A Strange Wilderness

by Amir D. Aczel

  In 1792 Monge briefly met Napoleon, who was then a young artillery officer. Four years later, when Napoleon became the general in command of the French army invading Italy, he wrote a letter to Monge thanking him for his kindness when they first met. It was the beginning of a very close, warm friendship.

  In 1798, when Napoleon embarked on his naval invasion of Egypt—Leibniz’s dream realized more than a century later through Napoleon—he took with him his two most favored mathematicians: Monge and Fourier. The two mathematicians were among a select group of only a dozen people whom Napoleon trusted with his plan of attacking, conquering, and then “civilizing” the Egyptians. A fleet of five hundred French ships left France in the spring of 1798, arriving in Malta on June 9 and taking three days to conquer the island. Here, Monge founded fifteen schools as well as a university modeled after the École Polytechnique. They continued toward Egypt, reaching Alexandria on July 1. Monge was reportedly first in the party to come ashore.

  In Egypt, Napoleon asked the mathematician to lead a cavalry regiment in an attack on the fortifications of the Nile, and at one point Monge was about to be killed by an Egyptian soldier. Anticipating the fatal blow to his favorite mathematician, Napoleon galloped as fast as he could toward the combatants and saved Monge. On July 20 the French won the decisive Battle of the Pyramids, and Napoleon was master of Egypt. After founding the Egyptian Institute, he put Monge in charge of determining which works of art and ancient artifacts would be brought back to France as war booty.14 So when we visit the vast Egyptology collection in the Louvre, or admire the obelisk at the Place de la Concorde in Paris, we have Monge to thank—or blame—for their presence in France. But the Egyptians apparently didn’t want to be “civilized” by the arrogant Napoleon, and massacred many of his troops. On his return to France, the emperor took Monge with him; as we recall, Fourier was to remain there for some time.

  During the 1798 Battle of the Pyramids, in which Napoleon’s army achieved a decisive victory over Egypt, French mathematician Gaspard Monge served loyally under his commander. He was rewarded handsomely by the emperor, but his honors were taken from him when the monarchy was restored.

  THE RESTORATION OF THE MONARCHY in France, following the fall of Napoleon, brought trouble to the mathematicians closest to the emperor. Carnot had to leave France for exile in Magdeburg, Germany, and Monge was stripped of his many honors and robbed of his position at the École Polytechnique. He died soon afterward. Lagrange had died a few years earlier, but Legendre remained politically quiet and continued to publish. Laplace, with his ever-shifting political allegiances, could easily survive under any regime.

  The period of unrest that followed the temporary restoration of the French monarchy affected the lives of many mathematicians. One of them was a very young genius whose involvement with anti-Royalist politics would ruin his career and destroy his life at a young age. In his very few active years of life, however, he was able to catapult algebra to the fore and solve key problems that had stumped many able mathematicians over the centuries. We will meet him next.

  Eugène Delacroix’s famous painting Liberty Leading the People has become a symbol of revolutionary fervor in France. Mathematician Évariste Galois’s father was strongly pro-Revolutionary, and as a young man, Galois benefited from Napoleon’s emphasis on the study of mathematics and science.

  TWELVE

  DUEL AT DAWN

  The story of the incredibly brilliant French mathematician Évariste Galois is one of the most romantic and powerful in history. Ever since the time of ancient Greece, mathematicians had been looking for ways of generalizing the method of finding solutions to equations of increasing order, in terms of their coefficients. For example, the simple linear (first-order) equation 2x – 4 = 0 is solved in terms of its coefficients 2 and 4 as x = 4/2 = 2. In this equation, 2 is the coefficient of x to the first power, and 4 is the coefficient of x to the zero power. We all know the quadratic formula used to solve quadratic equations in terms of the equation’s coefficients. Tartaglia, Cardano, Fior, and del Ferro have shown that similar formulas can be obtained for cubic and quartic equations as well. But no such general formula for solving an equation in terms of its coefficients had been known for the quintic, or fifth-order, equation—or for higher-order equations. The question that mathematicians had hoped to answer was: What is the general formula for solving a fifth-order or higher-order equation? And if no such formula exists, then why?

  The answer to this conundrum was provided by the work of Galois, and as a bonus it also led to an understanding of the impossibility of the three classical problems of antiquity—which, as we recall, are squaring the circle, doubling the cube, and trisecting an arbitrary angle. Galois’s work also launched group theory, an immensely important part of modern abstract algebra. Incredibly, all these breakthroughs materialized from the work of a very young person who blossomed intellectually between the ages of sixteen and twenty—the age at which he tragically died.

  ÉVARISTE GALOIS

  Évariste Galois was born on October 25, 1811, in the town of Bourg-la-Reine, a few miles south of Paris. During the French Revolution, the name of the town was changed (reine means “queen”). The revolutionaries detested royalty, so they changed the name of the town to Bourg-l’Egalité (town of equality).

  Galois’s father, Nicolas Gabriel Galois, was a politician, a one-time mayor of the town, and principal of the local school. After Napoleon’s demise and the return to power of the Bourbons under Louis Philippe of Orleans, the so-called King of the French, the elder Galois turned fiercely anti-Royalist. His town, however, had many Royalists who constantly attacked him through libelous statements in newspaper articles.

  Galois’s mother, Adélaide Marie Demante, was the daughter of a judge. She educated her son at home, focusing much of her lessons on the law, as well as philosophy, religion, and literature. At age twelve Galois entered the Royal School of Louis-le-Grand, where he proved to be a brilliant student, but teachers described him as having “bizarre manners” and as being “rebellious.” The school was originally mostly concerned with teaching the classics, and mathematics and science were considered nonessential. During Napoleonic times, however, this trend was reversed since, as we know, the emperor favored these disciplines; but the classical tradition was still much in place in the grade Galois was in. He took a course in rhetoric and performed so badly that he was sent back a grade.

  Galois’s failure in rhetoric was fortuitous, as it turned out. The lower grade was already designed according to the Napoleonic system, so he was allowed to study mathematics. On his own, the young Galois read beyond what was taught in class, including the mathematical works of Gauss, Euler, Lagrange, and Legendre. He was especially interested in Legendre’s book Elements of Geometry, which he read with ease and excitement when he was only fifteen. But it was the research papers by Lagrange that inspired him, while still a teenager, to try to apply Lagrange’s methods to equations that could not, at that time, be solved.

  In 1828 Galois attempted the entrance examinations to the prestigious École Polytechnique but failed, so he had to remain at Louis-le-Grand. Luckily for Galois, Louis Richard, the teacher of an advanced mathematics course he took, recognized that he had a genius in the class and kept all Galois’s homework assignments. (After Galois’s tragic death, Richard would give these papers to the mathematician Charles Hermite, who later became famous through his own work in various mathematical spheres.) Richard encouraged Galois to publish his early research results. After an article by Galois appeared, on April 1, 1829, in the journal Annales de Mathématiques, the seventeen-year-old wrote a more extensive paper and sent it to the French Academy of Sciences, hoping to receive recognition for his important discoveries in algebra from this great academic body.

  Galois was a mathematics prodigy at a young age; when he was seventeen, one of his teachers began to preserve his homework assignments, recognizing their future importance.

  There, Galois’s pape
r was supposed to be read by the eminent mathematician Augustin-Louis Cauchy (1789–1857), who did groundbreaking work on determinants, a name he coined, as well as complex analysis and other areas. The great Cauchy lost Galois’s paper, however. By then, Cauchy had already lost another paper sent him by a young genius working on very similar problems in the theory of equations: Niels Abel.

  THE ABEL-RUFFINI THEOREM

  Niels Henrik Abel (1802–29) was born in the village of Findö, Norway, to a very large family. His father, the pastor of the village, was also a politician involved in the writing of Norway’s new constitution. When he was sixteen, Abel’s teacher gave him Gauss’s Disquisitiones Arithmeticae, hoping the bright student would enjoy it. Not only did Abel enjoy the book, but he was able to fill in details that had been left out of Gauss’s proofs.

  When Abel was eighteen, his father died in disgrace after he had made false charges against political allies and followed it by hard drinking, which ended his political career. Since Abel was the eldest boy, the responsibility to help provide for the family fell on his shoulders. Still, he found time for mathematics, and eventually achieved a breakthrough for a problem that had occupied many mathematicians in history and would soon obsess Galois. After wrongly thinking he could solve the quintic (fifth-order) equation, he was able to prove that the quintic equation cannot be solved by an explicit algebraic expression involving its coefficients, as can be done with the quadratic equation. In 1799, the Italian mathematician Paolo Ruffini (1765–1822) had actually derived a proof of the unsolvability of the quintic equation that was overlooked for many years, so today this mathematical result is called the Abel-Ruffini theorem.

  Abel came to Paris in 1826 hoping that his work would attract the interest of members of the French Academy of Sciences; but to no avail. Cauchy—who was only interested in his own investigations—was assigned to read Abel’s paper, but he either ignored or lost it. Frustrated and depressed, Abel wrote from Paris to a friend: “I have just finished an extensive treatise … and Mr. Cauchy scarcely deigns to glance at it.” He returned home, becoming weaker by the day. He had contracted tuberculosis.

  In 1829, barely twenty-seven years old, Abel died. Sadly, just two days after his death, a letter arrived offering him an academic position in Berlin, where he had visited and impressed mathematicians before going to Paris.

  It was in April of the year Abel died that Galois sent his own paper to Cauchy. Galois went further than Abel in explaining why a quintic equation could not be solved in radicals (meaning in terms of its coefficients), and his ideas would pave the way for a whole new area of mathematics: group theory.

  On July 2 of the same year, an event with grave consequences to the life of Galois took place. His father, Mayor Nicolas Galois, had endured ruthless, ceaseless attacks from the Royalists in his hometown. They even resorted to dirty tricks that included crafting malicious poems, falsely attributing them to him, and publishing them in the local paper. The elder Galois, no longer able to endure such malevolence, committed suicide. At his funeral, a riot erupted between the two political factions.

  A few days later, the young Galois again attempted the entrance examinations to the École Polytechnique. One of the two examiners, whose names were Dinet and Lefébure de Fourcy, asked Galois to explain logarithms. Distraught by the death of his father and irritated by a question he considered trivial, Galois threw the blackboard eraser at the examiner (probably Dinet).1 Needless to say, he failed the test, to the great disappointment of his teacher.

  Louis Richard then suggested that Galois apply to the “lesser” École Normale Supérieure (which, at that time, was known as the École Préparatoire, or Preparatory School). Reluctantly, Galois followed his teacher’s advice, and then submitted the results of his research to the Academy of Sciences for consideration in their competition for the Grand Prize in Mathematics. The paper was received by Fourier, who took it home to read—and died. For Galois, it was yet another stroke of bad luck. The manuscript was never recovered, but some of Galois’s results were later published in the June 1830 issue of Baron de Férussac’s Bulletin des Sciences Mathématiques, Physiques et Chimiques.

  To Galois’s great disappointment, because the academy had lost his paper when Fourier died, the Grand Prize was awarded jointly to Abel, who had died the previous year, and the German mathematician Carl Gustav Jacobi (1804–51). After this devastating misfortune, Galois became increasingly bitter, and at the same time arrogant. He knew that his knowledge of mathematics was so good that his teachers could not even understand the new results he was deriving.

  Resigned, Galois joined the entering class at École Normale Supérieure. On July 27–29, 1830, the school was locked down to prevent students from taking part in riots on the streets of Paris against Louis-Philippe of Orleans, the post-Empire reinstated Bourbon monarch. The students of the École Polytechnique, on the other hand, had no such restriction placed on them, and they made history by battling the king’s soldiers in the barricades on the streets and boulevards of Paris.

  Galois was greatly agitated by this development. Having been raised a Republican, he became more ardent in his political beliefs and took a Republican leadership position within the student body. According to a member of his family, his stated goal was “to defend the rights of the masses.”2 On November 10, 1830, Galois joined the Society of Friends of the People, another political body opposed to the reinstated monarchy. Around that time, he began to publicly attack the president and the administration of the university, which had supported the king. Galois also criticized the university’s instruction, which he considered mediocre at best. His attacks on the administration led to his being “indefinitely suspended” from the school.

  On December 1, 1830, the journal Annales de Gergonne published a short abstract of Galois’s work, and this became the last scientific paper published in his lifetime. Four days later, on December 5, he wrote a scathing criticism of the school, accusing the professors of “mediocrity of teaching.” Since he had already been suspended from the school, this new offense caused his outright dismissal, and Galois found himself on the street, with no income. In order to try to make a living, he offered a private course in mathematics on a street corner near the Sorbonne every Thursday at 1:15 p.m. On the first day, thirty students attended, and then more came, but the enterprise soon ended as student interest waned after a period of time.

  In frustration Galois joined the artillery branch of the French National Guard, a Republican-leaning body of the French Civil Service. On May 9, 1831, he attended a banquet held by this unit in the Parisian restaurant Aux Vendanges de Bourgogne. Alexandre Dumas, who was also present, later described the event. Apparently, Galois raised his wineglass unex-pectedly in a toast. After gaining the guests’ attention by clanging a knife against his glass, he said, “To Louis-Philippe.” Later, he claimed that he had added, “should he betray,” but because his toast caused an immediate uproar, nobody heard it. “The fumes of the wine had removed my reason,” Galois later told the police, in his defense.

  The next day, Galois was arrested at his mother’s house and accused of threatening the life of the king. He was sent to the Sainte Pélagie Prison, near the famous Jardin des Plantes, by the Seine. He was tried on June 15, and at his trial, he testified as follows: “Here are the facts. I had a knife I had been using to cut my food. I raised the knife while saying, ‘To Louis-Philippe, if he betrays.’ But the last words were heard only by my nearest neighbors because the first part of the toast elicited whistles.” Galois was acquitted.3

  On January 17, 1831, the Academy of Sciences had asked Siméon-Denis Poisson and Sylvestre Lacroix to read a paper sent to the academy by Galois. It appears that Poisson had actually encouraged Galois to resubmit the paper that had been lost when Fourier died, but when it came to actually reading Galois’s paper, neither eminent mathematician understood Galois’s ingenious application of permutation groups to the understanding of the relationships among roots of a
polynomial equation and ultimately to the solvability of the equation by radicals. On July 4, 1831, Poisson and Lacroix issued their final report on the work of Galois: “We have made every effort to understand this proof. The reasoning is neither sufficiently clear nor developed enough to allow us to judge its correctness.” Galois was very disappointed and did not supply any further explanation of his groundbreaking work. On July 14, Bastille Day, celebrated in France in commemoration of the French Revolution, Galois was again arrested. This time he was at the head of a large group of anti-Royalist demonstrators that had congregated at the Pont Neuf on the Île de la Cité in the heart of Paris. He was carrying a loaded rifle, pistols, and a dagger. Because it was his second offense, this time Galois was convicted and sentenced to six months in prison.

  In December of 1831, Galois again tried to publish his work, though he was still angry that his previous papers had all been lost or misunderstood by the academy. “Egoism reigns in the sciences,” he wrote to a friend. “People should study together instead of sending sealed letters to the academy.”4 In early 1832 a cholera epidemic caused the closure of the Sainte Pélagie Prison, where Galois had been held, and the prisoners were transferred to other facilities. Galois was moved to a halfway house located near the Place d’Italie, which was then in a township called Gentilly but today is in the heart of the 13th arrondissement of Paris. This was a pleasant place, as compared to a prison. It even had a resident physician to look after the health of the convicts living there. The physician was Dr. Du Motel.

  In early May 1832, Galois apparently became romantically involved with a young woman named Stephanie. (We know this because “Stephanie” and “Stephanie D.” appear in the margins of a number of letters he later wrote, sometimes effaced or blotted out.) She has been tentatively identified as Stephanie-Félicie Du Motel, the daughter of the physician at the halfway house. It seems that on May 14, the affair ended, and a man claiming to be Stephanie’s lover challenged Galois to a duel. Alexandre Dumas identified the man as Perscheux d’Herbinville, although this identification has been disputed. Galois may have been in the National Guard and have held a leadership position with the campus Republicans at the École Normale Supérieure, but he was still a very young, inexperienced person, and naive about the ways of the world. He had never taken part in a duel before, and probably had no clear idea about how a duel was fought. Worst of all, we know from his correspondence that apparently he felt that he could not avoid a duel, even if he was certain that he had absolutely no chance of winning it. Galois seemed to know that he would die, and yet he followed through and walked into the dreadful trap laid out for him.