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Mathematics by Elling Holst
Elling Holst wrote the chapter on "Mathematics" in the book Illustreret norsk Litteraturhistorie edited by Henrik Jaeger. The reference is as follows:
E Holst, Mathematik, in Henrik Jaeger (ed.), Illustreret norsk Litteraturhistorie (Hjalmar Biglers Publishing, Kristiania, 1896).
https://runeberg.org/ilnolihi/4/0070.html
We give a translation of this chapter into English. We have made minor changes such as adding full names when Holst only gives a surname. We have also added a few "Notes" so that the modern reader might appreciate what might have been well-known to a Norwegian reader at the end of the 19th century.
Let us note that Holst wrote at a time of great progress in mathematics in Norway and he was friends with major figures whom he writes about. He was an excellent person to write on Norwegian mathematicians since he had much experience as a writer of biographies.
E Holst, Mathematik, in Henrik Jaeger (ed.), Illustreret norsk Litteraturhistorie (Hjalmar Biglers Publishing, Kristiania, 1896).
https://runeberg.org/ilnolihi/4/0070.html
We give a translation of this chapter into English. We have made minor changes such as adding full names when Holst only gives a surname. We have also added a few "Notes" so that the modern reader might appreciate what might have been well-known to a Norwegian reader at the end of the 19th century.
Let us note that Holst wrote at a time of great progress in mathematics in Norway and he was friends with major figures whom he writes about. He was an excellent person to write on Norwegian mathematicians since he had much experience as a writer of biographies.
Mathematics, by Elling Holst.
Culture is to humanity what heat is to the physical world. It has a melting and accelerating effect and sets the forces of life in motion. During its gradual progress, the rigid, fixed conditions of earlier times, such as we know them from the Middle Ages, have gradually become fluid. The currents of culture now glide with a previously unimaginable ease from society to society and within each society from layer to layer. Thoughts and ideas travel around the world at the speed of its own revolution or more, and in a time shorter than it took a hundred years ago to implement a single reform within a branch of administration, the entire educated world now changes its way of thinking. An epoch-making discovery, even a winged word, gives a shock that is immediately felt by everyone, and exercises a noticeable influence on us all, attracting some, repelling others, completely according to the laws of the forces that prevail in the physical fluids.
Nowhere is the difference between the heavy melting of the human spirit of the past and the fluid mobility that prevails in the present felt more than when one follows the historical development through the ages in the natural sciences and mathematics. And this is almost most felt when one considers this development in terms of the individual people, which is what we are going to do here, as far as mathematics is concerned among us.
The first great advance in the mathematical field which was bestowed upon Western Europe during the Middle Ages was the introduction by the Arabs of our now-usable numeral system, which, as is well known, goes back to the Indians. But not until a couple of whole centuries after the struggle had been between the new trend of the Algorithmists and the followers of the old arithmetic, the Abacists, there are certain traces among us of an initial use of the Arabic numerals. However, we must of course admit that such an initial use must be assumed to have occurred somewhat earlier than we can strictly speaking prove. Whatever this may be, it is in any case certain that our otherwise rich and quite diverse Old Norse book treasure only shows a single work of mathematical content, the lawman HAUK ERLENDSSON'S (died 1334) "Algorismus", a rather short and concise arithmetic book in the "Indian" numbers, which according to ENESTRÖM and CANTOR is essentially a translation of a similar work by the Parisian scholastic, the English-born JOHANNES de SACROBOSCO (John Holywood).
After the union with Denmark and especially after the introduction of the Reformation, all spiritual nerve fibres gradually gathered in Copenhagen. The few and isolated examples of independent scholarly research that one occasionally encounters in Norway from now on are, over the long centuries, essentially limited to what some zealous priest, half buried in his mountain or fjord village, can try to gain from his free hours, by using his eyes or to maintain his university knowledge.
Presumably, very few of these home studies have reached the literature. Even less of what has reached the literature concerns the natural sciences, such as, for example, the natural history records we owe to PEDER CLAUSEN FRIIS. In any case, there is no work left behind that can show that mathematics has ever been a solitary country priest's solace or pastime.
It is also not easy to see whether any kind of mathematical instruction was given in any Norwegian Latin school up to the time of Christian IV. In the synodal decree of 1546, issued under the auspices of PEDER PLADE, no other subjects are seen to have been imposed on the Latin schools than Latin, religion, and singing and music. However, before the end of the century, arithmetic, geometry, and even astronomy are explicitly mentioned as subjects at several Danish schools, so there is a possibility that something similar may have been the case for some Norwegians as well. It is not very likely; the little information that is currently available mostly reveals only the extremely poor and miserable conditions of the Norwegian schools. The perpetual wars and the remote location of the country as a whole and especially its distance from the capital and its influence also fully explain this.
Under Christian IV, a brief attempt was made to introduce somewhat more science education in the schools, by the establishment of gymnasiums, of which the newly established Christiania also got one. They were to have among their three professors also a professor of mathematics and physics, who was also to be a good physician. However, this institution in Christiania, as indicated, did not last long anywhere. It only existed for 25 years.
Under the circumstances outlined here, it is not to be expected that the mathematical talent that our people have revealed so remarkably in this century would be able to sprout within the borders of our own country. So much the more interesting it is, within the much narrower circle, who after having attained a higher education expatriated for life, to come across a couple of more representative mathematicians of Norwegian descent. The most remarkable of these up to 1750 are the Copenhagen professor JOACHIM FREDERIK RAMUS (1686-1769) and the professor of philosophy from the Sorø Academy JENS KRAFT (1720-65). Ramus from Trøndelag, became a student in 1707 and soon entered practical work in Denmark, became in 1718 city conductor in Copenhagen, in 1720 director of the navigation school on Møen, and in 1722 professor of lower mathematics at the university. His mathematical writings are not of much importance, but the public made much of his knowledge, his labour, and his skill. Of far greater importance was Kraft, who is in fact one of the most outstanding mathematicians in popular literature and was fully at the peak for his time.
Kraft was born in Fredrikshald, where his father Anders Kraft was a captain; but he died early, and the son came to Denmark as a child to live with an uncle. Already as a second-year student he published mathematical dissertations that rose considerably above the usual level, and at the age of 26 he became professor of mathematics and philosophy at the foundation of Sorø Academy. He was a prolific writer. His main work is his "Lectures on Mechanics" in two volumes, which is the best work of its kind in the Danish-Norwegian joint literature and enjoyed a well-deserved reputation also outside Denmark, where it was translated into German and Latin.
Only in the middle of the last century did our country obtain a few more permanent scientific institutions, to which more independent and advanced mathematical studies could also be linked. In 1750, Frederick V established the "Free Mathematical School" in Christiania "for the benefit of our military staff and those who had already taken or would subsequently take service in the same, and for the promotion of the sciences required thereby." This was the first beginning of the present military school.
In the same year, the strange "Seminarium Fredericianum", based on science and modern languages, was founded in Bergen, whose founder is none other than Dr ERICH PONTOPPIDAN. Unfortunately, this beautiful idea of his had less viability than his other gift to the Norwegian school, the explanation of the catechism. [Note. Erich Pontoppidan was a Danish author who became professor of theology at Copenhagen in 1738 and bishop of Bergan in Norway in 1745.]
Finally, in 1757 the Bergseminariet, the Kongsberg School of Mines, was established at Kongsberg, probably with the intention of roughly equalling the mathematical school in Kristiania in terms of the depth and scope of mathematical studies. But it is clear, and is also confirmed by the various efforts towards the end of the century to reform the institution, that it could only be little attended and therefore not become of any particular importance.
In addition to these higher schools, we also had a learned society. In 1760, the naturalist Bishop JOHAN ERNST GUNNERUS and the two learned historians, Councillor PETER FREDERIK SUHM and Rector GERHARD SCHØNING, founded the "Throndhjemske Selskab", which from 1767 became the "Royal Norwegian Society of Sciences and Letters", whereby for the first time the scattered scientific pursuits in our country actually had a natural centre within the country itself.
At about the same time, the wills of the well-known ANGELL family provide new evidence that the sense for mathematical teaching was on the rise, as the will of 3 October 1763 in one of its codicils establishes a permanent teaching post in mathematics and physics, attached to Throndhjem's orphanage. The first teacher here deserves mention. He was the rather strange Danish autodidact DIDERICH CHRISTIAN FESTER, cartographer, mathematician, navigation teacher, etc., born near Slagelse in 1732, who from 1768-94 lived in Throndhjem, first as a teacher at the orphanage, later also as a navigation teacher and teacher at the cathedral school; he died in Sorø in 1811. He has among other things carried out the extensive map work in Pontoppidan's "Danish Atlas" and was otherwise an extremely productive author in a whole range of branches from pure mathematics to pure fiction. In the writings of the "Throndhjem Society of Sciences" we find a number of works from his hand, including some of purely mathematical content.
Of greater importance, however, was his contemporary, the rector of Bergen, FREDERIK CHRISTIAN HOLBERG ARENTZ, the first more outstanding Norwegian-born representative of mathematical studies within our own country.
Born in 1736 in Askevold, where his father, the later bishop of Bergen Friderich Arentz, was a priest, he received a better upbringing than was customary at the time, as his wealthy father was not only able to provide him with a capable tutor at home, but after his studies in Copenhagen even paid for him to travel abroad, during which, among other things, he studied physics and mathematics prominently in Leyden. The purpose of this journey was actually to train himself as an official physicist to participate in the learned research company that Frederick V intended to send to Arabia; however, he had to withdraw for health reasons and, by the time the journey took place in 1762 with CARSTEN NIEBUHR as a physicist, he was already in another permanent position, since in 1760, without application, Arentz had been appointed as a lecturer at the aforementioned "Seminarium Fredricianum". Here he gradually rose to the position of vice-rector. In 1781 he was appointed rector at the Bergen Latin School, a position he held until 1825; he died on 31 December of the same year.
Arentz was a learned man in the eighteenth century style; he taught theology, Latin, Greek, Hebrew, philosophy, mathematics and physics with equal zeal and thoroughness, and wrote works in almost all the subjects mentioned. He enjoyed great esteem throughout his life for his versatile learning, and also for his zeal and skill as a schoolman. His mathematical and other works have been printed partly in the Danish Scientific Society and partly in the publications of the Throndhjem Scientific Society. He was a member of both societies. His works are of interest first as evidence that Norway too was gradually beginning to count representatives of the exact sciences, but also on their own merits, especially one of them, a fairly early study of the kind of quantity combinations that are now called determinants.
As late as 1824-25, the writings of the Throndhjem Society contain a treatise by Arentz, and in an interesting encounter, the old-fashioned scholar of the 18th century meets in this volume a truly modern spirit from the new, the last aging representative of the Danish era with a radiant youthful figure, whom the scientific community of free Norway had just produced. The volume ends with a work by NIELS HENRIK ABEL. Nothing can better illustrate the rapid pace of development, now that we are approaching the present, than the vast difference between these two works. But also in the course of a strangely short number of years a whole new era had arisen. A new era abroad in mathematics and a new era at home in our own circumstances, both brought forward, like so much else, by the enormous knot that the French Revolution had put in cultural development.
While mathematics had previously been the noble pursuit of great solitary scholars, who had piled their theories on top of each other as if they would rather storm the heavens than directly benefit everyday earthly life, it had been laid by the brilliant leaders of the revolution as the natural basis for practical education, and the young students of the polytechnic school had thrown themselves into its study with such enthusiasm that in a short time a whole new school arose, the foundation of modern geometry. And at the same time that mathematics in this way erected new structures of thought, it made an even more significant advance by having a whole new foundation laid for its old reasonings. Parallel to the new geometric school, a new critical school arose, both parts together constituting what we may call the "French Revolution" of mathematics; its effects we live in and under to this day.
Our fatherland, which was so late seized by the cultural current, seems in return to have yielded a much fresher and relatively more fertile soil. Not only do we have, as we shall soon see, in Abel one of the great pioneers of the "revolution", but the very last days have brought out from the old papers of the turn of the century the name of a hitherto completely forgotten Norwegian, whom we can proudly record in our literary history as one of the most remarkable pioneers of the new era, and whose place in the ranks therefore remains precisely here, where the new era is about to dawn, despite the fact that his main work falls before the end of the previous century. He is surveyor and cartographer for the Danish general staff CASPAR WESSEL (born in Vestby 1745, died in Copenhagen as surveyor 1818), brother of the poet JOHAN HERMAN WESSEL. Little is known about his education. He was a student when he began to draw maps at about the age of 20. He has mapped large parts of Denmark and the Duchy of Oldenburg and has on the whole made no small contribution to Danish surveying. It also seems that practical surveying and the associated calculations have introduced him to the strange geometric ideas he has put forward in his only published scientific work. This, which has only recently been brought to attention, is printed in the Royal Danish Scientific Society's Writings, New Collection, 5th part (1799) and contains a number of ideas which, if they had become known outside Norway and Denmark at the time of their appearance, would have made Wessel's name famous for all time, as probably the first discoverer of one of the most important advances in the mathematics of our century, the principle on which the geometric representation of imaginary numbers in particular rests. Wessel's thesis, which the Danish Scientific Society is considering honouring according to its worth by having it published in a new edition in French after this new discovery, will not fail to arouse the greatest attention. It contains, in consciously implemented theory with elegant applications, a number of theorems whose rediscovery during the last century has given a number of later mathematicians of rank their fame. And it has lain unknown, unheeded, for almost a century.
But if Norway's name thus failed to be linked to the historical development of these very advances, it was not long before our scientific knowledge intervened as strongly as we could only wish.
The main names of the new critical school are GAUSS, CAUCHY and ABEL. The first two men, who had approached or passed the mid-point of life and enjoyed the highest reputation throughout the scientific world, the third a young student from a country so far out on the horizon that it was barely known by name within the same world.
The conditions for being able to grasp the new era and its spirit had in the meantime also reached Norway. In 1811 the country had got its own university, and in 1814 the connection with Denmark was suddenly severed, which otherwise, despite our university, would probably have led to the final and highest education still being sought there. Mathematics was the science in which our young university first bore fruit for the great European cultural community, already 15 years after its foundation and in a way that is our honour and pride to this day.
The very first beginning was of course small. The first professor of mathematics, SØREN RASMUSSEN (born 1768, died 1850), former head teacher at Kristiania Cathedral School, was a practical businessman and a popular socialite, and this probably more than a truly independent mathematical scientist, but as one of those who also helped Abel financially in his first difficult time, he will live in grateful memory. Of greater importance was his student and successor, Abel's actual teacher, BERNT HOLMBOE (born 1795, died 1850), who also exchanged the position of head teacher at the Cathedral School for the chair at the university. He has the merit of providing our geometry textbooks, which through him were made on a relatively modern level, essentially after LEGENDRE. But his main merit is the truly scientific education that he, even though he was not much more than a youth, was able to give to his brilliant student, and the piety with which he guarded Abel's work and his memory after his death. The most outstanding of the university's first teachers in the related subjects was without a doubt CHRISTOPHER HANSTEEN, who, however, was more of an astronomer and distinguished physicist than a mathematician, but whose lively and energetic spirit, which has made so many marks in various fields, also benefited mathematics more indirectly through the zeal and interest with which he took care of Abel. The essential merit of our university's first work is to have produced Abel at all.
NIELS HENRIK ABEL, one of the most original mathematical geniuses who ever lived, was born on 5th August 1802 near Stavanger, on Findø, where his father SØREN GEORG ABEL was a priest. According to a tradition which, as far as is known, has not yet been published, he was born 3 months prematurely and had to be wrapped in cotton wool and treated with the greatest care in order to live. The family had little, and Abel learned early to know the hardships of life. He received his education from his father until he was 13 years old, when he was sent to the Kristiania Cathedral School, whose wealth of scholarships and bequests was then, as now, a welcome help for poor sons of civil servants. His mathematical talent was revealed early, but developed considerably from 1818 when Holmboe became his teacher. As mentioned before, Holmboe took his work helping Abel with a seriousness and skill that cannot be overestimated. The teacher-student relationship soon developed into a faithful friendship, and through Holmboe, Abel's reputation spread early within the circles of the then small Christiania, and the university teachers became acquainted with his first works long before his schooling had ended. One of these was the well-known attempt to solve the general fifth degree equation, which Hansteen sent to Professor CARL FERDINAND DEGEN in Copenhagen for evaluation. Abel himself found the error in his reasoning, and against Degen's advice not to delve further into this "sterile subject", he, on the contrary, from now on "applied all his acumen to it with the energetic intention either to find the solution or to prove its impossibility". Degen, who himself was a peculiar mathematical thinker and had found great pleasure in Abel's experiments, drew his attention to the elliptic functions, which until then had almost found only one cultivator, Legendre. The "Magellanic passage through an immense analytical ocean", which Degen here with brilliant foresight announced to Abel, were to become, next to the "quintic equation" and the theories connected with it, the two main topics that Abel had time to work through in his short life. But in doing so he also opened the way for posterity on two of the most difficult-to-reach points in mathematics, where after his death the modern theory of functions and the modern theory of equations have found their "passages".
In 1820 Abel lost his father and from then on was solely dependent on the support he could receive through scholarships or private assistance. His mother was left with several children in very difficult circumstances. In 1821 he became a student and continued his studies with increased zeal. Several of the university professors, including Hansteen, in whose house he frequented, and Rasmussen, joined together to entertain him, and the latter even gave him help for a short trip abroad, namely to Copenhagen (1823). This trip is remarkable in that it appears from one of Abel's letters from there that he was already busy with one of his most beautiful ideas, the so-called "reversal" in the theory of elliptic functions. He had tried to discuss a difficult point in this with Degen, but to no avail. "God knows how I shall get out of it!" he exclaims in the letter. On this journey he met a young Danish lady, CHRISTINE KEMP, who later came to Norway, where the acquaintance was renewed. She became his fiancée, and after Abel's death, Mineralogy Professor Baltazar Mathias Johansen Keihau's wife.
The college also took care of him and applied to the government for a travel grant for him and a monthly contribution until the journey could begin. He did not obtain the travel grant, but instead he received (1824) a home grant of 200 spd. per year for two years. [Note. The Speciedaler (spd.) was Norwegian currency from 1816 to 1875. One Speciedaler was divided into 5 ort or 120 skilling.] This happened at about the same time that he won his first really big victory, as he kept his promise to himself to prove the impossibility of solving the general fifth degree equation by roots. Abel was then unknown to PAOLO RUFFINI who had been working on the same subject since the beginning of the century, but, despite the fact that he thus actually came in second, would still be able to claim all the rights of a priority. His view is so new and fundamental that in it he brought himself and science into the extremely fruitful question of the criterion for equations that can be solved by roots. Posterity has shown its gratitude, simply by forever linking the important kind of equations that he specially singled out and considered in the related treatises, in his name. They are now called "Abelian equations".
In addition to this work, which he had printed in a small quarto booklet, during these two years he occupied himself mainly with elliptic functions, where, according to CARL BJERKNES, he must be assumed to have mastered both the inversion, the subject of his eager inquiries to Degen in Copenhagen, the double periodicity and finally even the far more far-reaching addition theorem. SYLOW and LIE leave this to a later date.
Abel's wait did not last two full years. With a travel grant of 600 spd. annually for two years, he was able to go south in September 1825. He was accompanied by, among others, BALTAZAR KEILHAU and CHRISTIAN BOECK. They were Norway's young hope, future leaders in our new society. The journey went via Copenhagen and Hamburg to Berlin. Degen had died, but Professor von SCHMIDTEN recommended Abel to meet for private discussions with AUGUST CRELLE in Berlin, a mathematician who was enthusiastic about the newer studies of the time, whose acquaintance was not only of the greatest importance to Abel, but whose living desire to be able to accomplish something significant for the emergence of mathematics in Germany found the right man in him.
With the acquaintance with Crelle, a new and important chapter in Abel's life begins. His stay in Berlin was certainly the happiest time of his life. Even their first meeting was strange. Abel himself describes it as follows:
"When I arrived in Berlin, I went to him as quickly as possible. It was a long time before I could make him understand what the real purpose of my visit was, and it seemed to be destined to have a sad end when I found courage, and he asked me what I had already read in mathematics. When I had mentioned to him a couple of the writings of the most excellent mathematicians, he became very affable, and as it seemed, really happy. He entered into a long conversation with me about many different difficult cases that had not yet been decided, and, when we came to talk about the higher equations, when I told him that I had proved the impossibility of solving them of the 5th degree in general, he would not believe it and said he would oppose it. I therefore handed him a copy. ..." The conversation then turned to the bad state of mathematics in Germany at the moment. "When I expressed my surprise that there was no journal for mathematics here like in France, he said that he had long had in mind to undertake the editorship of such a journal and would also put it into practice at once. ..." Here two men had met, from whose sympathy and joint efforts the most significant results were to emerge. Crelle became the best spiritual support that Abel came to possess; he appreciated from the first the immediate genius in him, and even did everything to tie him firmly to Berlin, thus offering him the very editorship of his journal.
This was really the end of the happiness that Abel ever achieved. He never got any further than offers he dared not accept. His piety for home, his sense of duty to his fatherland, which had funded his journey, his anxiety for his poor family, meant that he never dared to think of accepting any of these honourable positions, which perhaps, when all had come to an end, could have become the pride of his country and the best support of his poor family.
He now developed a restless activity. In a quarter of a year he had finished six works. Two of them were groundbreaking, his somewhat revised thesis on the fifth degree equation and his fundamental investigation on the binomial series. He wrote in French, and Crelle translated. He enjoyed his moments of rest among his Norwegian traveling companions and thus swung with fewer sorrows than at any other time in his life between his two existences, the esteemed and great mathematician despite his youth, and the childish, cheerful student despite his abstract learning.
This life lasted 5 months. In March 1826 the small Norwegian colony left Berlin and headed south. Abel went via Dresden, Vienna, Tyrol and northern Italy to Paris, where he arrived in July and stayed until the end of the year. The stay in Paris was not nearly as pleasant as the stay in Berlin had been. The great French mathematicians were not very accessible. There was no replacement for either Crelle or the faithful circle of Norwegian friends he had left. The painter JOHAN GØRBITZ was the only one with whom he had somewhat more regular contact among his compatriots. From this emerges the beautiful, gentle, friendly, melancholy portrait in ink, which forms the basis of all pictures of him. But he worked tirelessly. This particularly concerned the editing of a large treatise in which he wanted to set down the fruits of his study of elliptic functions. In October the treatise was finished and submitted to the Academy. Abel himself had the greatest expectations of it. He had no idea that when he handed it over to the Academy he buried it so that it would not see the light of day until long after his own death. LEGENDRE and CAUCHY were commissioned to judge. The first had shown Abel a fleeting goodwill, the second had hardly deigned to give a look at the work when he had tried to show it to him. Cauchy, the head of the French mathematical science, the distinguished grand dignitary of legitimacy, was far from suspecting how closely he would be linked in the history of mathematics to this embarrassed, poor young stranger. Bjerknes assumes that Abel's second great treatise on elliptic functions, "Recherches sur les fonctions elliptiques" was also begun in Paris.
Towards the end of Abel's stay in Paris, worries began to mount around him. His travel fund was running low, so he had to write to Holmboe for a loan, but the future at home looked bleaker. With the travel fund gone, his public support was actually over, and that is not to say that his home could afford or need him. The journey home was made in a natural longing for Crelle and his sympathy for Berlin, where the latter made the last, but in vain, efforts to keep him back. In the spring he was home again, to take up the fight against poverty and at the same time try to keep up with developments and get his works printed as best he could. For the time being he was almost relegated to giving introductions and guidance, and incurring debts.
The year that begins with his return is without comparison the hardest of his life. Worries about food and the depression of exchanging the outside world, where in Berlin he had at least found recognition and attractive offers that far exceeded his own modest expectations, with the small, shabby backwater that was Kristiania at the time, and our small and poor settlement conditions in general, which offered him nothing in the way of prospects. Added to this was the fear of never hearing anything about his great prize-winning thesis submitted to the French Academy, and finally the nervousness that suddenly struck him when he learned that he had a fellow suitor, JACOBI, who was also trying to penetrate the same exclusive niches, the elliptic functions, where he had so much unpublished work, the priority of which he had to secure. The last circumstance was fortunate insofar as it forced him to forcefully edit and print his new work. A very strange rivalry arose between the two rivals, which was carried out partly in "Crelles Journal" and partly in "Astronomische Nachrichten".
The questions of priority which have arisen in this connection for the historians of mathematical literature have been more often discussed, for example by LEJEUNE-DIRICHLET, KÖNIGSBERGER and most recently and thoroughly by Carl Bjerknes; they have always, more and more definitely, granted Abel the right of first discovery on all disputed points. The peculiarity of the alternating publications by Jacobi and Abel can be briefly described as follows: Abel once in the process appeared in possession not only of the proofs for the statements uttered by his rival, but also of others which far encompassed these.
At the beginning of 1828 it looked as if things were beginning to dawn on Abel. At the same time as Hansteen's Siberian journey, Abel had received a small fixed income in a docent position at the university, which was intended to be a temporary position, but which became his short-lived final position in life; rumours of his poverty and hardships had also reached outside the country and down to his admiring and capable friends in Germany. He had more of them than he himself knew. HUMBOLDT and GAUSS supported Crelle in his efforts to secure him a call to the University of Berlin. But the joy over this was for the time being short-lived. In the autumn, Crelle, who had had rather good hopes over the summer, could only send bad news. Although Abel still felt pain at becoming familiar with the idea of changing his homeland, this turn of events was a bitter disappointment to him. To all these sorrows and worries finally came his weak health. Abel had never been strong. Sanguine and sensitive, as he appears in his letters, he easily gave in to his emotions, was happy with those who were happy, but in return suffered greatly from the pressure and hardship of the times. The letters are full of the most immediate expressions of these moods. It is also not certain that he was always careful about his health. As I said, he was happy with those who were happy, and student life at the time was wild. Abel participated in many a merry party, and the card games often lasted through the night until the early morning.
The overexertion of his eager production and the many kinds of excitement and disappointments during the dark period after his return finally took their toll on him. In the autumn of 1828 he fell ill. He certainly recovered, but the Christmas trip down to Froland, where he had recently found a replacement for the vacated Hansteen home with the factory owner SMITH'S family, where his fiancée was a governess, gave him the decisive blow. He was poorly provided with outer clothing and caught a cold on the way. The Christmas visit turned into a sickbed. A stubborn pneumonia developed, which despite the most loving care turned into consumption.
Even on his deathbed his pen did not stop. It was especially the uncertainty about the fate of the Paris treatise that tormented him. And as if at the last moment to save a smouldering ember from the fire, he wrote down again the addition theorem contained therein, which he rightly considered one of his greatest discoveries, and sent it to Crelle. This is the last thing from his hand, his testament to science. The 6th April 1829 was his death anniversary. His grave, with a simple cast-iron pillar, is in Froland's cemetery. To this day, no other public memorial has been erected to Norway's first scientist.
"Premature and late" is the sad motto of Abel's life. He himself, like HENRIK WERGELAND's first butterfly, was premature with his fine and noble genius in our first, still too poor and harsh, time of independence, when the country had neither need nor affordability for geniuses of his abstract nature. [Note. Henrik Wergeland (1808-1845) was a Norwegian writer and poet. He was a leader in promoting Norwegian heritage and culture.] And late because he wasn't able to obtain the only position that the home had, which was rightly suitable for him, the professorship that his older friend and teacher Bernt Holmboe received in 1826, while Abel was in Berlin. And far too late also came the final recognition, the invitations and the praise of the French Academy. All this arrived, some on his deathbed, some only later.
The contrast between what he was and what he accomplished and the fate he reaped, between the feat of manhood he had truly accomplished and the young age, 26 years old, when the burden broke him, makes him one of the most sympathetic, but also one of the saddest figures in the history of science. The news of his death came completely unexpectedly both to his countrymen and his many foreign friends, and the attendance was large and general. The news reached Berlin immediately after the final decision on his calling had been made. The announcement also made a deep impression on the French Academy. The great prize, 3000 francs, which was awarded to him and Jacobi, was sent to the heirs of Jacobi and Abel to be divided equally. These rare honours from abroad only really made his countrymen understand what a loss we had suffered.
It was also the French Academy that gave the impetus to the public publication of his collected works (1831). Holmboe was commissioned to do this, and in 1839 he completed the edition in two quarto volumes. Among other things, Holmboe had spared no effort to find out about the forgotten Paris treatise and its fate. This is too strange to be ignored.
As mentioned above, the Academy had appointed Legendre and Cauchy to evaluate it; Cauchy had it with him, and there it lay forgotten or overlooked. It is not known how long this would have lasted if a quotation of the addition theorem in one of Abel's articles in "Crelles Journal" had not been accompanied by a footnote in which he mentioned this theorem as occurring in his thesis submitted to the Academy. During Jacobi's correspondence with Legendre, the strange thing happens that the latter draws Jacobi's attention to the addition theorem mentioned in Abel's article in his reply (14th March 1829), which he seems to have overlooked. Jacobi bursts out in the strongest exclamations of admiration at this discovery; "but", he continues, "how can this perhaps the most important mathematical discovery that has occurred in our century, have been communicated to the Academy and have escaped the attention of you and your colleagues!" The question was delicate, and Legendre must admit in his reply that the thesis still lies unread with Cauchy. In his apology he states that the thesis was so difficult to read because of the ink and the writing that a copy had been requested from the author at one time, which he had not yet been sent. But Legendre adds that he has already arranged for the manuscript to be sent from Cauchy. As for the demand for a copy, this can nowhere be seen to have come to Abel's knowledge.
This transfer of the manuscript from Cauchy to Legendre may have prevented its destruction at the eleventh hour, when Cauchy followed the legitimate monarchy into exile the following year at the July Revolution, but neither the manuscript itself nor the news of Abel's death had any further effect on the old Legendre to hasten its evaluation or printing. Legendre died in 1833, and it was not until Holmboe's preface to the edition of the collected works that the Academy seems to have been seriously reminded of its duty. Two years later the manuscript was rediscovered and printed in "Mémoires publiées de divers savants". A strange fate nevertheless befell the old treatise. Before the proof had been read, the manuscript disappeared again and has not been found since.
If we were to briefly summarise the main points of the services that Abel has rendered to science, it would be the following:
Fundamental work in the theory of equations, with the conclusion of the question of the fifth degree equation and the opening of the new question of the form in general of the equations which can be solved by roots;
His teachings on the inverse functions in the theory of elliptic functions; on the whole his quite new, clear and fundamental works on these, but notably:
The new path he broke for even more advanced studies, namely of the higher Abelian functions named after him, those about which he stated his great addition theorem, after Legendre his "monumentum aere perennius" [Note. This Latin phrase means "a monument more lasting than bronze". Originating from the Roman poet Horace, it is a metaphor for an achievement that outlasts physical monuments, granting the creator lasting fame];
His participation in re-founding higher mathematics, notably through his exemplary works on the convergence of series, his participation in what I called above the critical school at the beginning of the century;
to which finally comes:
His active participation in the founding and development of "Crelles Journal" and its reputation.
The work mentioned under the penultimate main point is recorded partly in his above-mentioned work on the binomial series, one of his best works in "Crelles Journal", partly and notably in his letters, where in a series of witty and instructive statements he attacks and judges the boldness with which the great mathematicians of the previous centuries reasoned. Of the letters, perhaps most have been printed partly separately here and there, partly in excerpts, notably in his collected writings (translated), and partly by Bjerknes. A complete edition should not be long in coming. Together, they give a faithful and lively picture of his own person, his childish naivety, his enthusiasm for his science and his struggle for existence. But they also contain valuable contributions to the characteristics of the time and its people.
A new, considerably enlarged and improved public edition of Abel's writings was prepared during the seventies [1870s], by LUDWIG SYLOW and SOPHUS LIE (published 1881). During the review of his old protocols for the benefit of the new edition, various discoveries were made, one of which showed that his participation in the criticism of the use of divergent series had taken him further than had previously been suspected, since he was seen to have come into possession of the entire complete theory of convergent series, which BERTRAND had presented later in the thirties [published 1841], but which Holmboe either did not find or did not know sufficiently to be able to understand what was recorded by Abel about it.
Abel was not allowed to teach his own subjects at the university, not even strictly speaking his own subject, and he only taught for a couple of semesters in total. He is thus isolated also in the sense that he was not allowed to found a school, and left no students behind at home. Rather, his only student was his own teacher Holmboe. Originally, there was also no subject group included in the university examinations, which included any particularly advanced mathematical teaching. Mathematics was, as it still is, a subject for the examen-philosophicum and one of the subjects for the berg-examen, but both parts were on a very narrow scale. Until 1851, when it was decided to establish the real teacher examination, those who wanted any teaching outside of the second examination in mathematics were referred to the berg-examen or abroad, and we therefore have the peculiar view that before the first real examination (1855) people who had completely different plans for themselves, the mathematician Bjerknes, the astronomer Fearnley, the physicist Christie and others, took the berg-exam and sometimes even got employment for a shorter or longer period at the Kongsberg mines, and up to the year mentioned, no Norwegian mathematics teacher had submitted to any real examination in his teaching subjects. Teaching was in the hands of philologists, theologians and officers. No wonder then that the time that followed Abel was poor in purely mathematical production. And basically pure mathematics has had few practitioners in this country. Of the university professors, Broch, Bjerknes and Guldberg have all almost exclusively cultivated the applications of mathematics. Only in Ludwig Sylow and Sophus Lie do we again have thinkers whose speculations are exclusively directed towards purely exact science.
OLE JACOB BROCH (born 1818, died 1889), an administrative and organizational talent of the first rank, mathematician, physicist, statesman and both practical and theoretical state economist and statistician, belongs to a family of which several representatives were outstandingly gifted in the same directions. He was born in Fredriksstad, where his father, later war commissioner JOHAN JØRGEN BROCH, was based as a company commander. He spent his childhood partly in Kristiansand, where his father was transferred, and partly in Kristiania. Even then his mathematical talent came to the fore, and his uncle, later major general THEODOR BROCH, began to study higher mathematics with him. But as one can conclude from his versatility later in life, he became equally good at all subjects. Already at the age of 17 he was a student and threw himself with energy both into his studies and into the struggles that then prevailed in the student community, a preschool for the great society where so many of our public men have received their initiation. Broch almost joined the Wergeland side. His closest teacher at the university was Holmboe, and his teaching fell precisely at the time when Holmboe was preparing the publication of Abel's collected writings. It is therefore beyond all doubt that these provided an important subject for the mathematical entertainment between teacher and student. As early as the autumn of 1840 Broch went with a public scholarship to Paris and Berlin, where he partly continued his mathematical studies of elliptic functions, and partly was introduced to mathematical physics by Cauchy, who had again returned from his exile, especially the theory of light. The only purely mathematical works that Broch published outside of textbooks in the proper sense date from this period, namely a series of treatises on elliptic functions, which were presented at the Academy in 1841 and enjoyed the honour of being published with many words of praise in the "Journal des savants étrangers". In the following years his attention was largely directed to optics, in which he took his doctorate after his return (1847). After a short period of activity as a university fellow and as co-founder and co-director of Nissen's school, he became seriously associated with the university in 1848 first established as an extraordinary and, after Holmboe's death, as an ordinary professor of pure mathematics.
With BROCH's appointment at the university, a series of reforms in our mathematics teaching began, which are very much linked to his name. Partly through his textbooks, but especially through a significantly toughened examination for the artium, he sought to have a stiffening effect on the old-fashioned breed of mathematics teachers around the country. People soon feared him as an examiner. "Ole Jacob's" famous somewhat nasal "Nok far, sjeks far!" made the two mathematical subjects for the artium a true Scylla and Charybdis, whose dangers for the fragile candidates made both teachers and parents tremble. Countless are the anecdotes that preserve the drastic forms of ignorance in his concise examination as a background, and many were probably also the annual victims of his zealous work for his subject. But the remedy for all this revealed ignorance, a teaching staff trained directly for their work, who had themselves learned mathematics, also emerged after Broch's initiative (law of 1851). As a university teacher, Broch developed a rare versatility. An extraordinary energy and work ethic, which he brought to everything that his restless mind and hand set as a task, also led him to gradually treat in his lectures, as it seems according to a calculated plan, almost all mathematical branches. The section catalogues show that he carried this out over a period of 5-6 years. He was not to fully succeed in this twice. His administrative and statesmanship talents were seized by the state and municipality on a scale that no one else had, it must have been Schweigaard. As early as 1846 he was appointed a member of a commission to revise the Widows' Fund's tariffs, which was followed by a succession of public assignments concerning the insurance system, state loans, the mortgage bank, the railway, the telegraph systems, the public engineering system, the Royal Tax Commission, commissions concerning weights and measures, money, coinage and banking legislation, technical education, and even purely military questions, and he frequently sat as chairman on these commissions. The municipality seized him from 1857 as a member of the chairman's or representative council, and from 1862, when he entered the Storting, he became one of its most active members and from there entered the government in 1869, setting the example that was only followed more often later of a civilian chief at the head of a military administration. It was one of the first signs of an incipient parliamentarism, which brought Broch into the government, and true to his principles he stepped out again when the State Council case of 1872 was refused sanction. He returned to the university as an extraordinary professor. But this time the university did not benefit from his activities for long. The state continued to exploit it in the most diverse ways in almost all the branches of our economy that can be mentioned. Soon, moreover, an international institution seized him in the most honourable way, which for the last years of his life moved his residence to France. Broch had frequently represented our country at congresses of various technical and economic nature, and in particular he had participated in the repeated congresses for the uniformity of coinage, measures and weights in a way that had drawn the special attention of foreign countries to his remarkable knowledge and skill. In 1879 he received an honourable testimony to this by being called to be the director of the international bureau of measures and weights in Sevres. In this trusted position, from which a number of new works have been published, partly by his own hand and partly under his auspices, he also repeatedly represented Norway at international congresses, both diplomatic and technical and economic. He lived in Sevres for the last ten years, until his death in early 1889.
As we have seen, of Broch's numerous and voluminous contributions to our scientific and professional literature, a relatively small part directly concerns mathematics. But his importance for this subject is no less for that. Most of what can be mentioned of mathematical learning in this country after his time is written by his students and the students of his students. The distinguishing feature of his mathematical writing is energy. In his lectures as in his writings he bore through the most violent calculations in a manner that is as determined as it is certain. On the other hand, he is not what mathematicians call elegant. His sense of artistic design, which is in fact of great importance in mathematics, the clarity and logic of which also require a clear and transparent form to be fully classical, does not seem to have been in proportion to the irresistible force with which he carried out his investigations. Nor is his language appealing; his books are therefore often heavy-handed and not popular in any sense of the word.
In addition to Abel and Broch, Holmboe directly educated another significant student, Bjerknes.
Professor CARL ANTON BJERKNES (born in Kristiania 1825) is a quiet scholar, a complete opposite to Broch who is preoccupied in all practical matters, but in his studies Bjerknes is just as firmly established in the applications of mathematics as Broch, although these applications still have a purely abstract and theoretical character. After a childhood whose memories fluctuate between the narrow streets of Vaterland and the adventurous river voyages with the "Røiertene" up and down the Drammen River between Drammen and Hoksund, where he had his family in Eker and Sandsvaer, he became a student in 1844. [Note. In the 1840s, Røiertene were specialised transport boats used on the waterways connected to the Drammen region.] He has already been mentioned as one of the examples of those who, in order to follow their desire for the exact sciences, had to make the detour via the mining exam and Kongsberg. Up there he was a "night guard": making sure that the workers did not keep illegal silver, but after a couple of years he became an assistant professor at Kongsberg Latin School and could from then on follow his inclination more fully. In 1852 he won the Crown Prince's gold medal for a work on trigonometric functions, in which it was a question of deriving their properties by series expansion. In 1854 he became a university scholar in pure mathematics and the following year, with a public scholarship, travelled to Paris, where he heard Cauchy and Lamé, and to Göttingen, where he heard the equally brilliant RIEMANN, but where DIRICHLET in particular became decisive for his entire subsequent field of study.
It was not just Dirichlet's superior teaching talent that was decisive here. An association of ideas, prepared by a strange coincidence, led Bjerknes to continue Dirichlet's work in a direction unexpected for him and the entire scientific community.
At an early age, whether as a student or a schoolboy he does not remember himself, he had come across an old book in his father's attic, a Danish translation of EULER'S famous letters to a German princess. Here this great mathematician polemicises and thinks strongly, although completely in vain compared to his contemporaries, against a view of nature which over the last few centuries had fought its way to the complete inviolability of a dogma. A view almost innate to naive humanity, which, for example, Aristotle had formulated in the simple theorem that every moving body must be in immediate connection with a moving body - as when the horse pulls the cart or the wheels mesh with the wheels of a machine - was suddenly shaken after Newton's discovery of the law of gravity and its application to the movement of celestial bodies. There was no connection to be discovered between the planet and the sun, and when attempts to construct a hypothetical intermediary to reconcile the law of gravity with the older view failed, it was finally decided to reject Aristotle's law, to replace it with the exact opposite: every body has the ability to act at a distance on another and set it in motion without the help of any intermediary. This law of action at a distance now gradually became the leading thought of scientists, it left its mark from the first hand on the new branches of physics which emerged at the end of the last and the beginning of this century (electrostatics, magnetism, electrodynamics, the theory of elasticity) and stood at the time we consider as a truth tested through generations and accepted by all.
It was when Dirichlet in his lectures, as an example of the application of his mathematical theories, gave the solution of the problem of the motion of an invariable sphere in an infinite fluid of invariable density, that the half-forgotten Eulerian ideas in all their irreconcilable contradiction to the thinking of the time emerged in Bjerknes's memory. He found that Dirichlet's result in all its simplicity pointed in a direction favourable to these Eulerian or, if you will, Aristotelian ideas, and he conceived the idea of resuming the struggle that Euler had fought in vain, and of resuming it not by polemicising, but by presenting new facts which he suspected would be found by the solution of a more general problem than Dirichlet's, by setting the number of spheres to any number and giving them variable volume. The processing of this and closely related problems therefore became from then on his life's work.
While other of Dirichlet's students, ERNST SCHERING and ALFRED CLEBSCH, continued Dirichlet's work in the direction indicated by the master himself, replacing the sphere with an ellipsoid, Bjerknes varied the problem by initially giving the sphere a variable volume. The first step was thus taken, although the result obtained was still too flimsy to give him any answer as to whether his suspicions would come true or not. This answer was to remain elusive for many years.
After his return from his trip abroad in 1857, he combined for a time, in accordance with his diverse education, the two positions as the newly appointed Professor Kjerulf's amanuensis at the mineralogical collection and as a teacher of mathematics at the military college (after Broch). In 1861 he was appointed and in 1863 he was made an associate professor in applied mathematics as Hansteen's successor in this subject. Three years later, the title of associate professor was replaced by the title of professor for all associate professor positions.
A more significant change in his position occurred when Broch entered the Stang ministry in 1869. The year before, Bjerknes had finally succeeded, despite writing cramps and a health that had broken down in many ways, in fighting his way to his first decisive victory in his hydrodynamic studies, as he had found the solution to Dirichlet's problem for an arbitrary number of spheres and had his hunch confirmed that interactions arose between the spheres that had the character of forces at a distance. But these learned studies were not in the taste of the then prevailing "practical" trend, and reluctantly yielding to pressure from some of his colleagues, he left his real profession, mechanics, and had himself appointed Broch's successor in pure mathematics. He continued his hydrodynamic studies, however, despite persistent physical frailty and new disappointments, as it proved more difficult than he had expected to pursue his victory. Among the tangle of interactions whose existence he had demonstrated between his spheres, he still tried in vain to separate out special classes which could be completely equated with the remote effects of nature. Nevertheless, he succeeded in bringing the purely mathematical treatment of the problem to a provisional final result in 1871.
In 1875, the definitive breakthrough finally came, as he succeeded, through a closer discussion of the analytical results that he had long been in possession of, in isolating the force phenomena named after him, which arise between the spheres when they oscillate or pulsate in the liquid, and recognised the analogy, which is feasible to a certain extent, between these and the electrical and magnetic force effects.
These discoveries, which were considerably expanded in the years 1879 and 1881, attracted the greatest attention, partly by the simplicity and novelty of the beautiful and unexpected phenomena, partly by the elegant experiments with which he presented them to the great European public at the electrical exhibition in Paris in 1881 and, by invitation, the following year in London and several other places, but above all by the powerful contribution they made to the possibility of a return from action-at-a-distance physics to the more original and simple ideas of older times. Bjerknes's collaborators for his first, even more imperfect experiments, were Professor OSKAR EMIL SCHIØTZ and cand. real S SVENDSEN; later he found an excellent helper in his son, then a student, now professor of mechanics and physics at Stockholm University, VILHELM BJERKNES, whose acumen and experimental talent celebrated their first beautiful victories here.
Professor Bjerknes, whose work on hydrodynamic, through these purely physical experiments, moved away somewhat from its original purely deductive character, has in recent years returned to the mathematical treatment of the problems. However, nothing new has yet been published.
In addition to this main work, Bjerknes has a great merit from his thorough studies of Abel's life and development history, published in various treatises, of which "Niels Henrik Abel. A description of his life and scientific activities" (Stockholm 1880) with its significantly expanded French translation (Paris 1885) is the most important and indeed one of the most important personal monographs in our literature.
Bjerknes's academic lectures and their method also set him in contrast to his robust predecessor Broch. He shows himself in this as a fine successor to Cauchy and Dirichlet, whose careful reasoning he everywhere implements in his developments and proofs. On the other hand, he has somewhat limited the scope of the mathematical disciplines at the university.
The first person who, after Abel's time, decidedly, wholeheartedly and with great success threw himself into pure mathematics was PETER LUDVIG MEYDELL SYLOW, born in Kristiania (1832), son of the then riding master, later state councillor THOMAS von WESTEN SYLOW. Like most Kristiania boys of good family at the time, he first went through the bourgeois school and from there entered the cathedral school. The mathematics teacher, assistant professor JENS ODÉN, was knowledgeable, but old and impractical as a teacher. Incidentally, Sylow learned well from him, but was one of the few chosen ones to whom this happened. A coincidence led him to go on his own and study higher mathematics. His sisters were allowed to study physics privately. As is well known, that subject was not offered at the Latin schools at that time. Caught off guard by this sisters' dominance, Sylow began to ask his father questions. But his father, who insisted on self-reliance for the longest time, referred his son to OSWALD MARBACH'S German physical lexicon, and what it did not explain to him, he had to seek information about in JURIJ VEGA'S voluminous mathematical handbook. [Note. Jurij (Georg) Vega was a Slovene mathematician, engineer and military strategist. He published his lecture notes in mathematics in the 3 volume "Vorlesungen über die Mathematik" (1782, 1788, 1788) with a supplement in 1790.] Sylow himself corrected the old-fashioned, not very correct reasonings, as best he could, when they did not satisfy his critical sense. In this way he acquired at a young age a rare ability for self-study of difficult mathematical subjects.
Sylow became a student in 1850, the same year that Broch first took his artium examination. Despite his excellent knowledge, he did not get better than a 2 for geometry, almost a victim of the disparity between Odén's teaching method and Broch's examination style. However, Sylow soon became good friends with Broch when he began his studies. If he had been among the first to get a feel for the new examiner, he was also among the first to benefit from Broch's energy as a teacher. He really got the new course in its first full fresh force. Broch did not skimp on the material. Hansteen laughed and said ironically that he was reading elliptic functions according to "Jacobi's great book" ("Fundamenta nova"). In return, the prize paper (on gnomonics) proposed by Hansteen, for which Sylow won the Crown Prince's gold medal (1853), was not particularly significant or instructive. But that was not Sylow's fault. [Note. The competition required a mathematical and historical treatment of the theory of sundials, a subject in which Hansteen had a professional interest as a professor of astronomy and applied mathematics.]
According to the strict rules then in force, the three-part civil service examination was taken in three consecutive semesters. Sylow graduated in 1855. He made a splash in his mathematics department. Broch examined him in the most difficult subjects, in Jacobi's transition from theta functions to elliptic integrals, etc., but he passed everything with flying colours.
Contrary to everyone's expectations, Sylow did not get a recommendation. Here, as so often in the lottery of life, he was unlucky. Professor LORENTZ CHRISTIAN LANGBERG, who in P A Munch's absence examined in physical geography, stubbornly held on to a 3.
After completing his graduation, Sylow received a teaching position at Nissen's school. His first attempt to obtain a scholarship abroad failed, as OLUF RYGH was preferred. However, here at home he laid the foundation for his studies in the difficult subject, where he gradually, and quite early on, succeeded in becoming one of the first researchers of modern times, the theory of equations in the footsteps of Abel and Galois. It was actually his intention to delve further into elliptic functions, and on that occasion, on Broch's advice, he completed the reading of Jacobi's "Fundamenta nova" and was very much absorbed by its magnificent content. Bjerknes predicted to him, however, that he would be even more captivated by the presentation of Abel, which he therefore threw himself into. Bjerknes's prediction came true, but with Abel the theory of equations began to captivate him even more than the elliptic functions, and in it he set about trying to interpret an unfinished and partly only fragmentary piece of work. Here the usefulness of the strict school at home and with Broch became evident. It had sharpened his ability to work independently, so that he completed the entire very difficult thesis in a short section of four lines, a feat that until then only one person, namely KRONECKER, had accomplished. During a conversation with Bjerknes, to whom he shared his studies, he was informed of GALOIS's theory, but also of its daunting difficulty. However, the difficulty no longer frightened Sylow, "but I must admit that it was unpleasant at the beginning," he has since said. But he managed. Some time later, an edition of SERRET's "Cours d'algébre" fell into his hands, and he read in it Kronecker's treatment of the same subject as Abel's "fragmentation treatise." It also contained for him the key to the four lines he had not deciphered before. It was clear to him: Abel had here given an essentially complete solution to his problem long before Kronecker, whose presentation, however, did not seem to want to acknowledge this.
During these studies, however, Sylow (1858) was transferred to Fredrikshald as head teacher. It was apparently a great stroke of luck that he did, but that place became for him like the magic chair in the mountains: whoever sits there stays seated. In external circles he has now, after almost 40 years, made no progress. However, this has not prevented him from both following along and from making his mark, so that it is not even improbable that for a certain period of his life he was simultaneously one of the least and one of the most esteemed of our mathematicians at home and abroad.
Sylow presented his view of Abel's fragmentary thesis at the natural scientists' meeting in Copenhagen in 1860 and sent his thesis in French for inclusion in "Crelles Journal". Kronecker himself was on the editorial board and most likely had the article reviewed. Whether he felt "cut too close" or the reason was something else: Sylow had his thesis returned, in accordance with a wish expressed for the occasion.It can essentially be read as a detailed annotation to the fragmentary thesis in the Lie-Sylow edition of Abel. The first time Sylow met Kronecker after the printing of the Abel edition, the latter took the blade from his mouth. There had certainly been a wound in an old wound.
Finally in 1861 Sylow received permission and a travel grant. This concerned Berlin and Paris, studies of equations, personal and literary knowledge. After his return he accepted Broch's offer to lecture for him at the university while he was at the Storting. Sylow then lectured, among other things, about Galois and had Sophus Lie as an audience. Here he probably came face to face for the first time with the concept of "group" which was to make him so famous.
In the first years Sylow published few things. A few treatises on those parts of the theory of elliptic functions which border on the theory of equations, and which served him precisely to test the correctness of his theories. But he already knew more about this subject than almost any other contemporary researcher. When CAMILLE JORDAN'S great "théorie des substitutions" was published (1870), relatively little in it was new to Sylow, who, on the other hand, had in his pocket the key to another important undisclosed secret. He presented this in 1872 in a treatise of a few pages which he sent to CLEBSCH'S "Mathematische Annalen", and was thus famous.
The so-called "Sylow's theorem" is based on a hitherto almost unnoticed and unused theorem of Cauchy and contains an explanation of the structure of the so-called "substitution groups", which are extremely important for the theory of equations. Sylow's theorem is now the first step for everyone who deals with these questions.
Jordan happened to visit Kristiania in 1872. Lie brought the two equation theorists together, and, on the return journey from Frognersaeteren, Sylow communicated to Jordan his theorem, which had not yet been published. Already from a Swedish canal steamer, Sylow received applications of his theorem from Jordan's hand, and Jordan, who, following PASTEUR's example after the war in 1870, had sent back his diploma as a member of the German Scientific Society, reopened the connection with Germany in order to hasten the publication of Sylow's thesis in a letter to Clebsch.
At this time, a new important task began to occupy Sylow, the new official Abel edition (1873-81), a testament to the fact that his solitary work had not escaped attention at home. During this time he enjoyed a four-year leave of absence from school, but from 1871 he had to attend to both tasks.
As an equation theorist and co-editor of Abel's writings, Sylow has two major achievements. His work is hopefully far from over; at 63 years of age he looks like a man between 40 and 50. His studies are also still continuing. He published his two most extensive works after the publication of Abel. "But," he says himself, "the dissertation in the Mathematische Annalen was something else, as short as it was." In 1894, Sylow was created an honorary doctor at the University of Copenhagen.
Twice there was a brief prospect for Sylow to come to the university, but it did not materialise: at Langberg's death in 1861, when Christie and Adam Arndtsen were competing for the professorship in physics, he was convalescent after a serious illness and was all the more prevented from participating in the competition, as the two fellow candidates each had a large travel stipend, and in 1869, when Broch became a state councillor, it was, as mentioned before, not the post in pure mathematics but in applied mathematics that became vacant, and Sylow did not feel suitable for it at that time.
With the training of teachers of science, there necessarily followed an opportunity for a scientific life in mathematics and the natural sciences, which was not previously conceivable, and the number of our natural scientists, but also of our mathematicians, has therefore grown according to a completely new standard. If the number of the latter has not increased, this is due to the few official uses that our country has so far been able to give to the scientific cultivators of pure mathematics. In particular, in the first twenty years, mathematical teaching posts, which also included higher mathematics, were restricted to the university, the war school, the naval officer school and the military college. Later, the technical schools were added to these. But in other uses of professionally trained mathematicians, e.g. in the insurance sector and the like, we are still far behind even our neighbouring countries, and the establishment of the technical college that has been debated for so many years has not yet been possible to unite the many conflicting interests. One of the first champions of this latter necessary and final link in our mathematical-technical education is Professor CATO MAXIMILIAN GULDBERG (born 1836), a mathematician of Broch's school and with his sense for practical business life, but considerably less occupied by heterogeneous tasks, so in return his scientific works are of all the greater importance.
C M Guldberg is the eldest of a group of brothers, most of whom have devoted their energies to the natural sciences or mathematics to a distinct degree. Their father was the strangely versatile genius, CARL AUGUST GULDBERG, who, Swedish by birth but Norwegian by schooling and university education, as a theological candidate founded the well-known "Guldberg & Dzwonkowskis bookstore", which deserved its extensive publishing activity, where he also followed his urge for literary work, as he, among other things, founded and edited "Skilling-Magazin" for a number of years, but he then left this busy and varied work and became a priest. [Note. Skilling-Magazin was published as an illustrated weekly magazine in Norway in the period 1835 to 1891. Carl August Guldberg was its first editor.] His son has clearly inherited the same drive for enterprise, practical sense and versatility.
After a few years of schooling in Fredrikstad, where his interest and ability for mathematics emerged early, C M Guldberg took advantage of the opportunity the garrison town provided to seek further education with the local officers. He came to Kristiania and became a student in 1854 and a realkandidat in 1859. In the same year he won the Crown Prince's gold medal for his dissertation: "On Touching Circles", later published as a university programme. Geometry, however, was not his subject, nor was pure mathematics at all. When he, at the same time as Sylow, received a travel scholarship, he focused his main work on mechanics and machine learning, and after his return took over the teaching position in the latter subject at the military college. At the same time (until 1867) he was a teacher at the military school and later also taught mechanics at the college. Rational mechanics was thus his subject, but he did not direct his own scientific research directly into this field, but rather into the questions that had been virtually unexplored by mathematicians up to that time, namely the mechanics of chemical forces. And here, together with the then associate professor PETER WAAGE, he achieved epoch-making results. These, the first of which were already published in the scientific society in 1864, were included in a university programme "Sur les affinétés chimiques" (1867). In the same year, Guldberg became a university fellow and two years later succeeded Bjerknes as professor of applied mathematics.
Guldberg had to suffer the same experience as most of our scholars: even in a foreign language, it is of little use to publish anything at home. A short extract of the main results, included in the French institute's "Comptes rendus", saved the important discoveries from being overlooked. The chemical law of mass action, now usually named after the discoverers (with the German pronunciation of Waage's name), was the result partly of numerous experiments, partly of a successful and correct point of attack for the mechanical theory. The correctness of the law was soon confirmed; in 1869 JULIUS THOMSEN'S thermochemical experiments were carried out in Copenhagen. Several researchers had gradually begun to adopt a similar line of thought, partly from the experimental and partly from rational sources. But the priority of the Norwegian researchers is fully recognised, and in the mechanical-physical conception of chemistry, which has now penetrated all the way down to textbooks, an important chapter in these is henceforth devoted to the Guldberg-Waage law.
It is on the border between chemistry and physics that Guldberg moves in his studies published without collaboration. These, which are not very voluminous, but on the other hand not few in number, all deal with molecular laws and have made important contributions to orientation on these obscure and difficult points in natural science. In the long tables of the physical quantities of the various substances, by looking them up in the large handbooks, one will find Guldberg's name almost more frequently than any other, and those calculated by him fit in excellently with those found by other researchers through direct experiments.
Apart from these molecular studies, mention must also be made of Guldberg's work (in collaboration with HENRIK MOHN) on the movements in the atmosphere, published in French in two university programmes, in which the mechanical theory of vortex centres is developed.
In addition to these purely scientific works, Guldberg has carried out considerable activity, partly in mathematical-technical treatises, partly on commission from the public sector, such as in the extensive work on the currents in our fjords, which had regard to the construction of the Drøbak Company, partly in textbooks, and finally also outside the literature in public administration, particularly concerning the technical and especially the railway system.
He has been the alternating president of the Kristiania Science Society for a number of years.
In 1860 or 1861, a young student entered the circle of science students who did not initially attract any particular attention as a mathematician, but who now, after his graduation, made his mark much more seriously again. It was SOPHUS LIE. Lie was born (1842) at Nordfjordeidet, where his father was a priest, but from where he was soon transferred to Moss. Lie grew up there. In contrast to most established mathematicians, he did not show any particularly pronounced or at least phenomenal ability for mathematics from the beginning, but was equally good in all subjects. Not even the choice of civil service studies, which came after some hesitation between science and philology, had any noticeable influence on him in this respect. On the other hand, his lecture-based teaching of astronomy in the student community attracted much attention due to its originality. Even after his civil service examination (1865) he felt uncertain for a while about his true calling and suffered from this, when during his somewhat scattered studies he was suddenly awakened to the awareness of his independent creative power by coming into contact with works of modern geometry (c. 1867).
The turn that took place, and the sudden revelation to himself and the world of a mighty awakening genius, are among the most peculiar events in the history of science. Barely a year later Lie is already a great force in his science, one of those that one begins to follow with great attention. His development also runs through the strangest phases with a rapid pace. After having debuted with a highly original imaginary theory, which already bears the clear mark of his penetrating eye for the essential and the fundamental ideas of the mathematical laws, and which brought him a travel scholarship (1868) and after his return (1871) the position of university fellow, a production followed step by step, which in quantity and richness of new and original ideas involuntarily leads the thought back to Abel. The starting point was modern geometry, and his grasp of this was so deep and decisive for him that his later purely analytical and algebraic new vision over the years has actually been geometrically viewed from the beginning. In accordance with a development parallel to that which time has otherwise gone through, in his later production he has more and more stripped away the geometric form and clothing and made it his business to let the purely algebraic truths spring out of purely algebraic fundamental considerations. Modern geometry was hardly known by name in Norway at that time. As far as I have been able to trace, there is only one small writing before Lie's time that can be attributed to it, namely by OTTO GILBERT DAVID AUBERT (1809-38): Echantillon d'une analyse sphèrique. (1833.) To all of Lie's great contributions to science, one must therefore also add that of bringing his country up to date on this important topic.
During his trip abroad, Lie received and gave powerful impulses. In Paris, where, incidentally, his stay came to an abrupt end at the outbreak of the Franco-Prussian war, he had established a very important and close acquaintance with two equally young geometers who promised significantly, which they also kept, the German FELIX KLEIN and the Frenchman GASTON DARBOUX.
The immediate fruit of Lie's foreign journey was his remarkable doctoral dissertation with the spherical geometry contained therein, the first ideas of which, incidentally, already dawn in his imaginary theory, and which is rightly considered one of the most beautiful geometric discoveries of the present day. His studies had moreover led him more and more into the extremely important field of differential equations, with its manifold unsolved problems, and already his older works here indicate a number of important advances. In particular, his peculiar geometric interpretation of the problems influenced the formulation and perception of the task by his contemporaries. Already then, his various advances in several places at the universities were accompanied by corresponding changes in the presentation. The greatest sensation in this first part of his scientific life was caused by his integration method for partial differential equations giving exact solutions, in which he gives this problem the exact and final solution.
The attention his brilliant output had attracted abroad did not escape attention at home, and when he applied for a vacant professorship in Lund at the beginning of 1872, a series of independent movements were set in motion to attach him to our university. The unusual nature of the whole situation, the sudden discovery of a talent of great dimensions among us, also created an unusual form for his appointment, as the Storting itself took the initiative and granted him an extraordinary professorship that same year.
However, from 1873 onwards he had already begun to formulate a new great idea, the emergence of which marks the second great period of writing in Lie's life. In 1876, together with OSSIAN SARS and JAKOB WORM-MÜLLER, he founded the "Archiv for Mathematik og Naturvidenskab" in order to achieve a final breakthrough in our narrow and, in the opinion of many, one-sided scientific conditions here at home, and Lie immediately began to develop a new series of ideas in this, in a long series of detailed treatises, his so-called theory of transformation groups. This is nothing less than a completely new mathematical subject, having application at once to a whole series of mathematical areas, especially to geometry and differential equations, and to the group theory devised by Galois for the theory of equations. At first, the new theory, published as it was in a new and unknown journal, far less respected than Lie's great name could have claimed, and it took a number of years before he got collaborators. However, this did not diminish his desire or courage, but rather gave him double the opportunity to formulate his great theory fully and entirely by his own efforts, which he soon presented to the general public in the much more convenient German journal "Mathematische Annalen". From now on, the theory of transformation groups attracted ever-increasing attention both among German and, not least, among French mathematicians.
In 1883 PICARD, a member of the French Institute, drew the attention of his countrymen to the new doctrine and its countless applications, and the following year a young German mathematician, FRIEDRICH ENGEL, was commissioned to travel to Norway and familiarise himself with the subject from Lie, in order later to lecture on it in Leipzig. He has since made no small contribution, as Lie's colleague, to bringing forth the great three-volume major work on the theory ("Theorie der Transformationsgruppen". I-III. Leipzig 1888-93).
But the most significant recognition came to Lie when in 1886 he received the honourable appointment of ordinary professor of mathematics and head of the mathematical institute in Leipzig, an event which again inevitably brings the thought back to Abel. This stay, which has now lasted nine years, is above all linked to the publication of the large, voluminous works on the "theory of transformation groups", on "continuous groups" and on "differential equations with known infinitesimal transformations" (1888-93), a series which is still being expanded with similar treatments of other fields of his colossal scope, and the entirety of which must at least be described with Picard's expression: a masterpiece.
During Lie's stay in Leipzig he has enjoyed no less public recognition from France than from Germany. Every year his audience is sought by official French pupils, and the circle of those who follow his path in French literature is growing every year. In 1892 he became a corresponding member of the Kronecker Institute, an honour all the greater because Germany was thereby represented by a foreigner.
Lie's production has been extraordinary. In aggregate, these numerous treatises and heavy volumes measure up fully to what our most productive writers in other genres have accomplished, and this is mathematics, a speculative science which narrows its results into the narrowest form. At first his form was almost too narrow. The amount of material that one or other of his earlier smaller treatises in reality contains is almost unbelievable. A small inserted sentence can contain the seed of a thesis.
Later he developed the exposition more, and several of his treatises have the most beautiful elegance; thus his well-known work on minimal surfaces is completely classical. His academic lectures are completely clear, highly original and lively.
Last year (1894) the Storting made 4000 kroner available as a personal supplement to the highest professor's salary, in case Lie should want to return. His response to this was a happy thank you. We can thus hope that Kristiania will again in the future be the place from which this Napoleon of mathematics leads his victorious theories into the field.
Lie has educated numerous students, although few of them belong to the our literature. The author ELLING BOLT HOLST (born in Drammen 1849) considers himself one of the oldest of these. After already having shown an aptitude and interest in mathematics at school, he was student from 1868 and graduate in 1874. Lie's lectures as a university scholarship made modern geometry his favourite subject. Privately supported by the warm-hearted rich man JAKOB BORCH in Drammen, he was able (1874-75) to follow his inclination during a period of study with Lie's close friend FELIX KLEIN in Erlangen and Munich. After his return, he was employed for some time at the Meteorological Institute and then (1876) for several years university fellow in pure mathematics and since then as a teacher at Aars and Voss's school. In 1878 he won the crown - the Prince's Gold Medal - for his thesis "On the Importance of Poncelet for geometry" and studied in the winter semester of 1879-80 with a public scholarship in Paris, where he prepared his doctoral dissertation: "A couple of synthetic methods especially for use in the study of metric properties". Here he makes a new use for the geometry of certain functional theories and methods, and showed the fruitfulness of this approach in a number of new situations. When Sophus Lie moved to Leipzig, he was given the position to provide the geometry lectures at the university, but the salary therefore set was so meagre that he was overwhelmed with extra work that he had to put his own studies aside. Since 1891 he has been head teacher at Kristiania Technical School and thus unites a position created for him (1894), a docent position at the university, with the obligation to prospective teachers to lecture on general school mathematics topics. This arrangement was initiated by application from his older and younger students, who wanted a more permanent arrangement of this a kind of teaching that had not been resumed since Broch's death, and who also feared that Lie's expected return could threaten his position at the university.
In recent years, his work has been mainly aimed at raising awareness among the general public of the importance of mathematics in school and life. This includes various activities in speaking and writing, including his lectures at the Kristiania Workers' Academy. His work in the mathematical seminar for students, which he founded in 1886, goes in the same direction. In 1889, Holst became a Norwegian member of the international committee set up by the Societé de mathématique de France to edit a general mathematical bibliography, where he still remains.
He has found the necessary recreation during his rather strenuous professional work in publishing books for our children and ABC literature.
Of the younger generation, several of the students of Bjerknes, Lie, Guldberg and Holst have made a fine beginning in our mathematical literature. Of these, who thus give hope for the future of our exact sciences, the more physical mathematicians DANIEL ISAACHSEN (born 1859), VILHELM BJERKNES (born 1862) and KRISTIAN BIRKELAND (born 1867) deserve mention, as well as the pure mathematicians, the head teacher in Trondheim AXEL THUE (born 1863), who is almost an logician, ALF GULDBERG (born 1866), a student of Sophus Lie, and the young promising CARL STØRMER (born 1874).
Last Updated July 2026