highly respected mathematics pub- their first two sons in 1800 and 1802. They named lisher as Springer-Verlag did not ask a mathemat- the second son Niels Henrik after his maternal ically trained person to change these mistransla- grandfather. Two years later the family moved back tions, which lend an unintentionally comical flavor to Gjerstad, where Søren Georg took over as vicar to the mathematical passages. after the death of his father. Niels Henrik grew up

796 NOTICES OF THE AMS VOLUME 49, NUMBER 7 in Gjerstad with four siblings. He was taught by his In 1821 Abel entered the university in Christia- father and later by a hired teacher. Only few con- nia that his father had helped to establish in 1811. crete facts and anecdotes about Abel are preserved Here he read the masters of mathematics and after from this period of his life. Still, Stubhaug succeeds two years began to publish small papers in a newly in weaving an interesting net of stories pertaining founded Norwegian scientific journal. Soon it be- to the Abel family and the region, allowing the came clear that Abel could not learn more mathe- reader to get a feeling for Abel’s childhood. matics in Christiania, and in 1823 he went to Den- In particular, Stubhaug gives a biography of mark for a few months to talk to Degen, who in Abel’s father, who was an interesting person in his 1821 had already examined Abel’s purported so- own right. All his life he subscribed to the positive, lution of the quintic and had made Abel realize his rational beliefs of the Enlightenment, which he had error. This led Abel to his first major mathemati- absorbed during his studies in Denmark. These cal result, the proof of the impossibility of the so- ideals saturated his social work as a vicar, his or- lution of the quintic by radicals. Moreover, Degen ganization of a local reading society (where even had directed Abel’s attention to the theory of el- books by Voltaire were read), his later political liptic functions, which Abel later revolutionized. work, and his widely read catechism, which stressed Degen recognized Abel’s genius but could not teach the “enlightened reason of Jesus”. He was active in him more. the Norwegian defence against a possible English In 1824 Abel was given a stipend that after a invasion and was consequently decorated as Knight year’s preparation allowed him to embark on a trip of the Order of Dannebrog by the Danish-Norwe- to Europe’s mathematical centers. Before entering gian king. When Norway was forced into a union university, Abel had begun to meet with other with Sweden in 1814 in the Treaty of Kiel, father young students interested in science, and now in Abel defended the country’s independence and its the summer of 1825 he embarked on his great new free constitution. He took an active part in the voyage together with four of these friends. He had applied for a stipend to visit Göttingen and Paris, condemnation of Nicolai Wergeland’s book about but, following advice from von Schmidten in Copen- Denmark’s transgressions against Norway, and yet hagen, Abel stayed a long time in Berlin, where he his own catechism was severely criticized by uni- became good friends with Crelle. This turned out versity professors who were inspired by the Dan- to be the most important result of his travel. Im- ish reformist Grundtvig. They accused him of being mediately recognizing Abel’s genius, Crelle invited “the father of falsehood”, who had led a whole Abel to publish in the new journal he founded generation of young Norwegians into eternal while Abel was in Berlin. After a very pleasant and damnation. The following year he was (probably productive stay in Berlin, Abel traveled with his unjustly) suspected of being the author of a very Norwegian friends through central Europe to Venice bold and infamous publication against the new and over the Alps to Paris. He was not particularly Swedish rule. When he was persuaded by a eager to visit the formidable Gauss in Göttingen, scoundrel to present defamatory accusations and he enjoyed the company of his Norwegian against some of the members of the Norwegian Par- friends and the prospect of seeing the wonders of liament, of which he was twice a member, he was Europe so much that he put off his promised vis- finished as a public figure. He died two years later. its to Paris and Göttingen as long as possible. After In the meantime, in 1815, Niels Henrik Abel had one year on the road he finally reached Paris, where been sent to the Cathedral School in Christiania he stayed for a rather miserable half year before (Oslo). At this point in the book, the focus gradu- returning to Christiania. ally shifts away from Abel’s family and toward the The last part of the book deals with Abel’s final academic and public life in Christiania, described two poverty-stricken years in Norway, where, in a vividly and with great insight. Abel lived for much competition with Jacobi, he composed within of this time with his elder brother, who also stud- twenty months 13 papers, of which the longest ied at the Cathedral School. His brother, however, was 126 pages! We read about how Abel obtained gradually became more introverted and indifferent a temporary university position as a substitute for to school work and in the end had to be taken out Hansteen and about his fatal attack of tuberculo- of school. Niels Henrik did not distinguish himself sis and untimely death on April 6, 1829. in any particular way at first. In fact, he was passed The person Abel becomes increasingly more vis- to the third year on probation. During that year ible in Stubhaug’s story, starting with the descrip- (1818) he got the young Michael Holmboe as his tion of Abel’s years at the Cathedral School. The mathematics teacher, and with Holmboe’s help he reason is obviously that there is more source ma- quickly developed a mastery of this subject. In terial available. In particular, Abel’s letters during other subjects Abel continued to be an average the voyage through Europe allow Stubhaug to fol- student, but in mathematics Holmboe declared low not only the route of the young mathematician him to be a genius. but also his good and bad experiences and his

AUGUST 2002 NOTICES OF THE AMS 797 changing moods. These letters provide a fine tes- the only one who took care of the family. He made timony of Abel’s personality, but in a certain sense sure that one of his younger brothers got an edu- the earlier parts of the biography can also be con- cation and that his sister got a good job. He also sidered as an attempt to explain his personality, took care to pay his family’s debt, in particular the which Stubhaug describes as being two-sided. Every- rather large amount that his father had pledged to one who met Abel seems to have liked his mild and pay annually to the university. This seems to sug- modest manners. He was rather shy when he was gest that although life often disappointed him, around people he did not know, but when he was Abel did not run away from its challenges, but in the company of friends, he often faced them and dealt with them quite successfully. acted as a jester. He loved to While the cultural historical context, described swim, to play cards, and to go so vividly in this book, throws light on Abel’s per- to the theater, and he eas- sonality, it does not explain his mathematical ge- ily charmed the ladies. He nius. Genius of this magnitude is probably inex- enjoyed company, and plicable by sociological means, and in Abel’s case when he was alone he only very meager input from his environment was often suffered from enough to bring it forth. Only during his first year odd and melancholic with Holmboe did he have a teacher who could moods. His moods properly teach him something. After that he was could change abrupt- in contact only with mathematicians who could ly from boyish high not really advise him. To be sure, his father’s en- spirits to utter de- lightened ideas and the enthusiastic atmosphere in pression. the new university and the new Norwegian nation Except in Abel’s provided fertile ground for investigations in nat- mathematical work, ural sciences. Abel’s friends were thus attracted to which seems always to the natural sciences, but Abel’s love for pure math- have made him happy, ematics is unexplained. In a sense he might be he was not a particularly called the first pure mathematician. His only con- deep thinker and probably tribution to applied mathematics contained a dev- did not analyze his own astating error, as pointed out by Schumacher. changing moods. Stubhaug as- A recurrent theme in Abel biographies is that he cribes them to the incongruence be- never got a permanent position as professor of tween the positive, optimistic, rationalistic view of mathematics and lived in poverty during the last life instilled by his father and the realities of Abel’s years of his life. This issue is dealt with in a sober own life, which turned out to be miserable on many way by Stubhaug. While explaining how Abel was accounts. His father died a broken man, his mother disappointed by and sometimes bitter about his disgraced herself by sleeping with a servant and lack of an appropriate job and salary and how he drinking excessively even on the day of her hus- had a hard time making ends meet, Stubhaug does band’s funeral, and his brothers did not do well in not point any accusing fingers or blame any spe- life. Abel’s upbringing did not prepare him to face cific person or organization. The issue would not such misery. Stubhaug interprets many things in have been a major one had Abel lived longer. Many Abel’s life as resulting from his inability to relate young talents have had to wait some years before to the realities of everyday life. He speculates about their scientific production earned them an appro- whether Abel’s early engagement to Christine priate job. If Abel had lived only a little longer, he Kemp, whom he met in Copenhagen in 1823, was would have been a professor. In fact, two days a way for him to avoid the complications involved after Abel’s death Crelle wrote a letter telling Abel in dating the fairer sex and whether his great love he would be offered a professorship in Berlin. It is for the theater was due to his longing for the reg- also likely that developments in Christiania and ulated, planned course of events that he found in Copenhagen would have resulted in job offers from the plays. the universities there. Moreover, Stubhaug suggests that to some ex- A comparison with Abel’s near-contemporary Ga- tent Abel gained so much pleasure from concen- lois is tempting. There are many similarities be- trated work in mathematics because the work al- tween their lives and work. Both died very young, lowed him to escape from the real world, which he both worked in pure mathematics and with simi- did not master, and to immerse himself in the clear lar theories. Both had ingenious memoirs neglected and unambiguous world of mathematics, which by the Paris Academy of Sciences and in particu- he mastered so perfectly. While this analysis may lar by Cauchy, who was so absorbed in his own work capture some essential elements of Abel’s per- that he did not report on the works of his younger sonality, it is somewhat at variance with the fact contemporaries. But there are also essential dif- that after his father’s death Niels Henrik Abel was ferences. Where Abel seems not to have inherited

798 NOTICES OF THE AMS VOLUME 49, NUMBER 7 his father’s strong political and social involvement, and this gives the author the occasion to summa- Galois was an uncompromising political radical. rize the most important events in Abel’s life and Where Abel was generally liked by people who met the main themes of the book. This stylistically him, Galois had a problematic personality. Where clever device prevents the reader from getting lost Abel wrote in a clear style, Galois’s papers were writ- in the details of the more comprehensive presen- ten in a highly inaccessible style. All these differ- tation that follows. In addition, the appendix con- ences go a long way toward explaining why Abel’s tains a brief chronology of Abel’s life and some com- genius was discovered early and why he became in- mentaries and complementary readings that ternationally recognized for his mathematical dis- provide material that for some reason did coveries during his lifetime, whereas Galois’s work not make it into the main presentation. was totally ignored during his short life (except per- Moreover, there is a list of Abel’s published haps by Cauchy, who did not find time to tell oth- works as well as the aforementioned anno- ers about it). Galois received little encouragement tated bibliography and a useful index of in his mathematical endeavors, whereas Abel was names. The book is beautifully illustrated encouraged and supported by many scientists such with 51 figures. as Holmboe, Hansteen, and some of the other pro- There is no doubt that this book is a fessors in Christiania, who privately paid Abel’s liv- great contribution to our knowledge of Abel ing expenses during his early student years, as and his time. For a historian of mathemat- well as by Degen and in particular by Crelle. Except ics it presents a great improvement vis à vis for the circle around Férussac’s Bulletin, the Parisian Ore’s book, both for its richer cultural con- mathematicians ignored Abel while he was in Paris text and because it lists the sources. For a and managed to overlook and even misplace his Scandinavian reader who is seriously inter- most wide-ranging memoir. Nevertheless, a num- ested in the cultural history of this part of ber of leading academics later took the unusual step the world, it is also to be preferred to Ore’s of writing to the Norwegian-Swedish King to en- book. Whether an international reader, in courage him to give an appropriate job to his most particular an international mathematical ingenious subject. reader who wants to celebrate the Abel bi- Stubhaug’s biography of Abel is a scholarly book centennial, would prefer Stubhaug’s biog- in the sense that it is based on a careful scrutiny raphy over Ore’s is a matter of taste. In fact, of a great number of primary and secondary the books are in many ways rather similar. sources and presents interesting analyses and com- They are both well written, they both focus binations of these sources. As far as the style is con- on Abel rather than his mathematics, and cerned, however, it is a literary work. That one they both put his life in the context of the does not find many lengthy footnotes or refer- times in which he lived. The main difference ences to sources certainly makes the book more is the amount of cultural history presented. pleasing to read. But the lack of references is a prob- For example, where Ore writes about Abel’s lem for a reader who would like to know the basis interest in the theater in rather general for the claims made in the book. The book contains terms, Stubhaug provides long descriptions a very useful and complete annotated bibliography, of various plays that Abel might have seen and in many instances it is rather clear where Stub- in Christiania or during his travels abroad. haug may have found a particular piece of infor- For a Scandinavian reader these summaries mation. In other cases, though, it is not so clear, of plays by Holberg, Wessel, Ewald, Oehlenschläger, for example in the many places where Stubhaug and others provide a valuable cultural background, writes: “It was (later) said that…”. In fact, it is some- but I am not so sure how interesting they are for times unclear whether Stubhaug is reporting a fact an international public. As another example one can that is unambiguously supported by the sources or mention the way the two authors deal with father whether he is inferring a probable course of events. Abel’s social work in Gjerstad. Ore mentions it For example, Stubhaug reports that father Abel’s rather briefly, whereas Stubhaug goes into more de- wedding was “most certainly...celebrated with all tail, describing father Abel’s attempts to improve the pomp and circumstance typical of a trading farming and industry and how he tried to reduce house of the time,” and then he continues with a famine during the English blockade by encourag- page-long description of the wedding. Is this based ing his parishioners to eat horse meat. Stubhaug on an eyewitness description, or is it Stubhaug’s re- also sets all these undertakings into perspective. construction based on his great knowledge of the Those who might find this much cultural history cultural history of the time? to be excessive may want to stick to Ore’s book. For The book is divided into eight parts. The first is those who want this richer cultural context, I an introduction and a long section entitled “A Short warmly recommend Stubhaug’s new book. Life”, which presents a summary of Abel’s life. Its point of departure is Abel’s death and his legacy,

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