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Karl-Gustav Jacob Jacobi



UDC 52

S.I. Aleshnikov KARL-GUSTAV JACOB JACOBI

The short biography of the outstanding German mathematician Karl-Gustav Jacob Jacobi whose scientific activity is closely connected with the Konigsberg university — Albertina is presented in article. The review of its works in the field of the theory of functions is given, to the theory of numbers and the algebraic geometry promoting formation of modern mathematical methods of information security.

In the article is presented the brief biography of the famous German mathematician Carl Gustav Jacob Jacobi the work whose is with the university Königsberg-Albertina bounded. Here is an review his works in function theory, number theory and algebraic geometry presented, which is the formation of modern methods of the data protection contributed.

It got almost into all fields of the science which expanded for 2000 to the improbable sizes. Everywhere, where its creative spirit directed, it received important and deep results, the new fundamental ideas are entered, the mathematical refinement is raised on higher step. Its scientific activity continued a little more than a quarter of the century — rather small term in comparison with the previous mathematicians of the first rank; only a half of that time during which Euler, one of the greatest mathematicians of all times created. Nevertheless, he was almost also mnogosto-ronen and is productive, as well as Euler. As well as Euler, he used all means modern to him sciences, every minute they were at his order.

So Gustav Dirichlet, the outstanding German mathematician, at an annual meeting of Prussian academy of Sciences spoke on July 1, 1852 about the colleague academician Karl-Gustav Jacob Jacobi concerning death of the last.

To. - G.Ya. Jacobi was born on December 10, 1804 in Potsdam. He was one of four children of the banker Simon Jacobi and grew in the most favorable

KGU bulletin. 2005. Issue 1 — 2. It is gray. Computer science and telecommunications. Page 12 — 18.

conditions, in wealthy family with a wide range of interests, in contact with all opportunities of education which that time gave. Karl had a sister, Theresa, and two brothers, Moses and Eduard. Jacobi's family belonged to upper the Jewish community of Potsdam. In their house spoke English and in French. The highest employees of the province were constant guests. After the death of the father his business was continued by the younger brother Eduard, but is unsuccessful. Further he directed the economic department of the Kreuzzeitung newspaper created by him. The elder brother Carla, Moses, studied at the Berlin and Goettingen universities upon termination of which he arrived on public service. Later he moved to the brother to Konigsberg. In 1835 he received the invitation as extraordinary professor in the university of Tartu (Dorpat), and in 1837 — in the university of St. Petersburg. Theresa Jacobi's fate is not known.

Karl got the first knowledge of mathematics and languages under the leadership of the uncle F.A. F.A. Lehmann who was his only teacher for five years. At the age of incomplete 12 years in November, 1816 he came to the second class of the Potsdam gymnasium, from where, after half a year was transferred to the first. In four years the training in a gymnasium successfully came to the end. Already in high school he got acquainted with eylerovsky Introductio in analysis infinitorum.

In 1821 incomplete 16 years old Jacobi became the student of the Berlin university. Before entering a university he bore a name Jacques-Simon. In student's years it changed belief with Christian, namely, on Protestant and respectively a name on Karl-Gustav-Jacob. Later new generations of students called his "uncle Jacques". It should be noted that time after Napoleon's defeat in 1812, despite "edikt about emancipation" (Emanzipationsedikt), was very adverse for the German Jews. Practice of oppression and restriction in the rights was common. Up to 1847 the Jews could not be "habilitirovana" in Prussia. The only chance of university career was given by change of belief. Jacobi claimed that he changed belief owing to the internal beliefs received when studying classical philology, history and philosophy. Actually left so that change of belief allowed it to become further the first Jewish mathematician who came to the forefront in Germany.

Originally Jacobi was fond of classic languages and for some time was an active participant of the university seminar on classical philology directed by professor Boeckh. The ideal of high pure scientific culture occurring in these circles and the system of teaching developed here played the defining role in its further pedagogical activity. His language skills, especially Ancient Greek, mathematicians and stories, were characterized by teachers as excellent and very thorough. It was called "the universal mind possessing extraordinary

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abilities and the high spirit covering and understanding everything tirelessly". It lacked ordinary university lectures, and then Euler, Gauss, Lagrange, Laplace whose works Jacobi studied became his teachers. Especially it was admired by Gauss's works. At 20-year age upon termination of the university with great success it passed a state exam and immediately began work on the doctoral dissertation.

Fall of 1825 Jacobi becomes a doctor. According to the examiner, he showed extraordinary independence and originality. Thereof it was authorized to it to combine protection of doctor's work and "habilitation", i.e. at the same time to receive also an associate professorship. Giving lectures in Berlin, he, according to his listeners of that time, showed so uncommon pedagogical talent that after half a year of work in 1826 at his desire it was sent as the privatdozent to the university of Konigsberg to the place of the died ordinary professor of mathematics Friedrich Wrede. At that time the Konigsberg university (together with its astronomical observatory) was one of the leading scientific centers of Germany. In it is mute the philosopher and the teacher Johann Friedrich Herbart, the astronomer Friedrich Wilhelm Bessel and other famous scientists worked. At the same time Crelle (Crelle) in Berlin based the well-known mathematical magazine of "Abstract and applied mathematics" and then Jacobi's cooperation with this magazine began.

In Konigsberg within seventeen years of Jacobi the associate professor, and then as extraordinary (1827) and ordinary (1831) professor developed grandiose activity at first as. As the stroke characteristic of Jacobi's behavior, we will note that at his taking office at the Konigsberg faculty there were known difficulties, "as he told each of members of faculty something venomous". But eventually the victory was gained after all by indisputable value of its scientific achievements. Thanks to efforts of Bessel, Jacobi and the physicist F.E. F.E. Neumann the Konigsberg university in the first half of the 19th century turned into the largest center of mathematical, physical and astronomical researches. Since 1829 Jacobi is the corresponding member of the Berlin academy of Sciences, since 1836 — the foreign member and since 1844 — the full ordinary member of academy.

on September 11th, 1831 Jacobi contracted marriage with Marie Schwink, the daughter of the wholesale merchant from Konigsberg. They had three daughters and five sons.

Jacobi had not only thirst for purely scientific knowledge, but also live requirement to state learned. This bent to influence others was expressed as brilliant pedagogical talent. He spent an essential part of the time for education of the pupils. Live communication of own scientific research with a training material was distinctive feature of his lectures. In lectures there was nothing complete. Delivered to them research for -

dachas triggered curiosity of listeners. It compelled them to hard mental work. To the same purpose served also the physical and mathematical seminar based together with F.E. Neumann with assistance of Bessel on whom students were engaged in own scientific work. Actually it was reform of a technique of teaching. The seminar became the base of so-called Konigsberg school of sciences and existed more than 100 years.

Jacobi was intended to create big school which was fated long prosperity. The so-called "Konigsberg school" founded by Jacobi and Franz Neumann was the first phenomenon of this sort in Germany which got long influence. The Konigsberg university turned into the center of the exact sciences. The powerful impulse proceeding from Jacobi extended far away from Konigsberg. Lindemann spoke "about a long time of revival of mathematics in Germany by which we are obliged to the Konigsberg school". All German universities were influenced by its influence. In Germany such famous scientists as Kirchhoff, Klebsh and Hesse were direct pupils of Jacobi. Moreover, at that time almost all departments of mathematics and mathematical physics of the German higher educational institutions were occupied by pets of the Konigsberg school.

Jacobi's influence extended also outside Germany. The leading mathematicians of France of the 40th years of the 18th century Hermite and Liouville, Cayley in England considered themselves Jacobi's pupils. Jacobi was a member of the London Royal society, the corresponding member of the Madrid and Parisian academies. In 1830 Jacobi became corresponding member, and then in 1830 and the honorary member of the St. Petersburg academy of Sciences. Supported close scientific ties with the Russian mathematicians: M.V. Ostrogradsky, I.D. Sokolov, O.M. Tikhomandritsky, M.D. Brashman and others. His brother Moritz (Boris Semenovich) Jacobi elected in 1837 the academician lived and worked in St. Petersburg, being engaged in a pilot study of the electric phenomena.

Exclusively vigorous activity of Jacobi in Konigsberg led it in 1843 to exhaustion of forces. He was forced to spend nearly a year on vacation, traveling around Italy. The climate of Konigsberg adversely affected his health therefore it accepted the invitation to Berlin where purely academic position without certain pedagogical duties was offered it. Together with Dirikhle, Steiner and Minding working in Berlin it promoted growth of the Berlin mathematical school. However the former working capacity more to it was not restored.

In the last years of life Jacobi was fond of political activity. In the summer of 1848 in the Constitutional club it spoke out in defense the speech of constitutional monarchy which was followed by a prolonged applause, than got displeasure of the minister Ladenberg of that time. The deviation of his candidate for a position ordinary profes-

was the first reaction of the authorities
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litter of the Berlin university. Soon he received the notice of refusal of the king to pay it a salary. As for health reasons it could not return to Konigsberg, it transported the wife and 7 juvenile children to the friend Hansen famous to the astronomer while itself it was forced to live in Londres hotel in Berlin. Only thanks to Alexander von Humboldt's mediation and generosity of the king at the end of 1849 his salary was restored and is even increased. Disorders of the last year finally affected Jacobi's health. After Christmas holiday at the beginning of 1851 he got the flu. After short recovery on February 11 he got sick again, but now with smallpox. Karl-Gustav Jacob Jacobi died on February 18, 1851

Jacobi conducted researches almost in all fields of mathematics. The thesis was its first publication. Its last publication is dated on January 10, 1851. Extremely fruitful mathematical competition of Jacobi to Niels Henrik Abel led to creation of the theory of elliptic functions. The central idea was to consider elliptic integral of the first sort not as function of a limit of integration, and on the contrary, an integration limit — as functions of integral. Taking a sine and a cosine of the received function, Jacobi received elliptic functions. Introduction together with Abel of imaginary sizes revealed double frequency of elliptic functions and gave theories extraordinary elegant look noted by Legendre. Jacobi entered and studied theta functions by means of which it was possible to express elliptic and which he considered the best creation in abstract mathematics. He removed formulas for elliptic integrals of the third sort. Studying addition of Abelian integrals at first in the hyperelliptic case, he proved Abel theorem generally. The problem of the address of the hyperelliptic functions belonging to Jacobi resolved in a general view only by G.F.B. Riemann is known.

It significantly progressed in the solution of a problem of division of a circle and its annexes to the theory of numbers, in particular to the theory of cubic and biquadratic deductions. It possesses generalization of a symbol of Legendre and the wording of the law of reciprocity for sedate deductions of the fifth and eighth degrees. Jacobi's sums serve as the most important tool of a research in the theory of numbers and arithmetic geometry till today.

The differential equations arising in calculus of variations bear a name of Jacobi. He entered and investigated a class of the orthogonal polynomials which are generalization of polynomials of Legendre — so-called polynomials of Jacobi, and applied them to the solution of the hyper geometrical differential equations. It possesses methods of integration of a system of the linear differential equations in private derivatives of the first order. His name is born by the variety having structure of group, corresponding to any algebraic curve. It put into practice functional oprede-

flew — yakobiana — and pointed to their role when replacing variables in multiple integrals and at the solution of the equations with private derivatives. It opened the law of inertia of square forms.

In the field of astronomy he conducted numerical researches of indignations of elliptic orbits of planets, progressed in the solution of a problem of three bodies in heavenly mechanics, having specified for it a number of new methods, entered the initial coordinates bearing his name, proved the theorem of an exception of knots, contributed significantly to the solution of a problem of definition of a form of a celestial body. After Bessel's death he was his scientific successor, having calculated the movement of the planet the Neptune and by that having anticipated its opening.

In physics it created Hamilton-Jacobi's theory which appeared mechanics, very fruitful for further development. It applied elliptic functions in the theory of a top and to calculation of geodetic lines on an ellipsoid. It possesses the proof of the theorem of Jacobi-Poisson of a conclusion of new integrals from the differential equations of mechanics, already known for any system. He formulated the principle of the smallest action in analytical mechanics. Jacobi was distinguished by a keen physical intuition. It was the only thing from the leading scientists of that time which managed to understand Helmholtz's work "About emergence of force".

During the travel across Italy in 1843 in one of after-dinner speeches of Jacobi it was called the representative of truly practical science what he objected to that the highest science, as well as art, are always impractical and that he always aspired to it. In other place he noticed that for science special honor — not to have practical benefits. To Legendre he wrote in the letter that the only purpose of science is the respect of human spirit and that at such requirement the question of numbers is so valuable, as a question of the Universe. In spite of the fact that he was engaged in annexes of mathematics to physics and astronomy, for physical reasons considered statement of mathematical tasks unnatural. For him, as well as for Euler with Lagrange, the mathematics remained the analytical art giving pleasure. Jacobi after Gauss considered that the mathematics is the center of science that concepts mathematicians are concepts of science in general that all scientists have to seek to become mathematicians. Once he wrote to the brother Moritz: "Life of gods is mathematics... my life is similar to life of gods". This understanding of mathematics as clean, not relying on experience and science, not dependent on applications, was postponed as a part of spirit of the Konigsberg school for other universities of Germany when mathematicians directed their departments the former residents of Konigsberg.

In the last third of the 20th age of the mathematician got the new person. The most abstract ideas and theories which arose as generation of pure logic, unexpectedly appeared in the center of new applications. Moreover, they were one of the main driving forces of development of these applications. Theory of elliptic functions and theory of theta functions,

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Jacobi's sums and circular fields, yakobiana of algebraic curves is very incomplete list of the created Jacobi that later since their opening formed more than 150 years the mathematical basis of modern methods of information security. In it, probably, the stamp of genius also consists.

List of references

1. Dirikhle P.G.L. Karl Gustav Jakov Jacobi: The biographic essay made by Lezhyen Dirikhle for the speech in Jacobi's memory delivered in a meeting of the Berlin academy of Sciences on July 1, 1852
2. Lavrinovich K.K. Albertina: Essays of history of the Konigsberg university. To the 450 anniversary since basis / Kaliningr. state. un-t. — Kaliningrad: Prince publishing house, 1995.
3. Königsberger L. Carl Gustav Jacob Jacobi. Festschrift zur der hundertsten Wiederkehr seines Geburtstages. - Leipzig: Druck und Verlag von B.G. Teubner, 1904.
4. Königsberger L. Carl Gustav Jacob Jacobi. Rede zu der von dem internationalen Mathematiker-Kongress in Heidelberg veranstalteten Feier der hundertsten Wiederkehr seines Geburtstages gehalten am 9.August 1904. — Leipzig: Druck und Verlag von B.G. Teubner 1904.
5. Pieper H. Carl Gustav Jacob Jacobi (1804-1851)//Die Albertus-Universität zu Königsberg und ihre Professoren: aus Anlaß der Gründung der Albertus-Universität vor 450 Jahren/hrsg. von Dietrich Rauschning; Donata v. Neree. - Berlin: Duncker Humblot, 1995.

About the author

S.I. Aleshnikov is a Cand.Tech.Sci., dots., KGU, cyber@albertina.ru

Walton Duane
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