b. Potsdam, Germany, 1804; d. Berlin, 1851
Theory of functions; number theory; algebra; differential equations;
calculus of variations; analytical mechanics; perturbation theory;
history of mathematics
Jacobi grew up as Jacques Simon Jacobi in a wealthy and well-educated Jewish family. He was baptized during his first year at the University of Berlin in 1821, probably in order to make it possible for him to follow an academic career at a time when Jews were ineligible for academic positions. Jacobi studied classics under the famous philologist Boeckh and philosophy under Hegel. Owing to the mediocrity of the mathematics staff in Berlin at that time, he was self-taught in the discipline, which soon became his favorite. He read EULER [VI.19], LAGRANGE [VI.22], LAPLACE [V1.23], GAUSS [VI.26], and, last but not least, Greek mathematicians like Pappus and Diophantus. In 1825, Jacobi was awarded his doctorate for a thesis, written in Latin, on the theory of functions. The subsequent disputatio (discussion) included critical comments both on Lagrange’s theory of functions and on his analytical mechanics. The following year Jacobi went to the University of Königsberg, where (in 1829) he got a full professorship. In 1834, he and the physicist F. E. Neumann founded the “Königsberg mathematical physics seminar,” which, because of the close connection between research and teaching that it fostered, soon led to Königsberg becoming the most successful and influential educational institution for theoretical physics and mathematics in the German-speaking part of the scientific world. By 1844, when Jacobi left Königsberg because of poor health and in order to become a member of the Berlin Academy of Sciences, he was recognized as Germany’s most important mathematician after Gauss. After seven more fruitful years of research in Berlin he died unexpectedly from smallpox.
Throughout his life Jacobi was an advocate of pure mathematics, conceiving mathematical thinking as a means of developing the human intellect and, indeed, of advancing humanity itself. He published his first paper in 1827: it was devoted to number theory (cubic residues) and was influenced by Gauss’s Disquisitiones Arithmeticae. Further investigations were devoted to higher residues, the division of the circle, quadratic forms, and related subjects. Many of Jacobi’s results in number theory were published in the book Canon Arithmeticus (1839). The extension of the concept of divisibility to algebraic numbers by Jacobi and Gauss paved the way for the later algebraic theory of numbers (by KUMMER [VI.40] and others).
Jacobi’s “most original achievements” (in the words of KLEIN [VI.57]) were his contributions to the theory of ELLIPTIC FUNCTIONS [V.31], which developed in competition with ABEL [VI.33] between 1827 and 1829. Starting with LEGENDRE’S [VI.24] work, Jacobi’s approach was analytical and focused on the transformation of elliptic functions, their properties (like double periodicity), and the introduction of the inverse function. Jacobi’s research on elliptic functions culminated in the book Fundamenta Nova Theoriae Functionum Ellipticarum (1829). Together with Abel he should be viewed as one of the founders of the theory of complex functions, which emerged in the second half of the century. In particular, his application of research on elliptic functions to Diophantine equations became important for the development of analytic number theory. Jacobi’s contributions to algebra include investigations into the theory of determinants (the “Jacobian” functional determinant) and their relation to inverse functions, into quadratic forms (“Sylvester’s law of inertia”), and into the transformation of multiple integrals.
Even Jacobi’s work in mathematical physics bears the stamp of “pure mathematics”: following the analytical tradition of Euler and Lagrange, he presented the foundations of mechanics in an abstract and formal manner, paying special attention to the relation between CONSERVATION LAWS [IV.12 §4.1] and symmetries of space and to the unifying role of variational principles. Jacobi’s achievements in this area, which he developed in close relation to the theory of differential equations and the CALCULUS OF VARIATIONS [III.94], include what is now called the “Jacobi-Poisson theorem,” the “principle of the last multiplier,” a theory for integrating HAMILTON’S [VI.37] CANONICAL EQUATIONS OF MOTION [IV.16 §2.1.3] by transformation (“Hamilton-Jacobi theory”), and a time-independent formulation of the principle of least action (“Jacobi’s principle”). His approach to these areas and the results he obtained are documented in two comprehensive books based on his lectures: Vorlesungen über Dynamik (1866) and Vorlesungen über Analytische Mechanik (not published until 1996). The former had considerable impact on the development of German mathematical physics in the last third of the nineteenth century. The latter reveals Jacobi’s criticism of the traditional understanding of mechanical principles (as laws that are firmly based on empirical observation or a priori reasoning) and shows strong parallels with the “conventionalist” viewpoint, which did not become popular in science and philosophy until half a century later, when it numbered H. Hertz and POINCARÉ [VI.61] among its adherents.
Jacobi not only promoted new mathematical developments, but also studied the history of mathematics: he worked on ancient number theory, was the advisor for the historical parts of A. von Humboldt’s great Kosmos (1845-62), and developed detailed plans for the publication of Euler’s works.
Koenigsberger, L. 1904. Carl Gustav Jacob Jacobi. Festschrift zur Feier des hundertsten Wiederkehr seines Geburtstages. Leipzig: Teubner.