**Carl Jacobi** made basic contributions to the theory of elliptic functions. He carried out important research in partial differential equations of the first order and applied them to the differential equations of dynamics.

- The University of Berlin, however, did not accept students below the age of 16, so Jacobi had to remain in the same class at the Gymnasium in Potsdam until the spring of 1821.
- Of course, Jacobi pressed on with his academic studies despite remaining in the same class at school.
- By the time Jacobi left school he had read advanced mathematics texts such as Euler's Introductio in analysin infinitorum Ⓣ(Introduction to infinitesimal analysis) and had been undertaking research on his own attempting to solve quintic equations by radicals.
- Jacobi entered the University of Berlin in 1821 still unsure which topic he would concentrate on.
- As he had done at the Gymnasium, Jacobi had to study on his own reading the works of Lagrange and other leading mathematicians.
- By the end of academic year 1823-24 Jacobi had passed the examinations necessary for him to be able to teach mathematics, Greek, and Latin in secondary schools.
- Jacobi presented a paper concerning iterated functions to the Berlin Academy of Sciences in 1825.
- Although this was not the best start for the young Jacobi, it did not hold him back for long and his publication record over the following years would be quite remarkable for both the number and quality of the works.
- Around 1825 Jacobi changed from the Jewish faith to become a Christian which now made university teaching possible for him.
- However prospects in Berlin were not good so, after taking advice from colleagues, Jacobi moved to the University of Königsberg arriving there in May 1826.
- Jacobi had already made major discoveries in number theory before arriving in Königsberg.
- Gauss was impressed, so much so that he wrote to Bessel to obtain more information about the young Jacobi.
- But Jacobi also had remarkable new ideas about elliptic functions (as Abel did quite independently and at much the same time).
- Legendre immediately realised that Jacobi had made fundamental advances in his favourite topic.
- One would have to say that Legendre reacted extremely well to the realisation that his position as the leading expert on elliptic functions had changed overnight with the new theory being developed not only by Jacobi, but also by Abel.
- Jacobi's promotion to associate professor on 28 December 1827 was mainly due to the praise heaped on him by Legendre.
- In 1829 Jacobi met Legendre and other French mathematicians such as Fourier and Poisson when he made a visit to Paris in the summer vacation.
- Jacobi's fundamental work on the theory of elliptic functions, which had so impressed Legendre, was based on four theta functions.
- However, despite Jacobi's brilliant contributions to elliptic functions he did not have the field to himself.
- Jacobi's reputation as an excellent teacher attracted many students.
- C W Borchardt, E Heine, L O Hesse, F J Richelot, J Rosenhain, and P L von Seidel belonged to this circle; they contributed much to the dissemination not only of Jacobi's mathematical creations but also the new research-oriented attitude in university instruction.
- The triad of Bessel, Jacobi, and Franz Neumann thus became the nucleus of a revival of mathematics at German universities.
- In 1834 Jacobi received some work from Kummer who was at this time a teacher in a Gymnasium in Liegnitz.
- In 1834 Jacobi proved that if a single-valued function of one variable is doubly periodic then the ratio of the periods is non-real.
- Jacobi carried out important research in partial differential equations of the first order and applied them to the differential equations of dynamics.
- He also worked on determinants and studied the functional determinant now called the Jacobian.
- Jacobi was not the first to study the functional determinant which now bears his name, it appears first in a 1815 paper of Cauchy.
- However Jacobi wrote a long memoir De determinantibus functionalibus Ⓣ(Functional determinants) in 1841 devoted to this determinant.
- He proved, among many other things, that if a set of nnn functions in nnn variables are functionally related then the Jacobian is identically zero, while if the functions are independent the Jacobian cannot be identically zero.
- The statement of this theorem is an afterthought to a paper in which Jacobi responds to the published correction by Thomas Clausen (1842) of an earlier paper by Jacobi (1836).
- In July 1842 Jacobi and Bessel attended the meeting of the British Association for the Advancement of Science in Manchester as representatives of Prussia.
- They returned to Königsberg via Paris where Jacobi lectured at the Académie des Sciences.
- In the following year Jacobi became unwell and diabetes was diagnosed.
- However, Jacobi was not a wealthy man and Dirichlet, after visiting Jacobi and discovering his plight, wrote to Alexander von Humboldt asking him to help obtain some financial assistance for Jacobi from Friedrich Wilhelm IV.
- A severe business depression throughout Prussia (in fact it was a Europe wide depression), had led to a bankruptcy in which Jacobi had lost all his money.
- Let us now return to Dirichlet and Alexander von Humboldt's attempts to help obtain support for Jacobi's trip to Italy.
- Jacobi had frequently corresponded with Alexander von Humboldt.
- Dirichlet's request to Friedrich Wilhelm IV, supported strongly by Alexander von Humboldt, was successful and Jacobi received a grant to allow him to spend time in Italy.
- The climate in Italy did indeed help Jacobi to recover and he began to publish again, his health having prevented him working for some time before this.
- Jacobi's interests in mathematics were very wide and while in Rome he took the opportunity to satisfy his interest in the history of mathematics working on manuscripts of Diophantus's Arithmetica which were kept in the Vatican.
- Lagrange's view that mechanics could be pursued as an axiomatic-deductive science forms the centre of Jacobi's criticism and is rejected on mathematical and philosophical grounds.
- Jacobi's criticism is motivated by a changed evaluation of the role of mathematics in the empirical sciences.
- As a consequence Jacobi's request to be allowed to join the staff of the University of Berlin was refused by the Prussian government.
- The Prussian government, still feeling aggrieved at Jacobi, took away the supplement to his salary which allowed him to live in Berlin.
- It was not a good deal for Jacobi and the fact that he accepted it means that he was strongly attached to his own country.
- Both were prolific writers and even more prolific calculators; both drew a great deal of insight from immense algorithmical work; both laboured in many fields of mathematics (Euler, in this respect, greatly surpassed Jacobi); and both at any moment could draw from the vast armoury of mathematical methods just those weapons which would promise the best results in the attack of a given problem.

Born 10 December 1804, Potsdam, Prussia (now Germany). Died 18 February 1851, Berlin, Germany.

View full biography at MacTutor

Algebra, Astronomy, Group Theory, Origin Germany, Physics

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive