Memorial speech for Carl Gustav Jacob Jacobi
A speech at the celebration of the hundredth anniversary of Jacobi's birth was delivered at the International Congress of Mathematicians in Heidelberg on 9 August 1904 by Leo Königsberger (Heidelberg). We give a version of this speech below.
Carl Gustav Jacob Jacobi
by Leo Königsberger.
In family circles, as in the life of people, it is a fine custom and a sacred duty to commemorate outstanding men in the moments of joy over happiness and prosperity of their own, in moments of joyful pride in acquired goods and national achievements, to whom we owe thanks for the construction, whose foundation they laid and which they helped to build with tireless work, supported by ideal humanity, by the enthusiastic dedication for the well-being of their nation to fight and defend everything that is dear and sacred to them, or that, gifted by the ingenuity of their spirit, guided by the love of truth and an indomitable research instinct, in the realm of spiritual and moral power stand as a sign of the progressive development of the human race in art and science. And so it was a beautiful thought and an act of piety worthy of a large scientific community that the International Congress of Mathematicians not only wanted excellent researchers to explain the modern development of individual parts of the exact sciences in broad outline in front of friendly-minded scientific circles, not only wanted to see all the outstanding collaborators in the advancement of mathematical science in the narrow circle of the initiates who want to explain the principles and results of their own work, but, as it were, as a solemn introduction to difficult and serious work, organised a commemoration of the brilliant founder of large and extensive disciplines in our science, which a hundred years ago, after the great mathematicians of France after Euler and the Bernoullis almost alone for a long time, steadily raising the flag of mathematical and mathematical-physics research, and as great and stimulating teachers of the next generation instilled the desire and courage to do hard work, who came to the aid of the Göttingen master, enthroned alone, with tremendous creative power at the height of exact science research, in order to have Germany also side by side with France in the development of mathematical science.
Carl Gustav Jacob Jacobi was born in Potsdam on 10 December 1804, the second son of the banker Simon Jacobi and his wife, from whose marriage two sons, Moritz and Eduard, and a daughter, Thérèse, sprang. After the mentally unusually active boy received his first instruction in the ancient languages and the elements of mathematics from his maternal uncle, "unique and beloved teacher", as he later called him, in November 1816, not yet twelve years old, he entered into the second class of the Potsdam Gymnasium and was admitted to the first class after half a year, in which he had to stay for four years since he could not be admitted to university until he was sixteen years old. As with most Gymnasiums in Prussia, after the state and intellectual upheaval of the people, teaching in the ancient languages and history was excellent, borne by the enthusiasm for the state and artistic life of the ancient peoples, and of moral seriousness with regard to upcoming cultural and intellectual work of the coming generation. But today we also want to commemorate the brave man who was responsible for teaching mathematics to the students of the Potsdam Gymnasium, and who also did literary work by writing textbooks. On the certificate attesting love, gratitude and veneration, which was presented to Heinrich Bauer by his students in 1845, for his 50th anniversary as a teacher, the greatest adornment is the name of Jacobi, who was then at the height of his fame, and who personally awarded the jubilee diploma to an honorary member of the German Society of Königsberg.
At Easter 1821, barely 16 years old, Jacobi passed his high school diploma; the mathematical work, which was judged to be very good, dealt in a clear and elegant manner with a problem of spherical astronomy, the derivation of which concluded with the confession: "I do not know whether this proof is found in any textbook, due to my lack of knowledge of mathematical literature." His graduation certificate states, "Blessed by God with rare faculties of the spirit, his knowledge of languages and mathematics is as thorough as excellent, quite unusual in Greek and in history," and when the Senate considered the praise for Jacobi from his teachers to be too extensive, the proud and frank men of their profession replied: "he has a universal mind, has unusual abilities and a high calmness of mind, seizes and embraces everything without being interrupted by fatigue; now he's studying philology and mathematics, but hardly wants to keep these subjects fixed forever, in any case he'll make himself famous one day."
Indeed, at first he devoted himself to the study of ancient languages with great enthusiasm and astonishing workmanship and attracted the attention of Böckh, the old master of Greek philology, in his seminar. "Miramini, commilitones suavissimi, philologum me vobis philologis dissertatiunculam proponere de Pappi Alexandri collectionibus mathematicis" ["You admire, sweet fellow, philology so I give to you my philological dissertation on Pappus of Alexandria's Mathematical Collection"] are the introductory words of his seminar paper, which was excellent in philological and mathematical terms, but was written at the time when his decision was made to devote himself entirely to mathematics. "By seriously studying philology for a while," he writes to his uncle, "I managed to take a look at the inner glory of ancient Hellenic life, so that I could at least not give up further research without a struggle because for now I have to give up completely. The immense colossus, which the work of Euler, Lagrange, Laplace created, requires tremendous strength and effort of thought, if one wants to penetrate its inner nature and not just dig into it externally. To become master over it, so that one does not have to fear every moment being overwhelmed by it, is driven by an urge that does not stop and rest until you can stand up and overlook the whole work."
The works of the great French mathematicians and the writings of Euler and Gauss were his teachers; lectures such as those held at that time in Berlin and at all other German universities were not challenging enough for him. Gauss, amazed and admired by everyone who devoted themselves to exact science research, lectured at a small university in front of a few listeners; what he presented was new and of ingenious originality, there were surprising points of view everywhere with a wide overview of the distant future of science, everything was exact in content, precise in form; but Gauss lacked the warmth and enthusiasm that also expresses itself externally, which must be mastered by the mathematical teacher if the sometimes so dry and sober truths are to fertilise a self-receptive mind, if the ideas emanating from the mind of a genius, however brilliant, should be echoed on a soundboard that is still fresh from youth. And in the second decade of the last century there was neither an important teacher nor an outstanding researcher in the lecture theatres of all the other German universities. When Dirichlet felt the urge to devote himself to mathematical studies, lectures on the elements of synthetic and analytical geometry, on the initial elements for algebra and on the all-encompassing combinatorics, the universities of his homeland could not offer enough, and he went to Paris to learn science and method, depth and clarity from the great French researchers at their famous university.
Dependent on his own strength, without any direction for his studies, Jacobi, taught by Friedrich Theodor Poselger, passed his state examination with excellent results after only one year of tremendous intellectual effort, barely twenty years old, and immediately prepared for his doctorate. After his oral examination, Enno Heeren Dirksen his examiner, said Jacobi was "equipped with commendable knowledge" and in the submitted work "meditationes analyticae" showed a "more than usual independence and a certain originality of treatment," so he was allowed to combine his habilitation with his doctoral thesis and to publish a part of the submitted dissertation as a sample under the title "Disquisitiones analyticae de fractionibus simplicibus", a work in which we already encounter the unusual depth and clarity of his mathematical views. By introducing infinite series to determine the coefficients of a general partial fraction decomposition of rational functions, he brings a function-theoretical element into algebraic studies, which later proved its fertility in the development of the theory of functions, and at the same time uses the principles applied to interesting transformations of infinite series.
So Jacobi, not yet 21 years old, stood at the Berlin lectern and, when he first appeared, according to the testimony of his listeners at the time, showed such a highly developed teaching talent that when, after just half a year, the Ministry had requested that he teach as a privatdocent at Königsberg, replacing the deceased full professor of mathematics Ernst Friedrich Wrede, in order to be able to offer the young lecturer more prospects of a possible promotion, he willingly accepted it. And while his transfer to Königsberg opened up a wide and fertile field for his teaching talent, it was a auspicious event for his writing that August Crelle in Berlin was already thinking about founding a mathematical journal, and in a very short time, even under the most difficult circumstances, to bring about this blessing to the mathematical sciences. As early as the summer of 1826, Jacobi sent Crelle two shorter works with the words: "I make you master over the life and death of these small creations", which contained explanations and simplifications of Gauss's studies on the approximate calculation of integrals, while at the same time he was working on an extremely difficult work on number theory, especially deep in the study of his disquisitiones arithmeticae.
At the end of the eighteenth century, Gauss, only 20 years old, had created a new number theory with astonishing depth and incomparable ingenuity, in which the division of circles was linked to the theory of the transcendent and laid down the basis for the later theory of functions, and in the first quarter of the last century created a surface theory, which gave the most abstract geometric truths an adequate form for mechanics and mathematical physics, not long after the theorems of today's potential theory were built up, in the history of physics and astronomy, through great and far-reaching discoveries, to which his name is permanently attached, and preserved after a creative work unrivalled in undiminished splendour until the middle of the nineteenth century. But no one had been able to follow his investigations, no one dared to join his research and continue it until Jacobi stepped into the arena of mathematical research in 1825, even if for a time alone and therefore forced, with the greatest effort of the spirit and the whole moral courage of a scientific pioneer, to make the ground of mathematical work in Germany fruitful, but with constant reference to the great Göttingen master.
First of all, number theory, the most difficult, because it is the most abstract, of all mathematical disciplines, had the greatest attraction for Jacobi, and he aroused to a great extent the interest of Gauss with his writings about the results he had reached in his studies of the 3rd and 5th power residues, and whom he shyly approached with the words: "Only the zeal for science could inspire an unknown young man to speak from his darkness to a mathematician who stands in such glory"; but his intimate dealings with Bessel, one of the greatest adornments of astronomical science, also led him away from the most abstract studies to dealing with the most varied applications of mathematics, to the astonishment of his older brother Moritz, who was studying in Göttingen at the time. Moritz, the later well-known St Petersburg physicist and inventor of galvanoplastic, writes: "If the transcendental universality of my mind, manifests itself in recognising and comprehending the qualities of the bricks and the masonry in general, you are assuring me of this universality, and with full rights, by showing through your letter the ease with which you enter a field that previously seemed to be alien to your innermost nature, astronomy and physics, pendulum experiments, triangulation networks and maps. I am pleased because it may avenge me, and you realise that only that which can be applied is of value."
Inspired by Bessel, he dealt with the question of the expressibility of the roots of an equation by means of certain integrals, based on a principle that is closely related to Cauchy's investigations, which determine the values of the zeros of a function by the contour integral round the completely enclosed space. Without knowing Gauss' fundamental idea of the introduction of complex quantities into number theory, he opened up, through a wonderful divination, generalised theorems on number division about the division of the circle and the cubic residues, and seemed to have new and fruitful principles precisely on the idea of initiating extensive research in this area when his mind was taken up by studies of a completely different kind, the fruits of which would bring the 22-year-old young man to the forefront of mathematicians as early as the summer of 1827.
"You are about to see me," Jacobi replied one day to a friend who found him strangely upset and asked why it was upset, "send this book (Legendre's Exercices du Calcul Intégral) back to the library, with which I have decided I have been unfortunate. Every other time I studied an important work, it always inspired my own thoughts and some ideas came me. This time I went away completely empty-handed and was not inspired by the slightest idea." But, it was precisely in this area that he went on to achieve everlasting laurels.
The excellent French mathematician Legendre had indeed created the basis for a comprehensive theory of elliptical integrals in that work; but for the further development of the transcendental doctrine it was only through the introduction of new and creative ideas that a mathematical discipline could develop, which should become the alpha and omega of modern analysis, and on the foundations of which the wonderful productive power of Jacobi was so brilliant applied. At the same time as Jacobi, the great Norwegian mathematician Abel, who was two years older, stepped onto the scene of scientific work in this same direction, somewhat earlier, but the one who knew nothing of the other, and both suspected that Gauss, after the hints in the Disquisitiones Arithmeticae, must have known the great secrets of these hidden truths for nearly 30 years; but Abel, like Jacobi, dared not to use his advice and knowledge in writing or orally, he, like Jacobi, felt alien to the strict and closed nature of that powerful mind, because Abel's mind and his mind were impulsive, as was Jacobi's nature.
Abel was already well ahead of Jacobi, in Königsberg, in researching transcendentals without him knowing about his discoveries in good time. Two short letters to Schumacher on 13 June and 2 August 1827, which laid the basis for the theory of transformations of the elliptical integrals, showed the mathematical world that Jacobi had conquered new ideas and principles in science; the infinite series of module chains and the principle of double periodicity will always appear in the history of mathematics in Abel and Jacobi as a steady leap in understanding in the study of mathematical truths. This discovery by Jacobi brought such a surprise to the old French master and tireless researcher in the theory of elliptical integrals that he did not want to believe him at all when the beautiful letter from Jacobi, which begins with the words: "Sir, a young mathematician dares to present to you some discoveries made in the theory of elliptical functions, to which he has been led by the assiduous study of your fine writings", gave the announcement of his findings, which clearly showed the immense scope of his principles for the whole further development of the transcendental doctrine. With these lines the correspondence between Legendre and Jacobi, which is so interesting for the history of the mathematical sciences, began, which dragged on until 1832, half a year before the death of Legendre [Legendre died in 1833], and at the same time that wonderful competition developed before the astonished eyes of the mathematical world between Abel and Jacobi, as he, far from envy and resentment, carried only by the purest love of truth, stands almost unparalleled in the history of mathematical science. "I can rest," Legendre writes a little later, "on the zeal of two tireless athletes like you and Abel; I nevertheless congratulate myself for having lived long enough to witness these generous struggles between two equally vigorous young athletes, who are turning their efforts to the benefit of science, the limits of which they are increasingly pushing back."
Jacobi's plan to publish a detailed account of his transformation theory and the further results of his research in the theory of elliptic functions as early as the autumn vacation in 1827 initially suffered a short delay due to the fact that he was led by a new and very heterogeneous series of thoughts to deepen Lagrange and Pfäff's investigations into the integration of total and partial differential equations; while developing the Pfäff method for the integration of a general total differential equation in an elegant and symmetrical form and establishing its connection with Lagrange's studies, he was led to the remarkable extension of the Lagrangian integration procedure for a linear partial differential equation to a simultaneous linear system of a very special form and in this way extended the Lagrange method for the integration of any first order partial differential equation with two independent variables to those with any number of independent variables.
So Jacobi stands next to Gauss as one of the leaders in the expansion of mathematical science; but he was still, although the only mathematics teacher at the University of Königsberg, a docent with a salary of 200 Talers, and when the Prussian minister asked the faculty about their opinion regarding Jacobi's appointment as an extraordinary professor, there was no doubt that the latter had excellent academic achievements, but nevertheless his appointment was rejected because he, such a young man, had inappropriately expressed himself about some older academic teachers and had sought his close contact essentially among younger lecturers - Heinrich Wilhelm Dove, Franz Ernest Neumann and others, names that today each of us recalls with piety and veneration. In delightful objectivity, the university's curator, in his report to the minister, emphasised that the concerns of the faculty were inaccurate, citing the favourable judgment that Bessel gave about Jacobi's academic achievements, as well as the enthusiastic report Legendre made to the academy about this work - "Legendre murdered jealousy and brought it to the gallows," his brother Moritz writes - determined the government should appoint him an extraordinary professor at the end of 1827; a few months later, the same promotion came for Neumann and Dove.
In the meantime the first part of the research by Abel had appeared and this had secured his priority of discovery in many points of the transcendental doctrine - and now that serious and difficult struggle began between the two young researchers, both inspired by the noblest scientific ambition and pure love of truth. We admire the mutual intermeshing and complementing of their investigations, the reliance of the one on the results and methods of the other, and so there was neither time for the planned elaboration of a coherent theory, which in all parts was well founded, because almost every day brought one a new discovery. While Legendre still did not diminish his astonishment at the infinite number of transformations of an elliptical integral into a "real analytic Proteus," Jacobi had already applied elliptical transcendents to the well-known geometric closure problem, the algebraic resolutions of the division equations of elliptic functions, as he gave Abel, simplified in terms of content and form, founded the theory of modular equations and, again guided by a wonderful divination, recognised in the analytical expressions of the elliptical inverse function the fundamental transcendent, the theta function, on which the theory of elliptical and Abelian transcendents should be based for all subsequent time.
But Abel had made many of these discoveries, either earlier or a little later than Jacobi; "did he plough with your calf or with his own powerful bulls?" Bessel asks Jacobi when the second part of "Abelsche recherches" came into his hands. Jacobi now realised that Abel solved the general transformation problem earlier than he did, and this drove him back to ever greater haste and unstoppable work; in September 1828 he began printing his great work, in which he wanted to design a coherent theory of elliptic functions, also published one work after another on the properties of his transcendents - but now the paths and methods of these two researchers were also increasingly moving apart, and, while Jacobi was led to very general investigations, some of them purely function-theoretical, Abel turned to the difficult theory of integrals of algebraic functions.
The time was approaching when the wishes of Crelle, who worked tirelessly for the development of the mathematical sciences, seemed to be realised in Berlin; Abel was supposed to leave Norway, to habilitate at the University of Berlin, to found a mathematical seminar there and to be permanently involved in the editing of Crelle's Journal. It was Humboldt's and Legendre's wishes that the seminar should be equipped with relatively rich resources and entrusted to Abel and Jacobi, in order to work with Dirichlet and Steiner, who were already lecturers at the university, to train a new generation of creative and enthusiastic young mathematicians; at the request of the university, however, Jacobi was "as worthy of promotion to a full professor in all respects" initially continued to hold his appointment in Königsberg in March 1829.
In the meantime, he had rescued Abel's fundamental work on the addition theorem of integrals of algebraic functions from oblivion, after it had been neglected by an unfortunate coincidence in the Paris Academy for almost two years; Jacobi calls this theorem of Abel "a durable bronze monument," "what a discovery by M Abel of this generalisation of the whole of Euler! have we ever seen such a thing!" exclaims Legendre, who, as he devoted the first supplement to his treatise to Jacobi's discoveries, chose the subject of his second supplement to treat that wonderful theorem for hyperelliptic integrals.
Abel's appointment to Berlin was signed by the minister, and the King's authorisation to form a seminar for higher mathematics and physics was granted, Jacobi was appointed full professor in Königsberg, and mathematical studies in Germany had a promising future. Then Abel suddenly died on 2 April 1829, 27 years old, "the hope that I had conceived of seeing him in Berlin was therefore cruelly disappointed, ... he has left us, but he left a great example," Jacobi reported in sadness and dismay after Legendre's report of the death of Abel.
A few days later, the Fundamenta nova funetionum ellipticarum by Jacobi appeared, a work worthy of Gauss's Disquisitiones Arithmeticae, and with all eyes on its author - the 24-year-old young man was undisputedly the leading German mathematician after Gauss, but he was physically and mentally exhausted from his enormous incessant work. He travelled to his parents in Potsdam, initially only to rest there for a short time; this was where the first personal acquaintance between Jacobi and Dirichlet was established, and on a trip they made together to Halle and from there in the company of Wilhelm Weber to Thuringia, they got to know each other better. The Ministry willingly gave him a vacation for the whole summer semester, and after spending a few more months with his relatives and friends in Berlin, he went to Paris, where he enjoyed, from the end of August to the middle of October, living for full enjoyment in nature and art, but at the same time also in constant scientific communication with Legendre, Fourier, Poisson and other outstanding mathematicians and physicists, some of their correspondence has survived.
Returning to Königsberg, he gave, for the first time at a German university, a lecture on the elements of the theory of elliptical transcendents but was also prompted by the constant communication with Bessel to gradually turn to problems of a different nature, which Stürm's publications on algebraic equations, with the help of the series resulting from his 'functio generatix' for the solutions of such, and interesting series expansions of functions of several variables, are closely connected with the theory of the Fourier series.
The recognition given to him by the excellent French mathematicians, Poisson's well-known report on his fundamentals, the honour he was given when the Paris Academy shared its Grand Prix between him and Abel's relatives, and the repeated recognition of "his meritorious effectiveness and acclaimed literary achievements" by the Prussian government made him look back with satisfaction and gratification at what he had accomplished; he was preparing for a large eight-hour lecture course on elliptic functions, which he envisaged for the next summer semester when the beginning of 1831 brought about a decisive change in his life.
"Dear Marie! It is a special matter that drives me to write to you, and I like to discuss the matter to you without further preparation," begins Jacobi's first letter to his future bride, the daughter of Schwinck, a wholesale merchant in Königsberg, who had just rushed to the hospital bed of her older sister, the wife of the regional president Wissmann in Frankfurt an der Oder. "From early on I was occupied with the most serious work, believing that the circle of my existence was exhausted in that, receiving full satisfaction from all that youthful lust for fame could dream of, I had to feel strange in myself when an apparently rich existence suddenly emptied for me, and glory and honour and science no longer seemed essential and of little value compared to a friendly look from your eye," and very soon afterwards he wrote to her:
"This step was done with such necessity, due to the innermost nature of your being and mine, such as the eternal laws of nature and spirit may be: there was no arbitrary resolution to be taken here and therefore no rush. And so it often happened to me at crucial moments in my life, as it certainly always happens to the unclouded spirit, that a necessary action is clearly in front of his soul, even if he can count on his fingers reasons against it, as well as everyone else."
The wedding took place on 11 September, and only then did Jacobi regain complete concentration on science. "My works are like this," he now wrote to his brother, "that for many years I only had to write by collecting the rarest results, for many things had already been worked out diligently, only the last touches were missing, but I have never been able to find the joy that is necessary for complete works. I am now as joyful as ever, for every undertaking and work, so there is hope for many things."
Before the end of the year, apart from his notes on the division problem and complex multiplication, the interesting and important work on the transformation of double integrals followed, some of which he used as a dissertation to take up the professorship. "The natural world and man himself are creations of Almighty God, the same eternal laws, the same nature, which is the condition without which the world would be unintelligible, without which nature is not understood ... The real cause for the necessary progress of mathematics is explained according to the laws which are forever planted in the human mind," is a passage in that beautiful inaugural speech, which was kept by Franz Neumann, and he also gave new dignity and a perfect shine through his work which has become so famous: Considerationes générales de transcendentibus Abelianis.
To what extent all of his earlier work in the field of transcendents was influenced by Abel's great creations until 1829 cannot be determined here; just as Abel, without having knowledge of Jacobi's work, opened up untracked and unimagined areas through his fundamental and far-reaching studies of the theory of elliptic functions so, on the other hand, we do not ignore the conviction that if Abel had not entered the competition, Jacobi alone would have created everything in this field that he had actually created from 1827 in the theory of elliptical transcendents. But would he have found the way without Abel's preparatory work to enter the area of the higher transcendents than the elliptical? In any case, we cannot answer this question in the affirmative. We know that, already at the height of his glorious productive activity, he was concerned again and again with the question of the inversion of hyperelliptic integrals, highly significant in his studies on the existence of functions of a variable with more than two degrees, but there had always been only negative results, but that his approach, which had been successful through a wonderful divination, the introduction of functions of so many independent variables than there are integrals of the first kind belonging to algebraic irrationality, would hardly have succeeded if it had not been for Abel showing the way for all time through his mathematical theorem dominating the whole theory of integrals of algebraic functions, on which alone a further expansion of analysis could have been made possible. Apparently, a greater person than Jacobi had failed because of this problem, and certainly for him, who was used to delivering everything to mathematicians in a finished, completed form, in its foundation as well as its structure, which is the reason why his discoveries, dating back to the 18th century, only became known the mathematical world long after his death. Nevertheless, the definition of higher transcendents built by Jacobi on the basis of Abel's theorem is one of his most brilliant discoveries, and we find no evidence to suggest that Abel himself, despite many unsuccessful attempts to invert hyperelliptic integrals, thought of introducing functions of multiple variables.
In addition to this work, Jacobi occupied himself with the most in-depth studies of number-theoretical questions, and he discovered the law for the number of classes in a extraordinary induction by comparing certain theorems on the division of the circle and the composition of quadratic forms of negative determinants; at the same time he applied the beautiful and elegant relationships between the different expressions of his elliptical transcendents to increasingly important and difficult problems of mechanics and astronomy. "I am now working," he wrote to his brother Moritz in December 1832, "on a large treatise on the attraction of the ellipsoids, about which I myself dealt with after the work of Newton, Maclaurin, d'Alembert, Lagrange, Ivory, and Gauss, and I found a lot of interesting things. But it is a tremendous effort for me, because it is difficult to do everything in the best way after it is done, and first one is asked." Unfortunately, all of these extensive investigations have remained unknown to us.
In addition to the most intensive productive activity, Jacobi's effectiveness as a teacher and head of the mathematics seminar was becoming increasingly successful; he not only pressed his listeners under the spell of his ideas, not only passed on a wealth of knowledge to them and opened up new perspectives for them in areas that were still unknown to them, but he also captivated their interest through the happy connection with the historical development of problems with the most varied solutions, taken from the most heterogeneous disciplines of mathematical science and critically examined them. By pointing out that there were always new problems that always had to be studied, he also drove his students to try their powers independently to continue and expand science, and awakened their ambition to feel themselves as a new generation of mathematicians, full of courage and self-confidence, but modest and not bursting with scientific jingles - because beside them stood their master, unreachable, the great mathematician, of course well aware of his strength and never denying it in his constant love of truth, that he counted himself among the most outstanding mathematicians of his time, but still humble around really great people, humane and appreciative of every aspiring talent.
"I was accused of being proud towards lowly people and only being humble with the great," he had said to a friend several years earlier. "But that infinite standard, which one applies to the world inside and outside of oneself, prevents overestimation of oneself by always keeping an eye on the infinite goal and ones limited powers. In that pride and in that humility I always want to persevere, and become ever more proud and ever more humble."
Although Jacobi's works were now gradually taking a significantly different direction by increasingly bringing the theory of differential equations and the field of mechanics into the area of his research, none of these studies initially come into public view. Apart from an extremely interesting hydrostatic work, in which, to the astonishment of all mathematicians, he demonstrated that a homogeneous liquid mass can rotate uniformly around a fixed axis while maintaining its external shape, if this shape is not only, as previously assumed, an ellipsoid of revolution, but that an uniaxial ellipsoid can also satisfy the conditions of equilibrium, his publications only report on basic algebraic "Jacobian" theorems, on which the theory of Abelian transcendents was later based, and on his pioneering studies on the theory of elimination and the closely related ones, famous theorems about algebraic lines and surfaces. No area of the mathematical sciences was left untouched by his discoveries, and therefore the need for personal contact with like-minded colleagues in the various areas of knowledge became ever more important.
The seclusion of Königsberg from the scientific centres of Europe sparked in Jacobi the desire to be transferred to another Prussian university, and after Wilhelm Adolf Diesterweg's death he asked the minister to be awarded the professorship in Bonn, the minister, however, considered it desirable in the interests of the case that Jacobi "for now continue to work in Königsberg and continue to justify the study of mathematical sciences there more and more." Bessel, too, was only able to support Jacobi's request in Berlin, out of easily understandable selfishness: "Although I would rather cut my finger off than say that it is appropriate in my view to let you out of Königsberg alive, I would rather cut my neck than say that you wouldn't shine as a treasure everywhere."
After Jacobi gave his 10-hour lecture on the theory of elliptical functions in the winter of 1835/36, which was edited by his student Johann Georg Rosenhain for publication, and which achieved more than any of his previous lectures in depth and elegance of presentation and in the richness of the material offered, and by the establishment of the well-known relations between the products of four functions became so important for the development of the transcendental theory that he stopped working on the theory of elliptic and Abelian functions for a long time - in fact he needed all the elasticity of his mind, to finally come out in public, if only initially in hints, with his far-reaching discoveries in mechanics, the calculus of variations and the theory of differential equations.
In June 1836, he made a short report to Bessel of his discoveries with the introductory words: "Should one be able to notice something new in something as common as moving a point on a plane?" And he writes to his brother Moritz in September: "I came up with some very abstract ideas about the treatment of the differential equations, which occur in the problems of mechanics, because these differential equations by their special form facilitate integration, which no one had noticed before. These considerations become the more important, I believe, because at the same time they extend to the differential equations that occur in isoperimetric problems and the integration of the partial first order differential equations."
And now, in November 1836, he finally summarised all the results found in a letter addressed to the Berlin Academy, proclaimed the theorems so well known to us mathematicians about the transformation of the second variation of integrals and the criteria of maxima and minima, highlighting the essential properties of the dynamic differential equations and the advantages that can be drawn from their special form for their integration, and shows the importance of Hamilton's partial differential equation in a concise form that is already characteristic for the entire scope of these investigations. At the same time, he designed the extremely deep, far-reaching and extensive results on the treatment of problems in mechanics and the methods for integrating partial differential equations, which we later came to know, and which were initially in his plan to compile into a large work; yet again, new interests, bigger and bigger problems got in the way of a systematic execution of all those investigations. He was caught up in number theory problems again; "As for my studies," he writes to his brother in December 1837, "for more than a year I have invented, that is, solved problems invented by others over a long time, in the area of analytical mechanics, calculus of variations, and number theory; in the latter I am in the process of finishing a large treatise of around 20 pages, and I have finally started to allow some of my theory of disturbances be put into calculations by good friends." But the abundance of discoveries that flowed from him made him more and more realise that he will initially find no time to process all the results obtained in the most diverse mathematical fields, and he therefore sent at least brief sketches of his fundamental theorems on division of the circle and his studies on the principles of mechanics to the Berlin and Paris academies; his introductory words in his lecture on the variational calculus were:
"In the history of mathematics and probably also in the development of all other sciences, it often happens that when a new discipline is discovered for the first time, bold and strong minds advance far beyond their time in one single point and make progress that only begins when understood and used by their descendants are,"
which apply to all of his work in the theory of transcendents as well as in number theory, in mechanics as well as in the theory of differential equations.
At this point it is impossible to give even an approximate idea of the large number of analytical truths and mechanical theorems with which Jacobi enriched mathematical science with astonishing genius. But the consequences of the immense mental effort did not fail to appear; headaches and nervous conditions made any further activity impossible, and Jacobi was forced to take a vacation for a summer spa treatment. He first travelled to his mother in Potsdam for a few weeks in March 1839 - he had already lost his father in 1832 - during this time he lived in Berlin in constant contact with Humboldt, Dirichlet and Steiner, and also visited his revered teacher Böckh, who had only one thing to complain about, that he was from Potsdam and, "there has never been a famous man from here" - Böckh couldn't know at that time that a young Potsdam student from the same high school, who a few years later, would show new paths by his principle of maintaining the power of all scientific research, was already in Berlin as a young student.
After Jacobi still found time to complete a work on the geodesic line of the uniaxial ellipsoid, which had become important due to a strange analytical substitution, and the submitted to the Academy, probably in view of the visit from Gauss planned in a few weeks, an investigation into complex prime numbers to be considered in the theory of the higher power residues, in which he made extremely interesting ideas about the Gauss's introduction of the complex numbers into arithmetic, he went to the spa town of Marienbad for a week's cure, visited Ferdinand Schweins in Heidelberg on his return trip, " I was interested in Steiner's teacher [Ferdinand Schweins] and because I read a lot of his work as a student," and after he also paid a short visit to Gauss, reluctantly returned after 7 months of absence with fresh strength to Königsberg, where he, as one of the leading stars of the university, the young king especially honoured at his high point in Königsberg and received scientific honours of the rarest kind from all sides.
"In the 7 months that I was absent, I completely forgot my little mathematics and have to start all over again," he writes to his brother. "I have been torturing myself for a long time with the elaboration and repeated revision of a large work entitled 'Phoronomia sive de solutionum finitarum problematum mechanicorum natura et investigatione'; as soon as the introduction, which is about ten sheets long, is ready, I will let the printing begin;"
but a few months later, after he had published a series of analytical and astronomical works, he reported to his brother:
"I have now given up writing a larger mechanical work called 'Phoronomie' because I don't have the staying power to hold back 20 papers, who knows how many years, until 20 others are written. In some form I will let everything that I have finished run off in batches, and if only the astronomical demon, who by the way has the right of priority, because these astronomical fancies are very old, would let go of me, then a true deluge of individual treatises should come."
Indeed, his systematic treatment of the theory of determinants immediately followed, as well as his famous work on functional determinants, for us the basis of algebra and function theory, and his fundamental "Dilucidationes" based on his multiplier theory set out the connection between a system of total and a partial differential equation in a form that is valid and fixed for all subsequent years.
Unfortunate family circumstances first of all forced Jacobi to interrupt his work; unforeseen, very large financial losses made the termination of his father's business necessary, and when he arrived in Potsdam he saw that not only he, but also his mother, had lost their entire fortune, he immediately made the decision, despite all the material difficulties, to take his mother to Königsberg. "Fortunately, such a talent cannot spoil," are the closing words of the letter which Bessel sent to Gauss on Jacobi's accident; "but I would still have wished him the feeling of freedom that property possesses."
"Whoever saw him at the time," says Dirichlet, "could not notice the slightest change in his mood; he spoke of scientific matters with the same interest as always and only complained about the fact that the unexpected journey had torn him away from an investigation that was keeping him busy."
The scientific world was soon to be informed of these investigations, which concerned the principle of the last multiplier, through his lectures at the British Association in Manchester and at the Academy in Paris, where he and Bessel visit in July 1842 at the request of the king; "I am convinced," wrote the minister, "that by fulfilling this wish you will be happy to grant the satisfaction that this trip brings to our fatherland."
His great winter lecture in 1842/43 on the integration of the differential equations, which, with Borchardts postscript and minor changes, was published as Jacobi's Vorlesungen über Dynamik edited by Clebsch in 1866, has become decisive and fundamental for all subsequent periods for our lectures as well as for the entire further development of mechanics. This was followed by important, highly interesting work on the three bodies problem and Abel's theorem; even during the Christmas holidays he wrote some reports about the geodesic lines of an ellipsoid of revolution and chain-like algorithms for the determination of the periodicity of the chain breaks also for cubic roots, which were only published after his death, and finally sent the Berlin Academy an investigation into new developments in error calculation - but now Jacobi, who had been feeling sick since the beginning of the year, collapsed completely. Driven by fear, Dirichlet hurried to his sickbed during the Easter vacation and stayed with his friend for three weeks. "Dirichlet's 16-day stay," Jacobi writes to his brother, "was a great refreshment - he took about 60 sheets of number theory from me to see how much remained to be done before it was published, because I am completely unable just to look at something like that now."
Dirichlet and Humboldt, together with Schönlein, were now looking for a longer vacation for him in Berlin and travel support for a stay in Italy, and after a few days the King from Sanssouci [the King's summer palace] wrote to Jacobi: "With great regret I became aware of your bad health, but to my reassurance I also received the assurance that you could expect complete restoration from a stay in a milder climate" and the King approved everything Jacobi's friends had asked for. After he had finished some minor work, he left from his family on 9 July, with whom he left behind his mother, but was very pleased to report to his brother: "The best thing about it is a young, amiable, talented, independent mathematician named Borchardt, who has become my excellent companion, who graduated with a doctorate yesterday. Dirichlet will also spend the whole winter in Italy with his family."
In letters to his wife, we have the most detailed accounts of Jacobi's stay in Rome and Naples, the tremendous impression that Italian works of art made on him, the months spent with Dirichlet, Steiner and Borchardt, and visits to the Natural Scientists' meeting in Lucca, where he received the most honourable ovation. His report to Bessel after the audience, which he and Dirichlet received from Pope Gregory XVI was granted, stated: "since you have a weakness for crowned heads - as Herr von Goethe rightly says in his Carmen: 'the superior power cannot be banished from the world. I like to deal with people and tyrants' - so you will share our admiration when you hear that he not only spoke of Newton, Kepler, Copernicus, Laplace with great sympathy, but knew how to state exactly that the squares of the orbital times behave like the cubes of the middle distances ... In short, you will approve of my kissing the hands of such an insightful man full of veneration, while Dirichlet, as a Catholic, made some unskilful attempts to kiss his feet that the Pope would not let it happen."
With the pleasingly progressive improvement in his physical condition in Italy, the desire for renewed scientific work awoke again immediately; during the five months of his stay in Italy he published several smaller articles in an Italian journal, wrote the first part of the important, very extensive treatise for Crelle's Journal on the theory of the multiplier of total differential equations for Crelle's Journal and undertook, in his leisure hours, the comparison of the Vatican manuscripts of Diophantus.
Jacobi returned to Berlin at the end of June 1844, and his friends' combined efforts finally made it possible for him to avoid exposing his fluctuating health to Königsberg's harsh climate; "With benevolent consideration of Professor Jacobi's suffering health," states the cabinet order of 20 August 1844, "which, according to the doctors' judgment, makes his current stay in Königsberg dangerous, I want to allow him to live in Berlin until he is completely restored so that he could live there entirely for science and only take part in lectures at the university to the extent that he himself considers this to be compatible with his physical strength and his scientific pursuits," and Jacobi, in addition to his salary from Königsberg, "with due regard to the greater inflation in Berlin and the extraordinary expenses caused by his sickness" allowed him 1000 Talers from the Royal Disposition Fund. "Your scientific career started here," says Bessel in his farewell speech to Jacobi; "but like the snowfall, which is growing rapidly, you swept away the mass that had rested until you came close to it. One year after your appearance here, the elliptical transcendents began to move, and what had appeared as a solid, impenetrable rock, followed your course, shattered and revealing its innermost nature. Your new foundations of elliptical functions had appeared as early as 1829, and Legendre, a strong predecessor of yours, a very strong one, willingly followed your triumphal march. But your trail will be characterised in all parts of mathematics by equal successes ... You have enriched analysis, number theory, geometry, and mechanics to the same degree. Whoever studied one of your papers without knowing the others would swear that one the subject of the ultimate purposes of all your efforts."
Relocated to Berlin in October 1844, he immediately began to complete his investigations into secular disorders and the second part of his extensive work, in which he developed a coherent theory of the multiplier and its application to the differential equations of dynamics and the derivation of the famous theorem provided of the tracing back of the last integral of the equations of motion to quadratures, theorems which he had mentioned years ago in letters to his friends and in his lectures. But the renewed mental exertion had a detrimental effect on his health, "in addition to my old, never-ending illness came a vertigo that struck me if I wanted to work for even a quarter of an hour," and he lost the courage to complete all of the works he had created on such a large scale. Just as he gave Dirichlet in Königsberg a comprehensive manuscript of theoretical content for review, he now repeated the wish he had expressed earlier to Hesse that he would make his 30-page manuscript about second-degree surfaces and the attraction of the ellipsoids ready for printing; all these manuscripts, as well as the parts of the planned great work on phoronomy, which had already been worked out, were neither published later nor found in his estate.
Although he had to impose restrictions on his productive activity in the interest of his health, it was beneficial in other directions by being everywhere to help his excellent mathematical friends and for the new generation of young mathematicians with his decisive influence in the leading scientific circles; a memoir addressed to the minister, borne by the feelings of noble independence and worthy of a great researcher, ensured Steiner a carefree position at the University of Berlin, and two letters, which he addressed to the King and to Humboldt, supported Dirichlet when he was appointed to Heidelberg in 1846 his previous so beneficial spheres of activity. Also he did not allow an interruption in his academic correspondence, and to his former students and friends Richelot, Hesse, Rosenhain and others, for example, the young, excellent French mathematician Hermite joined them, and Jacobi published, or at least was inspired by them, a series of interesting works on the properties of the elliptical and Abelian transcendents.
The great conditions in Berlin, the artistic aspirations of the educated circles, supported and nurtured by the young King's artistic, ideal views, also stimulated in Jacobi all the interests which he so enthusiastically pursued in Italy; "Nobody can listen to music with more understanding than he can," wrote an artistically knowledgeable lady, and his talk at the Singakademie [a Berlin music society] on Descartes was enthusiastically received; "I read your writing with the most lively interest," Minister Eichhorn writes to him, "because it speaks a judgment about theological zealots and political whirlwinds, which could serve as a corrective for similar spirits of our time." At the same time, we notice his growing interest in the development of mathematical physics, which was promoted by his dealings with Neumann and Weber, and which also motivated him to actively support the scientific promotion of the young Kirchhoff. A year after his lecture on Descartes, in which he already referred to the recent development of mechanics and physics, Helmholtz's "Erhaltung der Kraft" appeared; "The physical authorities," Helmholtz writes fifty years later, "were inclined to deny the correctness of the law and, in the eager struggle against Hegel's natural philosophy that they were leading, to explain my work as fantastic speculation. Only the mathematician Jacobi recognized the connection of my train of thought with that of the mathematicians of the last century, was interested in my attempt and protected me from misinterpretation."
In the autumn of 1846 the first volume of his opuscula mathematica was published with a dedication to King Friedrich Wilhelm IV, which was to play a fateful role in Jacobi's next few years. "We continued to fight in the regions of thought after the wars of freedom, supported by the holy alliance with the spirit, united the Prussians, and won some glorious victory in the sciences. And so we boast, even in mathematical science, we are no longer second."
Even though Jacobi published some very interesting and important works on the properties of the pentagonal numbers, the integration of the hyperelliptic differential equations and the representation of an uniaxial ellipsoid on one level, which was similar in the smallest parts, he was prevented from continuing intense mental exertion by the continuing weakness, and since he was unable to give a larger lecture course in the winter of 1846/47, so it was a desirable occupation and mental relaxation when Humboldt induced him in 1846 to give him fragmentary communications about the mathematics of the Hellenes to make use of it for his Kosmos - Entwurf einer physischen Weltbeschreibung; the records found in Jacobi's estate bear witness to the many years of in-depth studies he devoted to the history of the mathematical sciences. But when his state of health improved in the middle of 1847, publications related to the theory of the transcendent immediately followed, which concerned the different ways of breaking down a number into two factors and a particular solution of the Laplace potential equation; particularly noteworthy, however, is a note of 15 July 1847, which was presented to the Berlin Academy but not published, "About the history of the principle of least action", which was found in part in his posthumous papers. "Jacobi eliminates," says Helmholtz, " time from the integral to be varied; physically, its restrictive condition for a fully known and self-contained system is always to be regarded as valid; Hamilton's form, on the other hand, allows the equations of motion to be carried out even for incompletely closed systems on which changing external influences act that can be viewed independently of a retroactive effect of the moving system." We now know that Jacobi, in his lecture notes on partial differential equations, already gave that principle the form required by Helmholtz without expressly saying so.
In the summer of 1847 the mental depression brought on by illness and worry gradually began to subside, the old lust for work and labour returned, and he reported to his brother: "Finally I have come to write a large memoir about analytical mechanics, which hopefully Ostrogradski will have to study so that he will therefore learn German." Jacobi resumed his studies of the q-functions and the differential equation which they satisfy, developed the law of inertia of quadratic forms, which was later named after Sylvester, and in a large number-theoretical work established a new link between number theory and the theory of transcendents in which he was led to important expansions of Gauss's and Dirichlet's expansions, and designed a sketch on the history of the development of elliptical and Abelian transcendents, found only recently in the papers of Borchardt, full of the most lively interest in the works of Rosenhain and Goepel, whose studies are entirely in the area created by Jacobi. Again he tried to promote excellent young mathematicians at German universities, and in even the most phenomena further away from mathematical science, such as the computing genius Dase, he showed the liveliest interest and the most active support.
Violent political upheavals occurred in March 1848, and Jacobi, who did not have a permanent position at the university and who at that moment feared that a large part of his salary could be withdrawn in the violent transformation of the state, since it depended exclusively on the grace of the King, especially since he repeatedly expressed his liberal political outlook, in the interest of his numerous family members he had to be careful to take up a stable position coordinated with other scholars. He asked the minister to no longer leave him in his exceptional position and to appoint him a full professor at the Berlin University, but his request was rejected by the minister who was advised by the faculty: "Since three mathematicians already belong to the university, and also the way Professor Jacobi has recently been publicly heard don't even give certainty that his participation in the reorganisation work will be beneficial for the university."
Jacobi was asked by Dove to appear as a speaker for the constitutional monarchy in the Constitutional Club in the summer of 1848, and had received endless applause, according to Schelling, for a speech in which he is said to have achieved the greatest patterns of the ancient Greek speakers, but now after he had grown tired of the constant fluctuations in daily opinion, he had stayed away from political life for more than a year and had sought peace and satisfaction in a series of excellent works, largely belonging to the field of astronomy, when finally on 31 May 1849, after the increasingly strong reaction to Jacobi had not dared to be shown up to then, Minister Ladenberg asked him whether his state of health would not allow him to move back to Königsberg, and soon afterwards came the message that the grant given by the King was to be withdrawn from now on. Jacobi was now forced to move his wife and seven young children to Gotha, a much cheaper place to live, with his friend Hansen, one of the most important astronomers of his time, while he himself moved into a room at the Hotel de Londres in Berlin, which appears as his address on his subsequent famous works. At the end of 1849 he received a splendid appointment to the University of Vienna, and only now, after endless political pressures and humiliations, given to him by the Prussian government - "he would rather see everyone leave, except Jacobi, who was a natural force," the enlightened head of education Johannes Schultze wrote to the King - through the mediation of Humboldt and the nobility of the King, the previous support was not only approved as a salary, but also increased with backdating.
But the excitement of the past year had deeply shaken his health; his physical strength was completely broken, though his mind seemed to have retained the old resilience. In addition to a series of excellent works of astronomical content, which were essential for the perfection of his error investigations, he sent his famous work "Sur la rotation d'un corps" to the Paris Academy in March 1850, in which an excellent geometrical-mechanical with an incomparable ingenuity was found in the treatment of analytical forms which shows so wonderfully, investigations, which he supplemented and continued in many extensive reports with highly interesting theorems, finally expands his work on the number of double tangents of algebraic curves with arguments which add to Plücker, and gives formulas from first principles for the investigation of algebraic-geometric problems.
Returning from his Christmas vacation from visiting his loved ones in Gotha, he contracted the flu, but seemed to be recovering quickly when he collapsed again on 11 February; four days before his death he complained to Dirichlet about the misfortune that prevailed over many of his larger works; "But I understand that I can no longer hesitate to give those older works, to which I have devoted so much of my best strength, to the public if they are to successfully enter into the course of science." On 18 February in the evening at 23:00, eight days after his illness, he died without a fight.
Dirichlet, Borchardt and Weierstrass, supported by excellent younger scholars, have given the mathematical world the enormous treasure of his astonishingly productive work by publishing his collected works, his students have filled the following generation with the glory of their incomparable teaching, and Dirichlet was the one who also made a lasting memorial to him as a person in his magnificent remembrance speech, at a time when the waves of political passion for freedom of expression required all the courage of a personality borne by the noblest friendship, the purest love of truth and great scientific importance.
"Should I dare to try," says Dirichlet, "to describe him as he appeared outside the scientific sphere to those who are far from the mathematical sciences, I must describe it as the essence of his nature that he lived completely in the world of thought, and even that which most important people needed a special attempt to think about, had become habitual and second nature to him: the inexhaustible supply of knowledge and his own thoughts, peculiarly humorous, the sharpness of expression used to give the great mathematician an unusual meaning even in social dealings. As Jacobi's line of thinking manifested itself in the recognition of Abel's great discovery, so he showed a similar sense for everything spiritually significant. He assessed things according to how the human spirit manifested itself in them, and treated everything with equanimity that did not touch the world of thought."
Just as Jacobi's words ignited and enthused his listeners, so did his writings, through their content and form, arouse constant and loud reverberations in the minds of the new generation of mathematicians, and in this sense we can add that Hermite and Weierstrass, the two most distinguished representatives of mathematical research in the second half of the last century, were counted among his pupils - and we all, the pupils of these two excellent researchers, in whom we still heard, in veneration and piety, we heard the words and opinions of Jacobi as oracles and mathematical mysteries passed down to us, we who are gathered here to celebrate the memory of that great master, we are all students of Jacobi.
Highly respected Assembly!
I hope that I have succeeded, also those of you who are further away from our science, in not only gaining a feeling of admiration for the great master builder, who, from a mathematical, philosophical and aesthetic point of view, contributed to the construction of the enormous building, to which, rising from inconspicuous beginnings, our science has gradually developed; I hope I also have succeeded in providing you with a glimpse, if only fleeting, of the building of mathematical science itself, the pride and the adornment of human intellectual work. Certainly space is only the form of pure intuition, time is only the form of pure thinking, certainly the first foundations of our science, like those of thought in general, are taken from earthly experience, but these foundations are deeply immersed, so deep that we, the inhabitants of these places of human knowledge, can no longer see them, can no longer describe them, and therefore, if not always rightly, begin to doubt the firmness of the building even with the slightest swaying and trembling - but every storm, every violent encroachment on the beauty of the magnificent mathematical building, may be an enemy trying to shatter it, a clumsy construction worker uses a wrong stone, or an ingenious artist may burden the building with an all too heavy rock - the building will only become more beautiful, its structure only firmer, and grant the natural sciences, the more comfortable, luxuriant and nobler dwellings if it forms a protective shelter against any storm that disturbs their circles and shakes the foundations of their beautiful home - mathematics itself is the science of the nature of things in and around us, insofar as they reveal human knowledge at all.
The Sphinx sits there from the creation of the world, it sits forever, but joined in due time by Oedipus, sent by Apollosays Jacobi in his witty speech on the glory of mathematical science.
Last Updated January 2020