Joseph-Louis Lagrange

Giuseppe Lodovico Lagrangia was born in Turin on 25 January 1736, the eldest of eleven children in a family of French and Italian descent. His father, Giuseppe Francesco Lodovico Lagrangia, served as treasurer of the Office of Public Works and Fortifications for the King of Sardinia, but a series of failed financial speculations reduced the household to modest means ^[1]. Lagrange later remarked that this loss may have been fortunate, since a comfortable inheritance might never have driven him toward mathematics.

He enrolled at the University of Turin with the intention of studying law, the career his father had chosen for him. Classical literature occupied him at first, and the geometry of the ancients left him cold. The turning point came when he read a 1693 memoir by the English astronomer Edmond Halley on the use of algebra in optics. That short paper convinced the teenager that modern analysis, not classical geometry, was where the real work lay, and he threw himself into mathematics with almost no formal guidance ^[2].

Progress came astonishingly fast. Within roughly two years of serious study he had mastered the calculus of his day, and in 1754 he published his first paper, drawing an analogy between the binomial theorem and the successive derivatives of a product of functions. He soon discovered that Leibniz and Johann Bernoulli had anticipated the result, an embarrassment that pushed him toward genuinely new territory ^[1]. Anyone asking who was Joseph-Louis Lagrange should begin here: a largely self-taught adolescent in Turin who, by age nineteen, had been appointed professor of mathematics at the Royal Artillery School ^[3].

In August 1755 Lagrange sent Leonhard Euler, then the most celebrated mathematician in Europe, a letter outlining a purely analytical method for solving problems of maxima and minima of integrals. Euler had wrestled with such questions using semi-geometric arguments; Lagrange's approach, which became the foundation of the calculus of variations, was cleaner and far more general. Euler recognized the young man's superiority on this point, generously delayed publishing his own related work, and arranged for Lagrange's election as a foreign member of the Berlin Academy in 1756 ^[2].

In Turin, Lagrange gathered a circle of talented pupils and colleagues into a scientific society that eventually grew into the Royal Academy of Sciences of Turin. Its journal, the Miscellanea Taurinensia, carried much of his early output, including a landmark study of the propagation of sound in which he treated the vibrating string with new analytical rigor and engaged the ongoing dispute among Euler, d'Alembert, and Daniel Bernoulli over its solutions ^[3].

Prizes from the Paris Academy of Sciences built his international standing. He won the competition of 1764 for his work on the libration of the Moon, the slight rocking motion that lets observers on Earth see slightly more than half the lunar surface, and won again in 1766 for a study of the motions of Jupiter's satellites. Further prizes followed in 1772, 1774, and 1778, several shared or connected with the famous three-body problem ^[1]. When Euler left Berlin for Saint Petersburg in 1766, he and d'Alembert recommended Lagrange as his successor. Frederick the Great's invitation reportedly expressed the wish that the greatest mathematician in Europe should live near the greatest of kings, and Lagrange accepted, arriving in Berlin in November 1766 as director of mathematics at the Prussian Academy ^[2].

The twenty years in Berlin were the most productive of Lagrange's life, and they anchor any account of Joseph-Louis Lagrange achievements. In celestial mechanics he analyzed the gravitational three-body problem and identified special configurations, now called the Lagrange points, where a small body can maintain a fixed position relative to two larger orbiting masses. His 1772 prize essay on the problem predicted the equilateral triangle solutions long before the Trojan asteroids were discovered sharing Jupiter's orbit in the early twentieth century, a striking confirmation of the theory ^[4]. Space agencies now park telescopes such as the James Webb Space Telescope near these points, giving his eighteenth-century analysis a very modern afterlife ^[4].

Number theory occupied him as well. In 1770 he proved the four-square theorem, the statement that every positive integer can be written as the sum of at most four perfect squares, settling a claim Fermat had asserted without proof more than a century earlier ^[5]. He gave the first complete treatment of the general solution of Pell's equation, and a group-theoretic fact he uncovered while studying permutations of the roots of polynomial equations survives as Lagrange's theorem: the order of a subgroup divides the order of the group. His 1770 and 1771 papers on the algebraic solution of equations examined why formulas exist for degrees up to four and hinted at the obstruction for degree five, preparing the ground later worked by Abel and Galois ^[5].

The crowning work was the Mecanique analytique, published in Paris in 1788 after roughly a quarter century of development. In it Lagrange rebuilt the whole of mechanics on the principle of virtual work and the calculus of variations, deriving the motion of any system from a single analytical framework rather than from geometric diagrams. He noted with evident pride that the book contained no figures at all ^[2]. The Lagrangian formulation of mechanics, with its generalized coordinates and the differential equations now named for him and Euler, remains the standard language of theoretical physics, extending far beyond Newton's original setting into quantum field theory. In pure analysis he introduced much of the notation still taught today, including the prime symbol f'(x) for the derivative of a function ^[1].

Lagrange was reserved, gentle in manner, and famously indifferent to controversy. He avoided the priority disputes that consumed many contemporaries, and colleagues in Berlin and Paris alike described a man of even temper who preferred quiet work to public argument ^[1]. He suffered periodic episodes of ill health and melancholy throughout his life, some of them attributed by early biographers to overwork in his Turin years.

In 1767 he married his cousin Vittoria Conti in Berlin. The couple had no children, and by his own account the marriage was companionable rather than passionate. Vittoria's long illness and death in 1783 left him deeply depressed, and the loss, combined with the death of his patron Frederick the Great in 1786, loosened his ties to Prussia ^[3].

A second chapter opened in Paris. In 1792 the fifty-six year old Lagrange married Renee Francoise Adelaide Le Monnier, the young daughter of his colleague the astronomer Pierre Charles Le Monnier. Contemporary accounts agree that she was devoted to him and that the marriage, which surprised Parisian society given the difference in their ages, restored much of his spirits during his final two decades ^[2].

After Frederick's death, courts across Europe competed for Lagrange. He accepted the offer of Louis XVI, moved to Paris in 1787, and took up residence with apartments in the Louvre as a pensioned member of the Academy of Sciences ^[1]. The Mecanique analytique appeared the following year, though Lagrange, in a strange interlude of exhaustion, reportedly left his own masterpiece unopened on his desk for a long stretch afterward.

The French Revolution broke around him, and Lagrange navigated it with characteristic caution. A 1793 decree ordered foreigners out of France, but the chemist Antoine Lavoisier intervened to secure a specific exemption for him. When Lavoisier himself went to the guillotine in May 1794, Lagrange is recorded as saying that it took the mob only a moment to remove that head, though a hundred years might not produce another like it ^[2]. He served on the commission that created the metric system, arguing for the decimal base ten against proposals for base twelve, and his advocacy helped fix the shape of the units most of the world now uses ^[3].

Teaching duties came late but productively. He lectured at the short-lived Ecole Normale in 1795 and then at the new Ecole Polytechnique from 1794, and out of those courses grew two books, the Theorie des fonctions analytiques of 1797 and the Lecons sur le calcul des fonctions, which attempted to found calculus on power series rather than on infinitesimals or limits. The foundational program did not survive later scrutiny, but the books spread his methods to a generation of French engineers and mathematicians ^[5]. Napoleon, who admired him greatly, made him a senator, a count of the Empire, and a grand officer of the Legion of Honour. Lagrange was still revising an expanded edition of the Mecanique analytique when his strength failed. He died in Paris on 10 April 1813 and was buried in the Pantheon ^[1].

Few names recur across modern science as often as Lagrange's. Physicists write Lagrangians to describe everything from pendulums to elementary particles, engineers use Lagrange multipliers to solve constrained optimization problems, numerical analysts rely on Lagrange interpolation, and every calculus student meets the mean value theorem in the form he gave it ^[5]. In astronomy the five Lagrange points of the Sun-Earth and Earth-Moon systems host spacecraft, and the Trojan asteroids clustered at Jupiter's triangular points carry his prediction through the solar system ^[4].

His influence was as much a matter of style as of results. Lagrange insisted that mechanics could be treated as a branch of pure analysis, that elegance and generality were scientific virtues, and that a well-chosen notation was itself a discovery. Hamilton later called the Mecanique analytique a kind of scientific poem, and the analytical tradition it founded ran through Hamilton, Jacobi, and eventually into the variational principles at the heart of twentieth-century physics ^[2].

Both France and Italy claim him, with some justice on each side. He was born and educated in Turin, wrote his first great papers there, and signed early work in Italian forms of his name, yet he spent his mature career in Berlin and Paris, wrote in French, and died a count of the French Empire ^[3]. Streets, lunar craters, an asteroid, and institutions in both countries carry his name. For readers of any Joseph-Louis Lagrange biography, the essential Joseph-Louis Lagrange facts are these: a self-taught prodigy from Piedmont who unified mechanics under analysis, proved theorems Fermat only conjectured, helped build the metric system, and left tools that working scientists still reach for every day ^[1].