Joseph-Louis Lagrange

• 1. Introduction

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  Joseph-Louis Lagrange was born Giuseppe Lodovico Lagrangia in Turin on the 25th of January 1736, the firstborn of eleven children. At seventeen, he stumbled across a paper by Edmond Halley from 1693 and was immediately seized by mathematics. Within a year of solitary, relentless study, he had taught himself to a level of mastery that would astonish the leading minds of Europe. How does a boy destined for a legal career become, in the words of Frederick the Great of Prussia, the greatest mathematician in Europe? And how does a single man reshape mechanics, number theory, celestial mechanics, and the very foundations of algebra, all while navigating revolution and exile? The story moves from Turin to Berlin to Paris, through war, bereavement, and the guillotine, tracing how the son of a financially ruined court treasurer remade the mathematical world.

• 2. A Family of Lost Fortunes

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  Lagrange's father, Giuseppe Francesco Lodovico, held the post of Treasurer of the Office of Public Works and Fortifications in Turin and had charge of the King's military chest. He should have been wealthy. Instead, before his son was grown, he had lost most of the family property through speculation. Lagrange later reflected that this financial ruin may have been his greatest stroke of luck. Had his father remained rich, he reasoned, he might have become a comfortable lawyer and never discovered mathematics at all.

  His paternal great-grandfather had been a French cavalry captain from the region of Tours who served under Louis XIV and then entered the service of Charles Emmanuel II, Duke of Savoy, marrying into the noble Roman family of Conti. That French thread in his ancestry would later become politically significant. His mother was the only child of a wealthy doctor from Cambiano, in the countryside of Turin. Raised as a Roman Catholic, Lagrange would later describe himself as an agnostic. His favourite subject at the University of Turin was classical Latin, and he initially found Greek geometry dull. The Halley paper changed everything.

• 3. The Tautochrone and Euler's Courtesy

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  Starting in 1754, Lagrange worked on the tautochrone problem, a classical puzzle about the path a bead must follow to reach the lowest point in the same time regardless of where it starts. In solving it, he developed a method of maximizing and minimizing quantities called functionals, a method that would grow into an entirely new branch of mathematics. He wrote a series of letters to Leonhard Euler between 1754 and 1756, outlining what he called his delta-algorithm, which led directly to the Euler-Lagrange equations of variational calculus.

  Euler's response set a tone of extraordinary generosity. It has been recorded that, with characteristic courtesy, Euler withheld a paper he had already written covering some of the same ground, so that the young Italian might complete his work and claim the undisputed invention of the new calculus. Lagrange formally published his method in two memoirs of the Turin Society in 1762 and 1773. By 1756, Euler and Maupertuis had already tried to persuade Lagrange to come to Berlin, but he shyly refused. He would refuse again in 1765, telling d'Alembert it did not seem suitable to go to Berlin while Euler was still there.

• 4. The Turin Academy and the Miscellanea

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  In 1757, Lagrange joined the chemist Giuseppe Angelo Saluzzo and the physicist Giovanni Francesco Cigna to found a scientific society that met in Saluzzo's home. That society was later incorporated as the Turin Academy of Sciences, and its published transactions became the Miscellanea Taurinensia. The first volume alone contains a paper correcting Newton on the propagation of sound, a complete solution of the transversely vibrating string problem that identified gaps in the work of Brook Taylor, D'Alembert, and Euler, and a treatment of echoes, beats, and compound sounds that contemporaries described as masterly.

  The second volume presented the full calculus of variations, illustrated by deriving the principle of least action. The third covered dynamical problems, a solution to a problem posed by Fermat, and the general differential equations for three bodies moving under mutual gravitational attraction. The 1764 paper on the libration of the Moon, explaining why the same face always faces Earth, won Lagrange the prize of the French Academy of Sciences. The French Academy awarded him the prize again in 1766, and he would share or win the prizes of 1772, 1774, and 1778 as well.

• 5. Twenty Years in Berlin

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  In 1766, after Euler left Berlin for Saint Petersburg, Frederick II wrote directly to Lagrange. The letter expressed the wish of the greatest king in Europe to have the greatest mathematician in Europe at his court. Lagrange was finally persuaded, and he would spend the next twenty years in Prussia. He married his cousin Vittoria Conti in 1767. Frederick, who became a personal favourite of Lagrange's, frequently lectured him on the virtues of a regular life. Lagrange took this seriously, studying his own mind and body as though they were machines, measuring exactly how much work he could perform before exhaustion and setting himself a definite task for the next day every night.

  His productivity was extraordinary. He contributed between one hundred and two hundred papers to the academies of Turin, Berlin, and France, and except for periods of illness he averaged roughly one paper per month. Among the most celebrated results of these years: in 1770 he proved that every positive integer is the sum of four squares, a theorem stated by Bachet without proof. In 1771 he proved Wilson's theorem. His 1770 and 1771 papers on solving algebraic equations laid the groundwork that would later become Galois theory. In 1772 he found the special solutions to the three-body problem that today bear his name as the Lagrangian points.

  Vittoria's health was poor throughout the Berlin years, and she died in 1783 after years of illness. Lagrange fell into a deep depression. Frederick II died in 1786, and the atmosphere of Berlin became uncongenial. Spain and Naples both sent invitations, but Lagrange accepted Louis XVI's offer to come to Paris.

• 6. Paris, Revolution, and the Metre

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  Lagrange arrived in Paris in 1787 at the age of 51 and was given apartments in the Louvre. He became a member of the French Academy of Sciences. Yet the move that should have crowned his career plunged him into a paralyzing melancholy. The printed copy of his Mécanique analytique, on which he had laboured for a quarter of a century in Berlin, lay for more than two years unopened on his desk.

  In 1792, Renée-Françoise-Adélaïde Le Monnier, the 24-year-old daughter of his friend the astronomer Pierre Charles Le Monnier, saw his sadness and his timidity and insisted on marrying him. He became warmly attached to her and she proved a devoted companion.

  The Revolution sharpened the danger around him. When the decree of October 1793 ordered all foreigners to leave France, Lagrange was specifically exempted by name, partly through the intervention of Antoine Lavoisier. On the 4th of May 1794, Lavoisier and 27 other tax farmers were arrested, sentenced, and guillotined the same afternoon. On Lavoisier's death, Lagrange said: "It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like." Lagrange had been preparing to flee but was never actually in danger. Different revolutionary governments, and later Napoleon, continued to honour him.

  In the 1790s Lagrange played a central role in designing the metric system. After Lavoisier's death in 1794, it was largely Lagrange who influenced the commission of 1799 to adopt the metre and kilogram with decimal subdivisions. He was also a founding member of the Bureau des Longitudes in 1795. Appointed senator in 1799, he was the first signer of the Sénatus-consulte that in 1802 annexed Piedmont to France, which made him a French citizen. Both France and Italy continued to claim him as their own.

• 7. The Mécanique Analytique

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  The book that had sat unopened on Lagrange's desk was published in 1788 under the supervision of Laplace, Condorcet, Legendre as editor, and Cousin. At first no printer would take it. It was Legendre who finally persuaded a Paris firm to publish it. The Mécanique analytique offered the most comprehensive treatment of classical mechanics since Isaac Newton, and it did so without a single diagram. Lagrange prided himself on this. He viewed mechanics as a branch of pure mathematics analogous to a geometry of four dimensions, the three spatial coordinates plus time.

  The book's central idea was to reduce all of mechanics to the single principle of virtual work. From that one principle, using the calculus of variations, Lagrange derived the entire science of motion for both solids and fluids. The key innovation was the method of generalized coordinates: instead of tracking every part of a system individually as D'Alembert and Euler had done, Lagrange expressed the kinetic and potential energies of the whole system in terms of a set of variables equal to the number of degrees of freedom, then obtained the equations of motion by differentiation. Sir William Rowan Hamilton described the work as a scientific poem.

  In 1810, at the age of 74, Lagrange began a thorough revision of the Mécanique analytique. He completed roughly two-thirds of it before he died in Paris on the 10th of April 1813, at 128 rue du Faubourg Saint-Honoré. Napoleon had awarded him the Grand Croix of the Ordre Impérial de la Réunion just two days before his death. He was buried in the Panthéon, and his tomb inscription lists every honour he had accumulated: Senator, Count of the Empire, Grand Officer of the Legion of Honour, member of the Institute and the Bureau of Longitude.

• 8. The Lasting Points

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  Lagrange's name persists in the working vocabulary of mathematics and physics in ways that are easy to encounter without knowing the man behind them. Lagrange multipliers appear in every course on constrained optimization. Lagrangian mechanics underpins classical field theory, quantum mechanics, and general relativity. The Lagrangian points in orbital mechanics, those five gravitational equilibrium positions Lagrange identified in 1772 for the Earth-Sun-Moon system, are now used to park telescopes and satellites.

  His work on algebraic equations, published in 1770 and 1771, was foundational for the generation that followed: Cauchy, Jacobi, and Weierstrass all built directly on his lectures and his treatise Théorie des fonctions analytiques of 1797. His Recherches d'Arithmétique of 1775 developed a general theory of binary quadratic forms and influenced the direction of number theory for decades.

  Lagrange is among the 72 prominent French scientists commemorated on plaques at the first stage of the Eiffel Tower. A street in the 5th Arrondissement in Paris bears his name, as does a street in Turin where the house of his birth still stands. The lunar crater Lagrange and the asteroid 1006 Lagrangea carry his name into the sky. The second edition of the Mécanique analytique, issued in 1811, contains his final statement on infinitesimals, in which he concluded that once one has grasped the spirit of the infinitesimal method and verified its results, infinitely small quantities may be used as a sure and valuable means of shortening and simplifying proofs. It was, in its way, a mathematician making peace with the tools he had spent a lifetime refining.