Joseph-Louis Lagrange

French mathematician

• Born: January 25, 1736
• Birthplace: Turin, Kingdom of Sardinia (now in Italy)
• Died: April 10, 1813
• Place of death: Paris, France

One of the most brilliant mathematicians of the mid- to late eighteenth century, Lagrange accomplished astonishing syntheses of the mathematical innovations of his predecessors, especially in the systems underlying classic physics. Almost as remarkable for his winning personality as for his incisive intellect, Lagrange created the mathematical basis of modern mechanics.

Early Life

Born in what was then the kingdom of Sardinia of mixed French and Italian though predominantly French descent, Joseph-Louis Lagrange (zhoh-zehf-lwee lah-grahnzh) was the first son in an influential and wealthy family. His father, however, once a highly placed cabinet official, burned with the speculative fevers of the early eighteenth century and ended by losing everything. Typically, Lagrange took that in stride, remarking later that losing his inheritance forced him to find a profession; he chose wisely. Although early in his formal education he found mathematics boring, probably because it began with geometry, at age fourteen he chanced on an essay by the astronomerEdmond Halley, which changed his mind, and his life. In this essay, Halley, one of Isaac Newton’s disciples, proclaimed the superiority of the new analytical methods of calculus to the old synthetic geometry. From that moment, Lagrange devoted as much time as he could to the new science, becoming a professor of mathematics at the Royal Artillery School in Turin before the age of eighteen.

From the beginning, Lagrange specialized in analysis, starting the trend toward specialization that has since characterized the study of mathematics. His concentration on analytical methods also liberated the discipline for the first time from its dependence on Greek geometry. In fact, of his major work, Mécanique analytique (analytical mechanics), first conceived when he was nineteen but not published until 1788, he boasted that it contained not a single diagram. He then stated offhandedly that in the future the physics of mechanics might be approached as a geometry of four dimensions, the three familar Cartesian coordinates combined with a time coordinate; in such a system, a moving particle could be defined in time and space simultaneously. This system of analyzing mechanics reemerged in 1916, when Albert Einstein employed it to explain his general theory of relativity.

From the ages of nineteen to twenty-three, Lagrange continued as a professor at Turin, producing a number of revolutionary studies in the calculus of variations, analysis of mechanics, theory of sound, celestial mechanics, and probability theory, for which he won a number of international prizes and honors. In 1766, he succeeded Leonhard Euler as court mathematician to Frederick the Great in the Berlin Academy, the most prestigious position of the time. There, freed from lecturing duties, he continued to produce epochal studies in celestial mechanics, number theory, Diophantine analysis, and numerical and literal equations. He also found it possible to marry a younger cousin; the marriage was successful, and Lagrange was later devastated when his wife died of a wasting disease. Characteristically, he tried to overcome his grief by losing himself in his work.

Life’s Work

For most of Lagrange’s life, overwork was a habit. Yet it enabled him to achieve much at an early age. At twenty-three, Lagrange wrote an article on the calculus of variations, in which he foreshadowed his later unifying theory on the whole of mechanics, both solids and fluids. This integrated general mechanics in much the same way that Newton’s law of gravitation unified celestial motion. Lagrange’s theory proceeds from the disarmingly simple observation that all physical force is identical, whether operating in the solid or liquid state, whether aural, visual, or mechanical. It thus integrates a diverse array of physical phenomena, simplifying their study. In the same work, Lagrange applied differential calculus to the theory of probability. He also surpassed Newton by absorbing the mathematical theory of sound into the theory of elastic physical particles, becoming the first to understand sound transmission as straight-line projection through adjacent particles. Furthermore, he put to rest a controversy over the proper mathematical description of a vibrating string, laying the basis of the more general theory of vibrations as a whole. At this early age, Lagrange already ranked with the giants of his age, Euler and the Bernoulli family.

The next problems Lagrange attacked at Turin were those involved in the libration of the Moon in celestial mechanics: Why does the Moon present the same surface to the Earth at every point in its revolution? He deduced the answer to this special instance of the three-body problem, a classic in mechanics, from Newton’s law of universal gravitation. For solving this problem, Lagrange won the Grand Prix of the French Academy in 1764. The academy followed by proposing a four-body problem; Lagrange solved this, winning the prize again in 1766. The academy then proposed a six-body problem involving calculating the relative position of the Sun, Jupiter, and its four then-known satellites. This problem was not completely solvable by modern methods before the development of computers. Nevertheless, Lagrange developed methods of approximation that were superseded only in the twentieth century. After his move to Berlin, for further work on similar problems—the general three-body problem, the motion of the Moon, and cometary disturbances—Lagrange won further awards.

His career in Berlin lasted twenty years; during this career, he distinguished himself by unfailing courtesy, generosity to other mathematicians, and diplomacy in difficult situations—he was a stranger in a strange court, but he thrived. In addition to working on celestial mechanics there, he diverted himself by investigations into number theory, the humble matter of what his age considered higher arithmetic. Quadratic forms and Diophantine analysis—exponential equations—particularly interested him: He first solved the problem of determining for which square numbers x², nx² + 1 is also a square, when n is a nonsquare, for example, n = 3, x = 4. This problem was an ancient one; Lagrange’s paper is a classic, couched in his elegant language and supported by his equally elegant reasoning. He followed this by offering the first successful proofs of some of Pierre de Fermat’s theorems and the one of John Wilson that states that only prime numbers are factors of the sum of the factorial series of the next lowest number plus one—that is, p divides (p -1)(p -2)���3 � 2 � 1 + 1 only if it is prime. His most famous proof in number theory shows that every positive integer can be represented as a sum of four integral squares—a theorem that has had extensive applications in many scientific fields. He later did great work—which proved preliminary—on quadratic equations in two unknowns.

Perhaps the most important work of the Berlin period, however, relates to Lagrange’s work in modern algebra. In a memoir of 1767 and in later sequels, he investigated the theoretical bases for solving various algebraic equations. Though once again he fell short of providing definitive answers, his work became an invaluable source for the nineteenth century algebraists who succeeded in finding them. The essential principles—that both necessary and sufficient conditions be established before solution—eluded him, but his work contained the clue.

Eventually, Lagrange’s propensity for work broke both his body and his spirit. By 1783, he had sunk into a profound depression, in the grip of which he found further work in mathematics impossible. When Frederick died in 1786 and Lagrange fell out of favor in Berlin, he willingly accepted a position with the French Academy. Still, a change of scene brought no renewal of his interest in mathematics. When his monumental Mécanique analytique was published in 1788, Lagrange took no notice of it, leaving a copy unopened on his desk for more than two years. Instead, he turned his attention to various other sciences and the humanities.

It took the French Revolution to reawaken Lagrange’s interest in mathematics. Although he could have fled, as many aristocratic scholars did, he did not. The atrocities of the Terror appalled Lagrange, and he had little sympathy with the destructive practices of revolutionary zealots. Yet when appointed to the faculties of the new schools—the École Normale and the École Polytechnique—intended to replace the abolished universities and academies, Lagrange took up his professional duties enthusiastically. Because he became aware of the difficulties his basically unprepared students had with the theoretical bases of calculus, he reformulated the theory to make it independent of concepts of infinitesimals and limits. His attempt was unsuccessful, but he prepared the foundation on which modern theories are built.

Part of his duties at the École Polytechnique required Lagrange to supervise the development of the metric system of weights and measures. Fortunately, he insisted that the base 10 be adopted. Radical reformers lobbied for base 12, alleging superior factorability; it is still occasionally proposed as more “rational,” and for centuries it played an infernal role in the British monetary system. To suppress the reformers, Lagrange argued ironically for the advantages of a system with base 11, or any prime, since then all fractions would have the same denominator. A small amount of practice convinced the radicals that 10 was more functional.

Teaching and supervision alone, however, did not suffice to relieve Lagrange’s besetting melancholy. He was saved from despair at the age of fifty-six, by the intervention of a young woman, the daughter of his friend the astronomer Pierre-Charles Lemonnier. She insisted on marrying him despite their disparity in age, and, contrary to all expectations, the marriage proved a brilliant success. For the following twenty years, Lagrange could not bear to have her out of his sight, and she proved to be a faithful companion, adept at drawing him out of his shell. At the end of his life, he worked on a second edition of his masterpiece, Mécanique analytique, adding many profound insights. He was still improving it when death came, gradually and almost imperceptibly, on April 10, 1813.

Significance

Joseph-Louis Lagrange ranks with the outstanding mathematicians of all time; in his prime, he was widely recognized as the greatest living mathematician, and he is certainly the most significant figure between Euler and Carl Friedrich Gauss. Beyond the quality of his work, he was noted equally for the brilliance of his demonstrations and for his accessibility and personal charm. He is particularly celebrated as one of the classic stylists of mathematical writing, almost the incarnation of mathematical elegance. His composition combines exceptional clarity of description and development with remarkable beauty of phrasing. His language is supple, never stilted or contorted; he somehow seems to ease the effort of strenuous thought. Lagrange once remarked that chemistry was as easy as algebra; in his writing, he is able to make things seem transparent, especially those which seemed particularly dense before reading him.

Perhaps because of this ease of expression, Lagrange is more important for the stimulus he provided for others than for his own original work. Time after time, his contemporaries and descendants found inspiration in him. He made his foundations so complete that others were able to apply them to other cases. In some instances, he was simply ahead of his time; his ideas have had to wait for the ground to be prepared. At any rate, Lagrange’s work proved to be extraordinarily rich for those who labored after him.

Lagrange’s most important contributions lie in mechanics and the calculus of variations. In fact, the latter is the centerpiece on which all of his achievements depend, the insight he used to integrate the theory of mechanics. This calculus derives from the ancient principle of least action or least time, which concerns the determination of the path a beam of light will follow when passing through or refracting off layers of varying densities. Hero of Alexandria began the inquiry by determining that a beam reflected from a series of mirrors reaches its object by following the shortest possible route; that is, it is the minimum of a function. René Descartes elaborated on the theory by experimenting with the effects of various lenses on a ray of light, showing that refraction also produced minima. Lagrange then proceeded to demonstrate that the general postulates for matter and motion established by Newton, which did not seem to harmonize, also fit this scheme of minima. Thus, he used a principle of economy in nature—that physical mechanics also tended to minimal extremes—to unify the principles of particles in motion. This not only was revolutionary in his time but also gave rise to the further integrating work of William Rowan Hamilton and James Clerk Maxwell, and eventually blossomed in Einstein’s general theory of relativity.

Bibliography

Bell, Eric T. Men of Mathematics. New York: Simon & Schuster, 1937. Bell’s work is famous for three features: readability, accessibility to the general reader, and general historical background. This is the preferred reference work, though Bell does not provide the technical detail of other sources.

Burton, David M. The History of Mathematics: An Introduction. Newton, Mass.: Allyn & Bacon, 1985. Burton’s book has some very attractive features, especially the examples and practical exercises in real mathematics. However, readers should be aware that his focus is on major developments and broad concepts, so his treatment of Lagrange, while in one sense admirably concise, is also somewhat cursory.

Fraser, Craig G. Calculus and Analytical Mechanics in the Age of Enlightenment. Brookfield, Vt.: Variorum, 1997. This collection of essays written between 1981 and 1994 includes studies of Lagrange’s early contributions to the principles and methods of mechanics, and his problems in the calculus of variations.

Grabiner, Judith V. The Calculus of Algebra: J-L Lagrange, 1736-1813. New York: Garland, 1990. Grabiner describes Lagrange’s work in calculus.

Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. Kline offers a more thorough and more rigorously theoretical treatment than Burton (see above), but he requires considerable mathematical sophistication. Still, the book is not aimed at specialists, and Kline explains thoroughly, emphasizing the coherent evolution of mathematical thought. He highlights Lagrange’s consistency admirably.

Porter, Thomas Isaac. “A History of the Classical Isoperimetric Problem.” In University of Chicago Contributions to the Calculus of Variations. Vol. 3. Chicago: University of Chicago Press, 1933. Porter’s article is a study for professionals and scholars, with much detail and requiring knowledge of advanced mathematics. It does, however, contain the most extensive account of Lagrange’s most important work in the calculus of variations, with incidental reference to his other achievements.

Smith, David Eugene, comp. A Source Book in Mathematics. Reprint. Mineola, N.Y.: Dover, 1959. Smith’s work is for historians of mathematics, but his selections of extracts from Lagrange’s works are representative and reveal Lagrange’s clarity of exposition, making them quite accessible.

Struik, D. J., ed. A Source Book in Mathematics, 1200-1800. Cambridge, Mass.: Harvard University Press, 1969. This is an anthology of extracts from the original works, such as David Eugene Smith’s, but it is more extensive and representative of the entire body of Lagrange’s work. The introductions and notes are useful and thorough, and particularly good in helping the reader reach an appreciation of Lagrange’s accomplishments.