Why Beauty is Truth (24 page)

Galois, not surprisingly, was furious. He became convinced that what had happened was a conspiracy of mediocre minds to stifle the efforts of genius; he quickly found a scapegoat, the oppressive Bourbon regime. And he wanted to play a role in its destruction.

Six years earlier, in 1824, King Charles X had come to the throne of France, following Louis XVIII, but he was far from popular. The liberal opposition did well in the 1827 elections and even better in 1830, gaining a majority. Charles, facing the imminent prospect of forced abdication, attempted a coup; on 25 July he issued a proclamation suspending freedom of the press. He misread the mood of the people, who promptly rose in revolt, and after three days a compromise was reached: Charles was replaced as king by the duke of Orléans, Louis-Philippe.

The students of the École Polytechnique, the university Galois had hoped to attend, played a significant role in these events, demonstrating on the streets of Paris. And where was the arch antimonarchist Galois during this fateful period? Locked away inside the École Préparatoire along with his fellow students. The Director, Guigniault, had decided to play safe.

Galois was so incensed at being denied his place in history that he wrote a blistering attack on Guigniault in the
Gazette des Écoles:

The letter which M. Guigniault placed in the lycée yesterday, on the account of one of the articles in your journal, seemed to me most improper. I had thought that you would welcome eagerly any way of exposing this man.

   Here are the facts, which can be vouched for by forty-six students.

   On the morning of 28 July, when several students of the École Normale wanted to join in the struggle, M. Guigniault told them, twice, that he had the power to call the police to restore order in the school. The police on 28 July!

   The same day, M. Guigniault told us with his usual pedantry: “There are many brave men fighting on both sides. If I were a soldier, I would not know which to decide. Which to sacrifice, liberty or LEGITIMACY?”
   There is the man who next day covered his hat with an enormous tricolor cockade [a symbol of the republicans]. There are our liberal doctrines!

The editor published the letter but removed the author's name from it. The director promptly expelled Galois for publishing an anonymous letter.

Galois retaliated by joining the Artillery of the National Guard, a paramilitary organization that was a hotbed of republicanism. On 21 December 1830, this unit, very probably including Galois, was stationed in the vicinity of the Louvre. Four ex-ministers had gone on trial, and the public mood was ugly: they wanted the men executed, and were prepared to riot if they were not. But just before the verdict was announced, the Artillery of the National Guard was withdrawn and replaced by the regular National Guard, together with other soldiers who were loyal to the King. The verdict of a jail sentence was announced, the riot failed to materialize, and ten days later, Louis-Philippe disbanded the Artillery of the National Guard as a security risk. Galois was having no more success as a revolutionary than he had had as a mathematician.

Practical issues now became more urgent than politics: he needed to make a living. Galois set himself up as a private mathematics tutor, and forty students signed up for a course of advanced algebra. We know that Galois was not a good written expositor, and it's reasonable to guess that his teaching was no better. Probably his classes were laced with political commentary; almost certainly they were too difficult for ordinary mortals. At any rate, the enrollment rapidly dwindled.

Galois had still not given up on his mathematical career, and he submitted yet a third version of his work to the Academy, entitled
On the Conditions of Solvability of Equations by Radicals.
With Cauchy having fled Paris,
the referees were Siméon Poisson and Lacroix. When two months passed without any response, Galois wrote to ask what was happening. No one replied.

By the spring of 1831, Galois was behaving ever more erratically. On 18 April the mathematician Sophie Germain, who had greatly impressed Gauss when she first began her research in 1804, wrote a letter about Galois to Guillaume Libri: “They say he will go completely mad, and I fear this is true.” Never the most stable person, he was now verging on full-blooded paranoia.

That month, the authorities arrested nineteen members of the Artillery because of the events at the Louvre and put them on trial for sedition, but the jury acquitted the men. The Artillery held a celebration on 9 May in which about two hundred Republicans assembled for a banquet at the restaurant Vendanges des Bourgogne. Every one of them wanted to see Louis-Philippe overthrown. The novelist Alexandre Dumas, who was present, wrote, “It would be difficult to find in all Paris, two hundred persons more hostile to the government than those to be found reunited at five o'clock in the afternoon in the long hall on the ground floor above the garden.” As the event became more and more riotous, Galois was seen with a glass in one hand and a dagger in the other. The participants interpreted this gesture as a threat to the king, approved wholeheartedly, and ended up dancing in the streets.

The next morning, Galois was arrested at his mother's house—which suggests that there had been a police spy at the banquet—and charged with threatening the king's life. For once he seems to have learned some political sense, because at his trial he admitted everything, with one modification: he claimed that he had proposed a toast to Louis-Philippe, and had gestured with the dagger while adding the words, “if he turns traitor.” He lamented that these vital words had been drowned in the uproar.

Galois made it clear, however, that he
did
expect Louis-Philippe to betray the people of France. When the prosecutor asked whether the accused could “believe this abandonment of legality on the part of the king,” Galois responded, “He will soon turn traitor if he has not done so already.” Pushed further, he left no doubt as to his meaning: “The trend in government can make one suppose that Louis-Philippe will betray one day if he hasn't already.” Despite this, the jury acquitted him. Perhaps they felt as he did.

On 15 June, Galois was at liberty. Three weeks later, the Academy reported on his memoir. Poisson had found it “incomprehensible.” The report itself said this:

We have made every effort to understand Galois's proof. His reasoning is not sufficiently clear, not sufficiently developed, for us to judge its correctness, and we can give no idea of it in this report. The author announces that the proposition which is the special object of this memoir is part of a general theory susceptible of many applications. Perhaps it will transpire that the different parts of a theory are mutually clarifying, are easier to grasp together rather than in isolation. We would then suggest that the author should publish the whole of his work in order to form a definitive opinion. But in the state which the part he has submitted to the Academy now is, we cannot propose to give it approval.

The most unfortunate feature of this report is that it may well have been entirely fair. As the referees pointed out:

[The memoir] does not contain, as [its] title promised, the condition of solvability of equations by radicals; indeed, assuming as true M. Galois's proposition, one could not derive from it any good way of deciding whether a given equation of prime degree is solvable or not by radicals, since one would first have to verify whether this equation is irreducible and next whether any of its roots can be expressed as a rational fraction of two others.

The final sentence here refers to a beautiful criterion for solvability by radicals of equations of prime degree that was the climax of Galois's memoir. It is indeed unclear how this test can be applied to any specific equation, because you need to know the roots before the test can be applied. But without a formula, in what sense can you “know” the roots? As Tignol says, “Galois's theory did not correspond to what was expected; it was too novel to be readily accepted.” The referees wanted some kind of condition on the
coefficients
that determined solubility; Galois gave them a condition on the
roots.
The referees' expectation was unreasonable. No simple criterion based on the coefficients has ever been found, nor is one remotely likely. But hindsight cannot help Galois.

On 14 July, Bastille Day, Galois and his friend Ernest Duchâtelet were at the head of a Republican demonstration. Galois was wearing the uniform of the disbanded Artillery and carrying a knife, several pistols, and a loaded rifle. It was illegal to wear the uniform, and also to be armed. Both men were arrested on the Pont-Neuf, and Galois was charged with the lesser offense of illegally wearing a uniform. They were sent to the jail at Sainte-Pélagie to await trial.

While in jail, Duchâtelet drew a picture on the wall of his cell showing the king's head, labeled as such, lying next to a guillotine. This presumably did not help their cause.

Duchâtelet stood trial first; then it was Galois's turn. On 23 October he was tried and convicted; his appeal was turned down on 3 December. By this time he had spent more than four months in jail. Now he was sentenced to another six months. He worked for a while on his mathematics; then in the cholera epidemic of 1832 he was transferred to a hospital and later put on parole. Along with his freedom he experienced his first and only love affair, with a certain “Stéphanie D,” as his doodles identify her.

From this point on it takes a lot of guesswork to interpret the scanty historical record. For a time, no one knew Stéphanie's surname or what sort of person she was. This mystery added to her romantic image. Galois wrote her full name on one of his manuscripts, but at some later point he scrawled all over it, rendering it illegible. Forensic work by the historian Carlos Infantozzi, who examined the manuscript very carefully, revealed the lady as Stéphanie-Felicie Poterin du Motel. Her father, Jean-Louis Auguste Poterin du Motel, was resident physician at the Sieur Faultrier, where Galois spent the last few months of his life.

We don't know what Jean-Louis thought of the relationship, but it seems unlikely that he approved of a penniless, unemployed, dangerously intense young man with extremist political views and a criminal record paying court to his daughter.

We do know a little about Stéphanie's opinions, but only through some scribbled sentences that Galois presumably copied from her letters. There is much mystery surrounding this interlude, which has a crucial bearing on subsequent events. Apparently, Galois was rejected and took it very badly, but the circumstances cannot be determined. Was it all in his mind—an infatuation that was never reciprocated? Did Stéphanie encourage his advances? Did she then get cold feet? The very characteristics
likely to repel her father might have been distinctly attractive to the daughter.

As far as Galois was concerned, the relationship was certainly serious. In May, he wrote to his close friend Chevalier, “How can I console myself when in one month I have exhausted the greatest source of happiness a man can have?” On the back of one of his papers he made fragmentary copies of two letters from Stéphanie. One begins, “Please let us break up this affair,” which suggests that there was something to break up. But it continues, “and do not think about those things which did not exist and which never would have existed,” giving the contrary impression. The other contains the following sentences: “I have followed your advice and I have thought over what . . . has . . . happened . . . In any case, Sir, be assured there never would have been more. You are assuming wrongly and your regrets have no foundation.”

Whether he imagined the whole thing and his feelings were never reciprocated, or he initially received some form of encouragement only to be subsequently rejected, it looks as though Galois suffered the worst kind of unrequited love. Or was the whole affair perhaps more sinister? Shortly after the breakup with Stéphanie, or what Galois interpreted as a breakup, someone challenged him to a duel. The ostensible reason was that this person objected to Galois's advances toward the young lady, but yet again the circumstances are veiled in mystery.

The standard story was one of political intrigue. Writers like Eric Temple Bell and Louis Kollros tell us that Galois's political opponents found his infatuation with Mlle. du Motel to be the perfect excuse to eliminate their enemy on a trumped-up “affair of honor.” One rather wild suggestion is that Galois was the victim of a police spy.

These theories now seem implausible. Dumas states in his
Memoirs
that Galois was killed by Pescheux D'Herbinville, a fellow Republican whom Dumas described as “a charming young man who made silk-paper cartridges which he would tie up with silk ribbons.” These were an early form of cracker, of the kind now familiar at Christmas. D'Herbinville was something of a hero to the peasantry, having been one of the nineteen Republicans acquitted on charges of conspiring to overthrow the government. Certainly he was not a spy for the police, because Marc Caussidière named all such spies in 1848 when he became chief of police.