Carl Gustav Jacob Jacobi

Carl Gustav Jacob Jacobi, a German mathematician, made essential contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. He was born in Potsdam in 1804 to Ashkenazi Jewish parents, the second of four children of a banker. His older brother Moritz later became known as an engineer and physicist. Initially taught at home by his uncle Lehman, Jacobi learned classical languages and basic mathematics. At twelve, he entered the Potsdam Gymnasium, where his strong preparation and talent led to his promotion to the senior class within half a year, despite his youth. Because universities required students to be at least sixteen, he stayed in that class until 1821, using the time to deepen his knowledge across subjects and attempt 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 Euler, Lagrange, and Laplace. By 1823, he chose mathematics as his sole pursuit. That year he qualified to teach secondary school but aimed for a university post instead. In 1825, he earned his doctorate with a dissertation on partial fraction decomposition, defended under Enno Dirksen, then completed his habilitation and converted to Christianity. At twenty-one, he lectured on curves and surfaces at Berlin University. He 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, after which he recovered in Italy. He then moved to Berlin, living on a royal pension except for a brief interruption. During the 1848 Revolution, he was politically active and ran unsuccessfully for parliament on a Liberal club ticket. After the revolution, his grant was cut off but soon restored through Alexander von Humboldt’s intervention. Jacobi died of smallpox in 1851 and is buried in Berlin’s Kreuzberg district, near astronomer Johann Encke. A lunar crater bears his name. Originally named 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.

His most influential work was the theory of elliptic functions and their link 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 important in mathematical physics for solving inverse problems in periodic and quasi-periodic flows. Equations of motion for the pendulum, Euler top, symmetric Lagrange top, and Kepler problem are integrable using Jacobi’s elliptic functions. He also advanced differential equations and classical mechanics, notably through Hamilton–Jacobi theory. His strength lay in algebraic development, and he published many papers from 1826 onward. He advised students to “Invert, always invert,” believing that reversing known results opens new research fields—for instance, inverting elliptic integrals to study elliptic and theta functions. In an 1835 paper, he proved that a single-valued function with multiple periods can have at most two periods, and their ratio cannot be real. He discovered key properties of theta functions, including the functional equation, the Jacobi triple product formula, and results on q-series and hypergeometric series.