# Lagrange, Joseph Louis

### From Enlightenment Revolution

**Lagrange, Joseph Louis** (1736-1813): Italian-French. Mathematician and Natural Scientist.

Born into a well-to-do French-Italian family in Turin, Lagrange came to mathematics by accident rather than design. Classics were his first interest, but he was converted by chance reading Edmund Halley’s memoir on the methodical superiority of the calculus. He learned quickly and was teaching mathematics at Turin’s Royal Artillery School by the age of nineteen.

Lagrange’s unwavering commitment to analysis showed itself in the conception of his *Analytical Mechanics* (conceived c. 1756, published 1788)—which was to do for general mechanics what Isaac Newton’s law of universal gravitation had done for celestial mechanics—as a work without diagrams. This groundbreaking text presents the Lagrangian equations, which describe the motion of a dynamic system in generalized co-ordinates in terms of its kinetic energy, generalized forces and time. In the simplest case, i.e. a conservative system with potential energy *V* and kinetic energy *T* described in generalized co-ordinates , the system’s Lagrangian function is defined as , and the equations of motion given by .

Together with his best students, Lagrange founded the Turin Academy of Sciences. In the Academy’s *Memoirs*, he published papers on the calculus of variations, the propagation of sound, the theory of vibrating strings and the application of differential calculus to probability. Having attracted Euler, Leonhard’s attention, Lagrange was elected a foreign member of the Berlin Academy in 1759. He won the Grand Prize of the French Academy of Sciences in 1764 for his work on the libration of the moon, and again in 1766 for tackling the theory of motion of Jupiter’s satellites.

Lagrange moved to Berlin in 1766, having been named, through the good offices of Euler and Jean d’Alembert, the director of the physico-mathematical division at the Berlin Academy. During his residency there he published papers on the three-body problem, differential equations, algebraic analysis, number theory and mechanics, for which he was again awarded the Paris prize in 1772, 1774 and 1778. In 1768 he presented the first known proof that •Fermat’s equation, , can be solved in all cases where *x*, *y* and *a* are positive integers, a not being a perfect square and . His “Reflections on the Algebraic Resolution of Equations” (1770) paved the way for Evariste Galois’ development of group theory. In 1772 he proved Lagrange’s theorem: that every natural number can be written as the sum of four squares of integers, and that every natural number of the form 8*k*+7, or a power of four times such a number, needs four nonzero summands (alternatively, that the order of a subgroup of a finite group *G* is a factor of the order of *G*.)

Lagrange moved to Paris in 1787, taking numerous honors and academic posts, revolutionary upheaval notwithstanding. He was assigned to committees for the adoption of the metric system and the Bureau of Longitudes before being named a professor at the École Polytechnique. He taught analysis there until 1799, when Bonaparte, Napoleon’s coup d’état named him a grand officer of the Legion of Honor. During this period, Lagrange tried unsuccessfully to re-formulate the theory of the calculus in rigorously analytic terms— without recourse to infinitesimals or limits—in his *Theory of the Analytic Functions* (1797) and *Lessons on the Calculus of the Functions* (1801). Failure aside, this work is notable for inspiring later attempts (like those of Augustin-Louis Cauchy and Karl Weierstrass) to inject greater rigor into the calculus.

Lagrange was made Count of the Empire in 1808, and awarded the Grand Cross of the Ordre Impérial de la Réunion in 1813. He died shortly thereafter. Together with Euler, Lagrange is remembered as the greatest mathematician of the eighteenth century.

Further Reading:

Filippo Burzio, *Lagrange*, 1993.

**Ziad Elmarsafy**

University of York