Carl Gustav Jacob Jacobi

Carl Gustav Jacob Jacobi was a German mathematician whose foundational work spanned elliptic functions, dynamics, differential equations, determinants, and number theory. Born in Potsdam on 10 December 1804 to Ashkenazi Jewish parents, he was the second of four children of banker Simon Jacobi. His older brother Moritz later became known as an engineer and physicist. Jacobi was initially homeschooled by his uncle Lehman, who taught him classical languages and basic mathematics. At age twelve, he entered the Potsdam Gymnasium, where after less than half a year his abilities placed him in the senior class, though university rules barred students under sixteen, so he remained there until 1821. During this time he deepened his knowledge across subjects and attempted to solve the quintic equation by radicals.

In 1821, Jacobi began studies at Berlin University, splitting his focus between philology and mathematics. He attended philology seminars by August Böckh, impressing the professor, but found the mathematics courses too elementary and instead privately studied advanced works by Euler, Lagrange, and Laplace. By 1823 he chose mathematics over philology. That year he qualified to teach secondary school and was offered a post at the Joachimsthal Gymnasium in Berlin, but he pursued a university career instead. In 1825, he earned his doctorate with a dissertation on partial fraction decomposition of rational fractions, defended before a committee led by Enno Dirksen. He then completed his habilitation and converted to Christianity. At age twenty-one, he lectured on curves and surfaces at Berlin University in 1825–26.

Jacobi became a private lecturer in 1826, an extraordinary professor in 1827, and a tenured mathematics professor at Königsberg University in 1829, holding the chair until 1842. Overwork led to a breakdown in 1843; he recovered during a few months in Italy. Returning, he moved to Berlin and lived as a royal pensioner, except for a brief interim, until his death. During the 1848 Revolution, he was politically active and ran unsuccessfully for parliament on a Liberal club ticket. After the revolution, his royal grant was cut off, but Alexander von Humboldt’s personal intervention soon restored it. Jacobi died in 1851 from smallpox. His grave is in Berlin’s Kreuzberg district at Friedhof I der Dreifaltigkeits-Kirchengemeinde, near astronomer Johann Encke. The lunar crater Jacobi is named after him. Born Jacques Simon, his name was later Germanized to Carl Gustav Jacob Jacobi and Latinized as Carolus Gustavus Jacobus Jacobi; he is sometimes called C. G. J. Jacobi.

Jacobi’s most influential achievement was his theory of elliptic functions and their connection to the elliptic theta function, presented in his 1829 treatise *Fundamenta nova theoriae functionum ellipticarum* and later papers in Crelle’s Journal. Theta functions are crucial in mathematical physics for solving inverse problems in periodic and quasi-periodic flows. Equations of motion for the pendulum, Euler top, symmetric Lagrange top in gravity, and the Kepler problem are integrable using Jacobi’s elliptic functions. He also made key contributions to differential equations and classical mechanics, notably through Hamilton–Jacobi theory. His algebraic strength is evident in numerous papers from 1826 onward. He advised students to “Invert, always invert,” believing that reversing known results opens new research fields—for example, inverting elliptic integrals to focus on elliptic and theta functions.

In an 1835 paper, Jacobi proved that a univariate single-valued multiply periodic function can have at most two periods, and their ratio cannot be real. He discovered fundamental properties of theta functions, including the functional equation and the Jacobi triple product formula, along with results on q-series and hypergeometric series.