by E. T. Bell
After leaving home in September, 1825, Abel first visited the notable mathematicians and astronomers of Norway and Denmark and then, instead of hurrying to Göttingen to meet Gauss as he had intended, proceeded to Berlin. There he had the great good fortune to fall in with a man, August Leopold Crelle (1780-1856), who was to be a scientific Holmboë to him and who had far more weight in the mathematical world than the good Holmboë ever had. If Crelle helped to make Abel’s reputation, Abel more than paid for the help by making Crelle’s. Wherever mathematics is cultivated today the name of Crelle is a household word, indeed more; for “Crelle” has become a proper noun signifying the great journal he founded, the first three volumes of which contained twenty two of Abel’s memoirs. The journal made Abel, or at least made him more widely known to Continental mathematicians than he could ever have been without it; Abel’s great work started the journal off with a bang that was heard round the mathematical world; and finally the journal made Crelle. This self-effacing amateur of mathematics deserves more than a passing mention. His business ability and his sure instinct for picking collaborators who had real mathematics in them did more for the progress of mathematics in the nineteenth century than half a dozen learned academies.
Crelle himself was a self-taught lover of mathematics rather than a creative mathematician. By profession he was a civil engineer. He early rose to the top in his work, built the first railroad in Germany, and made a comfortable stake. In his leisure he pursued mathematics as something more than a hobby. He himself contributed to mathematical research before and after the great stimulus to German mathematics which his Journal für die reine und angewandte Mathematik (Journal for pure and applied Mathematics) gave on its foundation in 1826. This is Crelle’s greatest contribution to the advancement of mathematics.
The Journal was the first periodical in the world devoted exclusively to mathematical research. Expositions of old work were not welcomed. Papers (except some of Crelle’s own) were accepted from anyone, provided only the matter was new, true, and of sufficient “importance”—an intangible requirement—to merit publication. Regularly once every three months from 1826 to the present day “Crelle” has appeared with its sheaf of new mathematics. In the chaos after the World War “Crelle” tottered and almost went down, but was sustained by subscribers from all over the world who were unwilling to see this great monument to a more tranquil civilization than our own obliterated. Today hundreds of periodicals are devoted either wholly or in considerable part to the advancement of pure and applied mathematics. How many of them will survive our next outburst of epidemic insanity is anybody’s guess.
When Abel arrived in Berlin in 1825 Crelle had just about made up his mind to start his great venture with his own funds. Abel played a part in clinching the decision. There are two accounts of the first meeting of Abel and Crelle, both interesting. Crelle at the time was holding down a government job for which he had but little aptitude and less liking, that of examiner at the Trade-School (Gewerbe-Institut) in Berlin. At third-hand (Crelle to Weierstrass to Mittag-Leffler) Crelle’s account of that historic meeting is as follows.
“One fine day a fair young man, much embarrassed, with a very youthful and very intelligent face, walked into my room. Believing that I had to do with an examination-candidate for admission to the Trade-School, I explained that several separate examinations would be necessary. At last the young man opened his mouth and explained [in poor German], ’Not examination, only mathematics.’ ”
Crelle saw that Abel was a foreigner and tried him in French, in which Abel could make himself understood with some difficulty. Crelle then questioned him about what he had done in mathematics. Diplomatically enough Abel replied that he had read, among other things, Crelle’s own paper of 1823, then recently published, on “analytical faculties” (now called “factorials” in English). He had found the work most interesting he said, but _______. Then, not so diplomatically, he proceeded to tell Crelle that parts of the work were quite wrong. It was here that Crelle showed his greatness. Instead of freezing or blowing up in a rage at the daring presumption of the young man before him, he pricked up his ears and asked for particulars, which he followed with the closest attention. They had a long mathematical talk, only parts of which were intelligible to Crelle. But whether he understood all that Abel told him or not, Crelle saw clearly what Abel was. Crelle never did understand a tenth of what Abel was up to, but his sure instinct for mathematical genius told him that Abel was a mathematician of the first water and he did everything in his power to gain recognition for his young protégé. Before the interview was ended Crelle had made up his mind that Abel must be one of the first contributors to the projected Journal.
Abel’s account differs, but not essentially. Reading between the lines we may see that the differences are due to Abel’s modesty. At first Abel feared his project of interesting Crelle was fated to go on the rocks. Crelle could not make out what the young man wanted, who he was, or anything about him. But at Crelle’s question as to what Abel had read in mathematics things brightened up considerably. When Abel mentioned the works of the masters he had studied Crelle became instantly alert. They had a long talk on several outstanding unsettled problems, and Abel ventured to spring his proof of the impossibility of solving the general quintic algebraically on the unsuspecting Crelle. Crelle wouldn’t hear of it; there must be something wrong with any such proof. But he accepted a copy of the paper, thumbed through it, admitted the reasoning was beyond him—and finally published Abel’s amplified proof in his Journal. Although he was a limited mathematician with no pretensions to scientific greatness, Crelle was a broadminded man, in fact a great man.
Crelle took Abel everywhere, showing him off as the finest mathematical discovery yet made. The self-taught Swiss Steiner—“the greatest geometer since Apollonius”—sometimes accompanied Crelle and Abel on their rounds. When Crelle’s friends saw him coming with his two geniuses in tow they would exclaim “Here comes Father Adam again with Cain and Abel.”
The generous sociability of Berlin began to distract Abel from his work and he fled to Freiburg where he could concentrate. It was at Freiburg that he hewed his greatest work into shape, the creation of what is now called Abel’s Theorem. But he had to be getting on to Paris to meet the foremost French mathematicians of the day—Legendre, Cauchy, and the rest.
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It can be said at once that Abel’s reception at the hands of the French mathematicians was as civil as one would expect from distinguished representatives of a very civil people in a very civil age. They were all very civil to him—damned civil, in fact, and that was about all that Abel got out of the visit to which he had looked forward with such ardent hopes. Of course they did not know who or what he was. They made only perfunctory efforts to find out. If Abel opened his mouth—when he got within talking distance of them—about his own work, they immediately began lecturing about their own greatness. But for his indifference the venerable Legendre might have learned something about his own lifelong passion (for elliptic integrals) which would have interested him beyond measure. But he was just stepping into his carriage when Abel called and had time for little more than a very civil good-day. Later he made handsome amends.
Late in July, 1826, Abel took up his lodgings in Paris with a poor but grasping family who gave him two bad meals a day and a vile room for a sufficiently outrageous rent. After four months of Paris Abel writes his impressions to Holmboë:
“Paris, 24 October 1826.
“To tell you the truth this noisiest capital of the Continent has for the moment the effect of a desert on me. I know practically nobody; this is the lovely season when everybody is in the country. . . . Up till now I have made the acquaintance of Mr. Legendre, Mr. Cauchy and Mr. Hachette, and some less celebrated but very able mathematicians: Mr. Saigey, editor of the Bulletin des Sciences, and Mr. Lejeune-Dirichlet, a Prussian who came to see me the other day believing me to be a compatriot of his. He is a mathematician o
f great penetration. With Mr. Legendre he has proved the impossibility of solving x5 + y5 = z5 in whole numbers, and other very fine things. Legendre is extremely polite, but unfortunately very old. Cauchy is mad. . . . What he does is excellent but very muddled. At first I understood practically none of it; now I see some of it more clearly. . . . Cauchy is the only one occupied with pure mathematics. Poisson, Fourier, Ampère, etc., busy themselves exclusively with magnetism and other physical subjects. Mr. Laplace writes nothing now, I believe. His last work was a supplement to his Theory of Probabilities. I have often seen him at the Institut. He is a very jolly little chap. Poisson is a little fellow; he knows how to behave with a great deal of dignity; Mr. Fourier the same. Lacroix is quite old. Mr. Hachette is going to present me to several of these men.
“The French are much more reserved with strangers than the Germans. It is extremely difficult to gain their intimacy, and I do not dare to urge my pretensions as far as that; finally every beginner has a great deal of difficulty in getting noticed here. I have just finished an extensive treatise on a certain class of transcendental functions [his masterpiece] to present it to the Institut [Academy of Sciences], which will be done next Monday. I showed it to Mr. Cauchy, but he scarcely deigned to glance at it. And I dare to say, without bragging, that it is a good piece of work. I am curious to hear the opinion of the Institut on it. I shall not fail to share it with you . . .
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He then tells what he is doing and continues with a rather disturbed forecast of his prospects. “I regret having set two years for my travels, a year and a half would have sufficed”
He has got all there is to be got out of Continental Europe and is anxious to be able to devote his time to working up what he has invented. “So many things remain for me to do, but so long as I am abroad, all that goes badly enough. If I had my professorship as Mr. Kielhau has his! My position is not assured, it is true, but I am not uneasy about it; if fortune deserts me in one quarter perhaps she will smile on me in another.”
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From a letter of earlier date to the astronomer Hansteen we take two extracts, the first relating to Abel’s great project of re-establishing mathematical analysis as it existed in his day on a firm foundation, the second showing something of his human side. (Both are free translations.)
“In the higher analysis too few propositions are proved with conclusive rigor. Everywhere we find the unfortunate procedure of reasoning from the special to the general, and the miracle is that after such a process it is only seldom that we find what are called paradoxes. It is indeed exceedingly interesting to seek the reason for this. This reason, in my opinion, resides in the fact that the functions which have hitherto occurred in analysis can be expressed for the most part as powers. . . . When we proceed by a general method, it is not too difficult [to avoid pitfalls]; but I have had to be very circumspect, because propositions without rigorous proof (i.e. without any proof) have taken root in me to such an extent that I constantly run the risk of using them without further examination. These trifles will appear in the journal published by Mr. Creller.”
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Immediately following this he expresses his gratitude for his treatment in Berlin. “It is true that few persons are interested in me, but these few are infinitely dear to me, because they have shown me so much kindness. Perhaps I can respond in some way to their hopes of me, for it must be hard for a benefactor to see his trouble lost.”
He tells then how Crelle has been begging him to take up his residence permanently in Berlin. Crelle was already using all his human engineering skill to hoist the Norwegian Abel into a professorship in the University of Berlin. Such was the Germany of 1826. Abel of course was already great, and the sure promise of what he had in him indicated him as the likeliest mathematical successor to Gauss. That he was a foreigner made no difference; Berlin in 1826 wanted the best in mathematics. A century later the best in mathematical physics was not good enough, and Berlin quite forcibly got rid of Einstein. Thus do we progress. But to return to the sanguine Abel.
“At first I counted on going directly from Berlin to Paris, happy in the promise that Mr. Crelle would accompany me. But Mr. Crelle was prevented, and I shall have to travel alone. Now I am so constituted that I cannot endure solitude. Alone, I am depressed, I get cantankerous, and I have little inclination for work. So I said to myself it would be much better to go with Mr. Boeck to Vienna, and this trip seems to me to be justified by the fact that at Vienna there are men like Littrow, Burg, and still others, all indeed excellent mathematicians; add to this that I shall make but this one voyage in my life. Could one find anything but reasonableness in this wish of mine to see something of the life of the South? I could work assiduously enough while travelling. Once in Vienna and leaving there for Paris, it is almost a bee-line via Switzerland. Why shouldn’t I see a little of it too? My God! I, even I, have some taste for the beauties of nature, like everybody else. This whole trip would bring me to Paris two months later, that’s all. I could quickly catch up the time lost. Don’t you think such a trip would do me good?”
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So Abel went South, leaving his masterpiece in Cauchy’s care to be presented to the Institut. The prolific Cauchy was so busy laying eggs of his own and cackling about them that he had no time to examine the veritable roc’s egg which the modest Abel had deposited in the nest. Hachette, a mere pot-washer of a mathematician, presented Abel’s Memoir on a general property of a very extensive class of transcendental functions to the Paris Academy of Sciences on the tenth of October, 1826. This is the work which Legendre later described in the words of Horace as “monumentum aere perennius,” and the five hundred years’ work which Hermite said Abel had laid out for future generations of mathematicians. It is one of the crowning achievements of modern mathematics.
What happened to it? Legendre and Cauchy were appointed as referees. Legendre was seventy four, Cauchy thirty nine. The veteran was losing his edge, the captain was in his self-centred prime. Legendre complained (letter to Jacobi, 8 April, 1829) that “we perceived that the memoir was barely legible; it was written in ink almost white, the letters badly formed; it was agreed between us that the author should be asked for a neater copy to be read.” What an alibi! Cauchy took the memoir home, mislaid it, and forgot all about it.
To match this phenomenal feat of forgetfulness we have to imagine an Egyptologist mislaying the Rosetta Stone. Only by a sort of miracle was the memoir unearthed after Abel’s death. Jacobi heard of it from Legendre, with whom Abel corresponded after returning to Norway, and in a letter dated 14 March, 1829, Jacobi exclaims, “What a discovery is this of Mr. Abel’s! . . . Did anyone ever see the like? But how comes it that this discovery, perhaps the most important mathematical discovery that has been made in our Century, having been communicated to your Academy two years ago, has escaped the attention of your colleagues?” The enquiry reached Norway. To make a long story short, the Norwegian consul at Paris raised a diplomatic row about the missing manuscript and Cauchy dug it up in 1830. Finally it was printed, but not till 1841, in the Mémoires présentés par divers savants à l’ Académie royale des sciences de l’ Institut de France, vol. 7, pp. 176-264. To crown this epic in parvo of crass incompetence, the editor, or the printers, or both between them, succeeded in losing the manuscript before the proof-sheets were read.II The Academy (in 1830) made amends to Abel by awarding him the Grand Prize in Mathematics jointly with Jacobi. Abel, however, was dead.
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The opening paragraphs of the memoir indicate its scope.
“The transcendental functions hitherto considered by mathematicians are very few in number. Practically the entire theory of transcendental functions is reduced to that of logarithmic functions, circular and exponential functions, functions which, at bottom, form but a single species. It is only recently that some other functions have begun to be considered. Among the latter, the elliptic transcendents, several of whose remarkable and elegant properties
have been developed by Mr. Legendre, hold the first place. The author [Abel] has considered, in the memoir which he has the honor to present to the Academy, a very extended class of functions, namely: all those whose derivatives are expressible by means of algebraic equations whose coefficients are rational functions of one variable, and he has proved for these functions properties analogous to those of logarithmic and elliptic functions . . . and he has arrived at the following theorem:
“If we have several functions whose derivatives can be roots of one and the same algebraic equation, all of whose coefficients are rational functions of one variable, we can always express the sum of any number of such functions by an algebraic and logarithmic function, provided that we establish a certain number of algebraic relations between the variables of the functions in question.
“The number of these relations does not depend at all upon the number of functions, but only upon the nature of the particular functions considered . . . .”
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