A Strange Wilderness

by Amir D. Aczel

  Among mathematician Joseph Lagrange’s many contributions to the field is his groundbreaking book Mécanique Analytique, published in 1788—thirty-three years after it was written.

  The Lagrange multiplier, which allows one to find maximum and minimum points of functions given some constraints, is another important discovery in applied mathematics that extends calculus ideas to far more general and complicated situations than previously achieved. Lagrange also pioneered the use of the determinant, which he used to find the areas of triangles and the volumes of tetrahedra.7

  When Frederick the Great died in 1786, a rise in nationalism that had been kept at bay by the monarch brought pressure on foreign-born members of his academy to leave Germany. Lagrange was a favorite of the academy, however, so when he resigned he promised to keep sending papers to be published in the academy’s proceedings. From the Royal Prussian Academy in Berlin, Lagrange moved to the Royal Academy of Sciences in Paris at the invitation of Louis XVI himself. The king’s wife, Marie Antoinette, admired him as well, and he lived for a while in the Louvre as special guest of the royal family.

  Lagrange was now feeling his age—he was in his fifties—and the move to Paris brought on a deep depression, which Marie Antoinette tried to ameliorate with lively conversations. Lagrange spoke little at the lavish parties he was invited to and was described as staring blankly out the window with his back to the other guests.8 He also lost his interest in mathematics. The only high point during this dark period was his friendship with the great French chemist Antoine Lavoisier (1743–94). Lagrange had begun to believe that the new age belonged to chemistry and other sciences and that mathematics had lost its glitter. When told of a fellow mathematician’s great discovery, he answered, “All the better; I began it; I won’t have to finish it.”9

  When the French Revolution began, he was warned that it was in his best interest to return to Berlin, where he was always welcome, to avoid the dangers at home. Lagrange refused to leave France, and felt no sympathy for either the Royalists or the Revolutionists, but when the Terror ensued and Lavoisier was guillotined for trumped-up charges, Lagrange became angry and even more depressed. Despite his disillusionment with mathematics and with life in general, he was appointed president of the committee that determined the new weights and measures system, and it was his personal triumph when it was decided that the system would be metric rather than duodecimal.

  Among those who were guillotined during France’s 1793–94 Reign of Terror were chemist Antoine Lavoisier, who helped Lagrange avoid persecution by the authorities, and King Louis XVI—the only king of France ever to be executed.

  Thanks to a young woman almost forty years his junior, Lagrange’s depression finally lifted. Renée-Françoise-Adélaide Le Monnier—the daughter of an astronomer friend, Pierre-Charles Le Monnier (1715–99)—met Lagrange when he was fifty-six years old and, feeling sorry for the depressed genius, offered to marry him. They wed, and his life changed overnight. He was so much in love with his very young wife that he couldn’t bear to be away from her—he accompanied her to balls and banquets he would have done anything he could to avoid prior to their marriage.

  When Napoleon came into prominence, he would often talk to Lagrange about mathematics and its place in society whenever the general was in Paris between military campaigns, and when Napoleon became emperor, he bestowed on Lagrange many high honors. By the time Lagrange had completed the last revision of Mécanique Analytique for a second edition, he had begun to suffer from dizzy spells. His body had finally gotten tired, and he died in 1813 at the age of seventy-five.

  PIERRE-SIMON LAPLACE

  Lagrange’s fellow mathematician on the weights and measures committee that gave the world the metric system was the mathematician Pierre-Simon de Laplace (1749–1827). A mathematical astronomer, Laplace was often called the Newton of France.10 Laplace’s origins are shrouded in mystery, due in part to the fact that he was born poor and tried hard to hide where he came from once he obtained noble status. We know he was born in the Calvados region of Normandy, in northwest France. He started off as a theologian but later in life turned atheist. As a young man, he absorbed mathematics quickly and, with strong recommendations from his teachers, left for the big city: Paris.

  In the French capital, Laplace tried to get an audience with Jean Le Rond d’Alembert (1717–83), a successful mathematician of equally humble origins. D’Alembert was the illegitimate son of Louis-Camus Destouches, a French artillery officer, and Claudine Guérin de Tencin, a well-known writer, former nun, and sister of a cardinal. As his father was abroad when the baby was born, Madame de Tencin decided she would just as well get rid of the unwanted child by placing him on the steps of the chapel of Saint Jean le Rond—where he was baptized and from which he took his name—near the Cathedral of Notre Dame in Paris. He was then taken to an orphanage, and when his father returned to France, he arranged for him to be adopted by a glassworker named Rousseau and his wife.

  Though Chevalier Destouches kept his paternity a secret, he helped pay for his son’s upbringing and education. After he died, when Jean was nine, his family continued to support the officer’s illegitimate son. When Jean enrolled at the Collège des Quatre Nations, he assumed the fake name Jean-Baptiste Daremberg, since he could not use his father’s name and did not want to use the name of his adoptive parents. He later changed it to something slightly more noble sounding: Jean Le Rond d’Alembert. Meanwhile, d’Alembert’s birth mother found out who had taken her baby, and when it became clear that he was brilliant and on his way to becoming a major mathematician, she tried to get him back. “You are only my stepmother,” the young man told his wealthy and aristocratic biological mother, insisting that the poor couple who had been raising him were his real parents.

  The mathematician Jean Le Rond d’Alembert, among his many other achievements, wrote more than one thousand articles for Denis Diderot’s famous Encyclopédie, published between 1751 and 1772. He also served as coeditor of the publication.

  Having made it in life, d’Alembert faced Laplace, a young man from the provinces who dropped several glowing letters of recommendation on his desk. D’Alembert read the letters but seemed unimpressed, and the young mathematician left. A few days later, trying again to gain an entry to Paris academia through d’Alembert, he sent him a short paper he had written on the mathematical principles of mechanics, the area studied so assiduously by Lagrange. Some time later a letter arrived back from d’Alembert. “Sir,” he wrote, “you see that I paid little enough attention to your recommendations; you don’t need any. You have introduced yourself better. That is enough for me; my support is your due.”11 A few days later, on d’Alembert’s recommendation, Laplace was appointed professor of mathematics at the École Militaire.

  Laplace began his research while teaching at that school, laying the groundwork for his world-famous masterpiece, Mécanique Céleste (Celestial Mechanics). He decided to apply Newton’s physics and the mathematics of his theory of universal gravitation to a study of the entire solar system, with all its known planets.

  It was a monumental work but borrowed heavily from Newtonian principles, which Laplace acknowledged. However, the book also relied on the work of Lagrange—most significantly, the idea of a gravitational potential, which is a key concept in the physics of gravitation today—and methods of analysis developed by Legendre. Laplace did not reference or acknowledge the latter two mathematicians.12

  Pierre-Simon Laplace’s five-volume masterwork, Mécanique Céleste, was published between 1799 and 1825.

  In Mécanique Céleste, Laplace attacked the problem of how bodies mutually affect each other gravitationally. This grand extension of the work of Lagrange to the entire system of planets and the sun involved many questions, including: Could one of the planets veer off into space and leave our solar system? Could Mercury slip from its orbit and fall into the sun? Could the moon begin to move away from Earth and crash into Mars? All these questions comprise one overar
ching riddle: Is our solar system stable, and if so, why? Laplace addressed this great set of questions by analyzing the gravitational forces acting between and among the bodies in the entire known solar system. In the end he was able to prove that the solar system was stable.

  Mécanique Céleste went far beyond Newton in its mathematical study of gravitation. Newton had explained the force of universal gravitation but assumed that the solar system was held together by divine intervention, thus reconciling his deep religious beliefs with the scientific and mathematical principles he had discovered.

  Thanks to his great achievement, at the age of twenty-four Laplace was inducted into the French Academy of Sciences. He spent his entire life perfecting his theory of mathematical astronomy, but he also did work in probability theory, as applied to astronomy. He is known for inventing what we call the Laplacian operator, given by the sum of the second-order partial derivatives of a function, which is very useful in physics and other applications of mathematics.

  WHEN THE REVOLUTION OCCURRED, Laplace and Lagrange are said to have escaped the guillotine because they were expedient in their abilities to calculate trajectories of artillery shells. The mathematician M. J. Condorcet fared much worse. In jail, taken in by a sweep of all nonworkers and aristocrats, he asked for an omelet. A nobleman, Condorcet had never seen an omelet being cooked. “How many eggs do you want in your omelet?” his jailers asked. He thought about it for a minute and then said, “Twelve.” The jailers then asked to see his hands, and when they confirmed to their satisfaction that the man probably never did any hard manual work in his life, they sent him to be guillotined.

  When the Terror passed, Laplace decided to become a politician. Having attained high status through his achievements and membership in the prestigious French Academy, the great mathematical astronomer decided to try to obtain real power. He was somewhat mercurial, however, changing from Republican to Royalist whenever the political winds shifted. When Napoleon came to power, he heaped honors on Laplace, who as a scientist had brought glory to France. In addition to receiving the Grand Cross of the Legion of Honor and assuming the title of Count of the Empire, the humbly born mathematician eventually became a marquis.

  Laplace made a gift to Napoleon of his Mécanique Céleste. Perhaps to tease him, Napoleon commented, “You have written this huge book on the system of the world without once mentioning the maker of the universe.” Laplace answered, “Sire, I had no need for that hypothesis.” Napoleon later repeated his exchange with Laplace to his other mathematician friend, Joseph Lagrange. The latter, ever tactful and diplomatic, responded, “Ah, sire, but it’s such a beautiful hypothesis.”

  When Napoleon was defeated at Waterloo, it was Laplace—now an official in the French government—who had to sign the decree banishing Napoleon to Saint Helena. There the fallen emperor summed up what he thought of the mathematician-turned-politician: “A mathematician of the first rank, Laplace quickly revealed himself as only a mediocre administrator; from his first work we saw that we had been deceived.”13

  JOSEPH FOURIER

  During his reign, Napoleon enjoyed the friendship of several other French mathematicians. One of the most important among these was Joseph Fourier (1768–1830). Fourier was the son of a tailor in Auxerre. When he was young, both his parents died, so he was brought to the care of the bishop of Auxerre, who arranged for him to be adopted and admitted to the local military college.

  Fourier was a troubled boy who refused to listen to teachers, and he nearly ended up on the streets, but then he discovered mathematics, which kept him fascinated and occupied until graduation. His low social status squashed his chances of becoming a soldier, and the French Revolution prevented him from entering the priesthood, so he concentrated on studying mathematics.

  At age twenty-one Fourier arrived in Paris and presented to the French Academy of Sciences his work on numerical solutions of equations used in physics. He was involved in revolutionary politics but abhorred the Terror that followed. He then studied at the École Normale Supérieure in Paris and excelled. One of his favorite professors was Lagrange, and he took courses from Laplace as well.

  His early success presenting a paper to the academy, his studies of mathematics at the École Normale Supérieure, and his connections with Lagrange and Laplace helped him to attain a position at the prestigious École Polytechnique in Paris. There he met Napoleon, who was seeking mathematical help in determining artillery trajectories while planning his early campaigns. When Napoleon went on his conquest of Egypt in 1798, he took Fourier with him.

  In his attempt to “liberate” the Egyptians from their “uncultured” state, Napoleon founded the Egyptian Institute, a Cairo-based offshoot of the Institut de France and its associated Academy of Sciences. As cofounder and secretary of the institute, Fourier was involved in archaeological excavations carried out by its staff. The tides of war eventually turned against the French, however, and in 1799 Napoleon felt compelled to depart, leaving Fourier behind to help administer the territory.

  Fourier returned to France only three years later, in 1801. After resuming his post at the École Polytechnique in Paris, Napoleon asked him to take an administrative post headquartered at Grenoble, in Isère, the prefect of which had just died. There he created his masterpiece on the mathematical study of heat. This beautiful piece of applied mathematics describes how heat is conducted, using differential equations, an element of advanced calculus. In particular, Fourier simplified the study of heat conduction by representing functions as trigonometric series—a method we refer to as Fourier analysis.

  Fourier found a way of deconstructing a set of data into its frequencies (as in music) and analyzing it. The new techniques Fourier developed in mathematics are extremely important today. Similar methods are used in economic and stock-market analyses, and Fourier’s derivation of the heat equation later led to other advances in physics, including methods in quantum mechanics.

  At Grenoble, Fourier discovered the remains of his great-uncle, Pierre Fourier, who had been canonized. When Napoleon escaped from his imprisonment on Elba in 1815, he passed with his followers through Grenoble and came to see his old friend. When he did not find him, the deposed emperor suspected that, in his absence, the man he had once made a top administrator in Egypt had switched alliances and gone on to support the Bourbons, the traditional rulers of France. Nevertheless, he appointed Fourier to the prefecture of Rhône. Napoleon planned an offensive against the British and Prussian armies, who hoped to unseat him from power. Fourier warned him that his plan would not succeed, but Napoleon ignored the advice of his prized mathematician. Waterloo would prove Fourier right.

  Fourier became the Permanent Secretary of the Academy of Sciences and never missed an opportunity to regale his fellow scientists with stories about his adventures with Napoleon in Egypt. The man who explained mathematically how heat is conducted had apparently developed the conviction that the desert heat of Egypt was salutary, so he tried to emulate it in cold and humid France, sleeping in a sweltering bedroom that no one else could survive in for long. Eventually this habit took a toll on his health and he died of heart problems when he was sixty-two.

  GASPARD MONGE

  Another French mathematician, Gaspard Monge (1746–1818), developed a particularly close relationship with Napoleon, accompanying both him and Fourier on the 1798 conquest of Egypt. Monge was born in Beaune, Burgundy, to a family of peddlers and knife grinders. His father held education in high esteem, however, and sent all his sons to college. But among all the successful Monge sons, Gaspard was the brightest. When he was fourteen he built a fire engine, to the astonishment of everyone around him. A couple of years later he drew a very accurate map of the region where he lived. He later explained that it was his spatial intuition that helped him accomplish both tasks. In fact, this uncanny ability to visualize shapes and forms in three dimensions allowed him to develop descriptive geometry: a way of capturing three-dimensional objects on two-dimensional paper
using ingenious graphing techniques that he invented. His ingenuity won him awards at school, and his invention of descriptive geometry earned him an invitation to lecture at the École Normale Supérieure in Paris, which had recently been founded. One of the people who sat in on his lectures was the great Lagrange, who was fascinated by this new, practical geometry. At the school, he also met Fourier. Eventually, Monge was offered a professorship at the University of Mézières.

  Once, at a party, Monge heard a nobleman vulgarly disparage a young widow who had rejected his advances. Monge rushed to defend the unknown lady’s honor, punching her insulter in the mouth. At another party later on, he was introduced to the woman whose honor he had jumped to defend, Madame Horbon, and fell for the beautiful young widow of twenty. He proposed to her then and there, and after taking time to put her late husband’s affairs in order, she agreed. Monge and Horbon married in 1777. It proved a good match—Monge’s young wife even saved his life during the Revolution, while he was in Paris working with d’Alembert and Condorcet. It was a frequent occurrence in times of terror for people to be denounced for various fabricated crimes. One day Monge’s wife happened to discover that her husband had been denounced by the Revolutionaries and rushed to Paris to warn him. The couple escaped to the countryside, where it was safe.