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Radical Solutions • Damn interesting

Paris, May 29, 1832. A young French man named Évariste Galois stayed awake all night, pen in hand, and desperately scribbled notes and equations over dozens of sheets. He had only been studying mathematics seriously for a few years, but he had proven to be a real child prodigy. After quickly exhausting his teachers' knowledge, he was committed to his own research, which was extremely forward-looking.

Galois should have been praised and recognized by the scientific community for his work. Above all, it should have been recognized and rewarded by the renowned French Academy of Sciences. But Galois, at least in his own estimation, had received little discharge from the math community. Now he was feverishly scribbling a letter to his best friend, trying to put as many of his latest ideas on paper as possible. After all, Galois had sketched out most of what he thought was possible early in the morning. "You know … that these are not the only issues that I have examined," he wrote. "But I have no time". The 20-year-old Galois fully expected that he would be shot.

Évariste Galois was born on October 25, 1

811 in the town of Bourg-la-Reine, which is now one of the southern suburbs of Paris. Although his parents ran a prestigious boarding school themselves, they sent the young Évariste to Paris shortly before the age of 12 to improve his social opportunities.

Galois & # 39; new school, the Collège royal de Louis-le -Grand was and is prestigious, but in the early 19th century there wasn't just an unprecedented list of alumni – including sizes like Voltaire and Charles-Marie de La Condamine – but also a terribly draconian atmosphere. Meals were poor, facilities failed, the cold was constant, the rats were regular, and the punishments were painful. The students were constantly monitored. The criminal environment – plus homesickness and health problems – took its toll on Galois. In his third year his grades dropped and he made a name for himself as a loner and troublemaker. One teacher described him as "chatter", who "I think has taken on the task of tiring me out". Nevertheless, Galois' school days would soon lead to two major upheavals in the 14-year-old's life – one political and one intellectual.

  Evariste Galois, from his brother's memory
Evariste Galois, pulled from his brother's memory

Galois grew up at the end of France's revolutionary and Napoleonic years, despite attempts by the restored monarchy to combat this inappropriate radicalism , the school had developed a liberal reputation. A new headmaster, Nicolas Berthot, took this as a course correction and set about opening up new rules that go back to the school's hard old Jesuit roots. In response, the students launched a non-violent resistance campaign – when asked to sing hymns, speak in class, or toast the king at dinner, they remained silent. Berthot, a apparently not for half things, but simply expelled the students – over 100 of them. Although Galois was not directly involved, he was horrified by Berthot's reaction. In this atmosphere of injustice and oppression, the young Galois began to change from a merely liberal-minded student to a full-blown anti-authoritarian.

In contrast, Galois' academic break was due to a happy convergence between an academic restructuring and his own academic failure. When his grades dropped, he had to repeat his entire third year. This may not have been welcome news, but there has been an unexpected upward trend. The curriculum for the third year had just been changed and should now introduce the students to further study of the classics in arithmetic and geometry. Galois was about to get to know mathematics. Few blind dates went so well.

Ignited immediately, the 15-year-old devoured entire textbooks on algebra, calculus and geometry. Galois & # 39; other classes that have already been somewhat neglected fell almost entirely off his radar. The faculty soon gave up hope of getting another topic in his brain.

Galois jumped straight into a very intuitive, original approach to tackling big unanswered questions. His concerned teachers suggested that he should methodically outline his problem solving, or at least follow the basics to show his work, but he was not interested in such an elementary trap. Within a few months he had grown out of his academic achievements and had reached the extreme limits of contemporary mathematical knowledge.

Galois was fully aware of his supernatural talent and was keen to attend the École polytechnique – France's leading technical school – as soon as possible. When he approached the minimum age of 16, he immediately registered for the entrance exam. He failed almost immediately. Still, it was nothing to be ashamed not to pass the entrance exam. Many students made their first attempts just to get a feel for how the exam worked – which made sense since it was a quick verbal questioning on a blackboard. Only every third candidate passed his first attempt. Although Galois' failure forced him to stay in Louis-le-Grand, he could at least console himself that this was just a temporary setback.

Another consolation appeared in the form of an enthusiastic new math teacher who understood the scope of Galois ideas and quickly became his mentor. Under the guidance of this mentor, in April 1829, the 17-year-old Galois published a short paper on the repetition of fractures in a prestigious specialist journal. While impressive, for Galois this was just a side project that was overshadowed by breakthroughs in the field of polynomial equations. It was a big deal. Mathematicians had hit a wall with polynomials at the end of the 18th century, and Galois wanted to suggest a way out.

For people who remember algebra classes, the best known type of polynomial is the quadratic equation . This can be useful for calculating areas, solving some accounting tasks, and addressing some physical problems. It is generally written as:

ax 2 + bx + c = 0

"X" is the only unknown value to be solved. There are several ways to find out the possible values ​​for x but a reliable way is to take and and c and put them in a formula that the Spanish mathematician Abraham Bar Hiyya Ha-Nasi discovered around AD 1100. It is aptly known as quadratic formula :

What makes a quadratic equation "quadratic" is that x is quadratic or the power of 2. So a quadratic equation is supposed to be have a degree of 2. An equation in which x is 3 (but nothing more) has a degree of 3, and these are called cubic equations that look like this:

ax 3 + bx 2 + cx + d = 0

The quadratic formula does not help with cubic equations, but a working cubic formula was developed by three in the 16th century discovered by Italian mathematicians one after the other. A student of one of them literally went one step further and literally managed to find an enormously complicated general formula for quarterly equations – those of the 4th degree.

The open question that Galois wanted to address was: is there a formula for solving fifth equations ie those of grade 5? Prominent mathematicians of the time suspected that there was simply no general solution for fifth equations. A mere failure to find such a formula was not evidence. The Italian mathematician Paolo Ruffini almost proved that there was no such general solution for 5th degree equations, which he published in an article in 1799.

The 17-year-old Galois more or less set off where Ruffini had left to prove that there was no general formula for fifth equations. In a broader sense, Galois was interested in an overarching question: What determined whether a general formula for a certain degree exists or not?

Galois' approach to unraveling the matter was amazingly original. He tied equations to several important new conceptual frameworks. He identified groups : sets of entities connected by a certain set of certain properties. Then permutations : all the different ways of ordering the members of a set. And Symmetries : ways in which entities look like other parts of themselves. There was so much new territory here that Galois invented a brand new approach to algebra. The system he developed to describe the subtle internal characteristics of groups – Group Theory – was almost unprecedented, but so versatile that it could capture not only the behavior of numbers, but all types of grouped elements and types Components showed self-similarity. When it came to polynomials, Galois' most important finding was that the solvability of a given equation ultimately has much less to do with the degree of the equation than with its internal properties related to symmetry. The astrophysicist and popular mathematician Mario Livio offers this analogy:

“The classification of equations according to their degree corresponds to the grouping of the wooden blocks in a toy box according to their size. Galois & # 39; classification according to symmetry properties corresponds to the realization that the shape of the blocks – round, square or triangular – was a more important characteristic. “

using his own new group theory to break down a complex polynomial equation into smaller parts and Galois tested those for solvability and definitely showed that quintic equations that are considered to be set would not solve this path . His results gave mathematics an opportunity to determine whether can solve a particular polynomial using a formula. All grade 2, 3, and 4 polynomials are qualified. From grade 5 onwards, however, some and others didn't – so there was no way to find a general solution for everyone.

  An illustration of Paris, 1829, by Giuseppe Canella "title =" An illustration of Paris, 1829, by Giuseppe Canella "style =" width: 476px; "/> </div><figcaption style= A representation of Paris, 1829, by Giuseppe Canella

Galois & # 39; mentor Louis Richard was aware of the boy's ideas Richard decided to help his brilliant protégé, in the spring of 1829 two works at the Academy of Sciences to get the attention of the Academy at that time was an essential step for any aspiring mathematician in France It was the most respected scientific organization in France The country and the adoption of a paper would be a spectacular feather in Galois & # 39 It would not hurt if it was about his second attempt to be admitted to the Polytechnique.

Only members of the academy were able to present new findings on the August body, so Galois had to find a willing sponsor to help him Presenting name papers.Richard turned to the revered Augustin-Louis Cauchy, probably because Cauchy 15 years ago n published two articles on general permutation theories. Cauchy, one of the most prolific mathematicians in history, rarely had time to read and support others' work – but in this case, against all odds, he agreed. In May and June 1829 he submitted two complementary works by Galois to the Academy of Sciences and planned to present a third in January 1830.

But the thrill should be brutally put down. In Galois & # 39; hometown Bourg-la-Reine a new priest had been appointed who immediately clashed with his liberal and warm-hearted mayor – who happened to be Galois & # 39; Father Nicolas-Gabriel Galois was. Determined to save the souls of his congregation from the insidious influences of openness and inadequate monarchism, the priest devised a plan to undermine the older Galois and to force him out of office. The popular Nicolas-Gabriel had a penchant for playful writing at times – he inspired the city dwellers with short coupled rhymes. The priest took note of this peculiarity and began to write his own rhyming couplets in the characteristic style of the mayor, making them mean rather than playful, and signing them with Nicolas-Gabriel's name. The slanderous forgeries spread. Ultimately, the conspiracy turned out to be even more successful than the priest had hoped. In July 1829, Nicolas-Gabriel Galois, devastated by the change of identity and the loss of his good name, committed suicide. He had been mayor of the city for 15 years.

The truth quickly became clear. Amazingly, the priest tried to attend the funeral service – only to flee from a crowd of angry city dwellers and rock volleys. Galois has witnessed the whole scene, and his grief and anger are unimaginable. He had already had reason to be annoyed and opposed to the religious right wing of his country: the political had now become very personal.

And then, with spectacularly bad timing, the next round of entrance exams for the École Polytechnique came up. Galois – at best angry, working under the emotional toll of his father's death, convinced of his own genius, already bitter about the perceived injustice of being rejected for the first time – was unable to deal with a second High stakes exam on a blackboard. He was never at his best when it came to explaining ideas orally, and he had spent the year working on original research rather than preparing for the exam. In addition, Galois 'Examiner was a man who was known to ask extremely simple questions – not to test candidates' knowledge but to measure their response to being asked. Galois faced an examiner who found him completely ignorant and did not do well. An unconfirmed legend has long claimed that the exam ended when the candidate threw the eraser in the examiner's face.

  The Gates of Polytechnique at Galois 'Times
The Gates of Polytechnique at Galois' Times

Whether it was his math or temperament that sank him, Galois became again denied admission to the Polytechnique. He turned his attention to a safeguard option: the École préparatoire (preparatory school), whose main task was to train teachers. She had her own entrance exams – across a range of subjects, including several that weren't maths. After school was over, Galois wrote a letter to the administration asking her to let him apply anyway. His letter suggests that recent events have not affected his high spirits. The examiners allowed him to continue, but the science judge in particular was less than impressed and dryly noted that Galois "knows absolutely nothing":

"I was told that this student had a talent for mathematics; This surprises me very much, because based on his examination I think that he has very little intelligence or is at least so well hidden that I could not discover it [.]. "

Nevertheless, Galois' grades for math were high enough – and his answers were clear enough for a change – that he was admitted to the École préparatoire in November 1829.

Galois had a way forward. His young math career, however, got into turbulence shortly after the start because a Norwegian mathematician named Niels Henrik Abel died prematurely. Kind, but unhappy, Abel had unsuccessfully called for the attention of French mathematicians for years. He had even traveled to Paris to seek recognition and had sent Augustin-Louis Cauchy one of his own papers. Unfortunately, Cauchy hadn't been able to read it. Even Abel's defeated letter to Cauchy a few years later asking him to return his manuscript remained unconfirmed. Then, only 26 years old, Abel succumbed to tuberculosis.

When the news of Abel's death reached the academy in June 1829, Cauchy tried to defend himself – clumsily and not convincingly – because he had neglected the young Norwegian. He hurried to read Abel's three-year manuscript and present it to the Academy, just a few weeks after his second presentation on Galois' work. It was immediately apparent that Abel had drawn many of the same conclusions that Galois had and had done earlier – by providing evidence that there could be no general formula for polynomials with an individual degree of 5 or higher. This fall, the Bulletin de Férussac published both an obituary for Abel and a detailed analysis of one of his earlier works. For connoisseurs, this underscored the fact that while Galois had made some phenomenal breakthroughs, Abel had first gained some of the same insights. Their methods were completely different, but when the conclusions were reached, Galois – nine years younger – was demonstrably anticipated. Not surprisingly, Cauchy's planned third presentation of Galois' research did not go as planned in 1830 – although the normally irascible Galois, as science historian René Taton emphasizes, did not complain about what indicates that he is doing his work voluntarily withdrew due to the unintentional overlap with Abel's findings.

  Niels Henrik Abel "title =" Niels Henrik Abel "style =" width: 403.2px; "/> </div><figcaption style= Niels Henrik Abel

That is, Galois still could not save the considerable original parts of Abel had dealt him with group theory. While keeping up with his regular schoolwork, Galois put his research together into a manuscript and submitted it to the first prize of the Academy of Mathematics at the end of February 1830. Despite some setbacks, Galois made a respectable place in the mathematical world. If he won the prize, this would consolidate his status as the leading light of his generation.

On June 28, 1830, the results of the main prize were announced and Galois' name was nowhere to be seen, and the jury decided to award the prize to two mathematicians for their separate work on elliptic functions, one of which was the German Carl Gustav Jacob Jacobi; the other was posthumously Abel. A clause in the competition rules provided that di e jury was able to award the prize for every paper published in the previous year, not only for official entries. The academy has thus made up for the memory of the young man whom she had ignored in life.

However, another young man was available to feel ignored in Abel's place. For mathematical reasons, Galois could not have contradicted the winners; he identified strongly with Abel, and Jacobi appears to have been one of the few other mathematicians he respected. What annoyed Galois was that his manuscript wasn't even listed as an official competition entry. In fact, it could not be found when he asked for his paper to be returned. This turned out to be a simple bad luck. The Academy's permanent secretary, Joseph Fourier, had served on the jury for the main prize and had taken Galois' manuscript home for reading. Unfortunately, he died in mid-May; Galois & # 39; manuscript was lost somewhere in the jumble of his papers and never reappeared.

Galois already had a considerable chip on one shoulder and was developing quickly on the other. He was increasingly convinced that the academy was deliberately avoiding him. Perhaps, he thought, they were too incompetent to understand his dramatically forward-looking mathematical ideas and too cumbersome to endorse his dramatically forward-looking political beliefs. Galois' resentment could only be heightened when he passed an exam on differential and integral calculus at the end of July 1830 and only took the fourth of eight students – a shockingly low rank for someone who had published articles in the same journal as Cauchy . In addition, he was still irritated by the fact that he was twice denied the opportunity to study at the Polytechnique, both because of the quality of their math classes and because of its opportunities on the path to anti-monarchist political activism. The second of these was due to be fully exhibited in the summer of 1830, when the tectonic plates of French politics shifted.

  King Charles X of France "title =" King Charles X of France "style =" width: 351.4px; "/> </div><figcaption style= King Charles X of France

King Charles X was, to say the least, extremely conservative. The elected ministers were even more so. All of them remained traumatized by the French Revolution of 1789 Mass beheading of her colleagues – including quite a few members of Charles' family – as a result, they sought to eliminate the ridiculous ideas of freedom, equality, and fraternity that the revolution had propagated.

As the results of an election in In the summer of 1830, when the liberal left favored, the government's ossified leaders returned their favor with a particularly bony decision: a series of decrees dissolved the newly elected legislature before it even met, and reduced the size of the assembly by 170 years in favor of the wealthy, censored opposition publications drastically restricted and in September ne over-election required. The government must have expected the liberals to be unhappy, but when the prime minister reported that he had received regular visits from the Virgin Mary and she told him that everything was fine, they went on anyway. What they did not take into account was that essentially the abolition of the printing industry would also make typographers unhappy. After their livelihood was gone, they took to the streets, followed by their working-class colleagues and then the angry middle class.

Riots became the lightning-fast July Revolution, which was quickly mythologized as Trois Glorieuses the "Three Glorious Days", on which all classes had come together to eliminate a tyrannical regime. Members of the democratically organized National Guard, which had been founded during the revolution but had been abolished by Charles, pulled their old uniforms out of the closet and spontaneously reformed the militia. They became known as the heroes of the hour – along with the Polytechnique students who also jumped into the fight.

Her colleagues at the École préparatoire, however, were excluded. Or more specifically, when the battle raged on the streets of Paris, the school administrators locked the doors to keep their students from participating. The new headmaster, Joseph-Daniel Guigniault, twice threatened to call in the military to maintain order among his students. (This might not have worked. The royalist army was a bit busy trying to control the streets while residents of buildings upstairs showered them with furniture, including the occasional piano.) Guigniault made matters worse by giving condescending comments the revolutionaries expressed. As it was, he could at least say that he was bringing his students to safety. The only one of them who appeared to be at risk of injury during the Trois Glorieuses was Galois, who, according to most reports, was so keen to engage in the riots that he tried to climb over the schoolyard wall.

His participation was unnecessary. Charles X faced the facts and went into exile, while France proclaimed a new constitutional monarchy under a new king, Louis-Philippe. The National Guard became an important part of the images surrounding the change. Their blue-white-red uniforms matched the newly restored tri-color flag that was waving everywhere.

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Not everyone liked the new regime: it was a compromise for which it was not far enough there was an abomination to the left and right. Paris, which set the tone for the whole country, remained a cramped medieval city with an underpaid, malnourished working class. There were ongoing uprisings of both unrest and disease. For Galois, the revolution also meant loss his best source of support at the Academy of Sciences: Cauchy, a right-wing Catholic and die-hard follower of the old monarchy, had left the country instead of taking the required oath of allegiance to the new king.

Galois' own path was of course diametrically opposite He joined the newly founded Société des a an mis du peuple a republican group that was so radical that it was e was completely banned in October 1830, whereupon it became a (theoretically) secret society. While the Polytechnique was honored for its active liberalism, Galois was stuck behind closed doors with its despised director Guigniault – his best chance of fighting the establishment was suppressed by the establishment. For Galois, the director's behavior was annoyingly hypocritical. At first, Guigniault had mocked and sacked the revolutionaries – only to show off in the tricolor of the same revolutionaries at the end of the battle. Guigniault also made the blank gesture to change the name of the school itself back to the Napoleon era name of École normal while still insisting that "good students should have no interest in politics".

Galois exchanged an increasingly irritable series of letters with school administrators and criticized Guignault's reaction to the riots and the revolution. When Galois published an "anonymous" letter in a prominent, educational magazine that brought the matter to the press, and hence to the public, Guigniault simply expelled it.

When the school was off the table, nothing prevented Galois from getting involved in the fame of the National Guard. The young mathematician entered a Guard artillery battery known to be a hotbed of Republican sentiment. The Republicans had a good chance to demonstrate their continued dissatisfaction with current affairs. Several of Charles X's ministers were on trial, and the consensus on the left was that anything but death sentences would be an acquittal – and another uprising.

Galois, who was wearing a newly purchased uniform, would have been among the artillerymen stationed at the Louvre on December 21, 1830, when the sentences were announced. Ministers were sentenced to life imprisonment rather than death. The situation was precarious for days and several members of the artillery were arrested for seditious behavior. König Louis-Philippe, kein Dummkopf, erkannte, dass es nicht der klügste Weg sein könnte, ein paar radikale Republikaner für Kanonen verantwortlich zu machen. Am 31. Dezember löste er die Artillerieeinheiten der Nationalgarde auf und verbot das Tragen ihrer Uniformen. Galois &#39;Militärkarriere hatte höchstens drei Wochen gedauert.

Galois war nicht an alle Institutionen gebunden und zappelte, aber die Akademie der Wissenschaften erweiterte einen Olivenzweig. Einer seiner angesehensten Mathematiker, Siméon Denis Poisson, der sich wenige Monate zuvor den Galerieraum mit Galois und Cauchy geteilt hatte, bat Galois, ihnen ein neues Papier zu schicken. Galois schrieb ein neues Manuskript – wahrscheinlich eine Nachbildung seiner verlorenen Einreichung beim Hauptpreis – und reichte es am 17. Januar 1831 ein. Nachdem zwei Monate ohne ein Wort vergangen waren, schrieb Galois einen kurzen Brief an den Präsidenten der Akademie. was darauf hindeutet, dass Poissons Verspätung etwas faul war. Galois wischte seine Erfahrung gezielt auf und krönte sie mit einem direkten Vorwurf von Hintergedanken:

„… der Prüfungsausschuss entschied a priori dass ich dieses Problem nicht hätte lösen können, zum einen, weil ich Galois hieß, und außerdem, weil ich Student war. Und das Komitee hat mein Papier verlegt. Und mir wurde gesagt, dass mein Papier verlegt wurde. Diese Lektion hätte mir bis heute genügen sollen […]meine Forschung hat mehr oder weniger das gleiche Schicksal erlebt […] Wird die Analogie bis zum Ende verfolgt? Bitte […] fordern die Herren Lacroix und Poisson auf, zu erklären, ob sie mein Papier verlegt haben oder ob sie beabsichtigen, der Akademie darüber Bericht zu erstatten. “

Unnötig zu erwähnen, dass dies nicht gut war Strategie für den Aufbau einer Karriere. Andere Mathematiker schüttelten bei Galois &#39;Verhalten den Kopf. Es gab Gerüchte, dass er den Verstand verlieren würde.

Am 16. April 1831 wurden neunzehn von Galois &#39;Republikanerkollegen wegen Volksverhetzung freigesprochen. Überglücklich trugen ihre Verbündeten sie triumphierend nach Hause und beschlossen schnell, zu ihren Ehren ein feierliches Bankett abzuhalten. Zu dieser Zeit gab es in Frankreich kein Recht auf freie Vereinigung, aber selbst die diktatorischsten Regime wussten es besser, als zwischen die Franzosen und ein gutes Essen zu kommen. Ein Bankett war daher eine der wenigen legalen Möglichkeiten, eine große Gruppe von Gleichgesinnten zusammenzubringen, um Ideen auszutauschen (oder gemeinsam zu entscheiden, eine Büste des Königs oder beides zu verteidigen). Der Veranstaltungsort war ein Restaurant, das jedoch nicht für seine Küche bekannt war. Es war vielmehr der größte Raum in Paris, den eine Gruppe problemlos buchen konnte. Ein Jahr zuvor hatte es ein liberales Bankett veranstaltet, das den Grundstein für das Ende der Herrschaft Karls X. gelegt hatte. Jetzt freute sich Galois auf ein ähnliches Ereignis – und in Vorbereitung besuchte er einen örtlichen Messermacher und bestellte sehr eifrig einen „Faltdolch“.

 Alexandre Dumas ca. 1831 "title =" Alexandre Dumas ca. 1831 "style =" width: 364px; "/> </div><figcaption style= Alexandre Dumas ca. 1831

Einer der anderen Revolutionäre, die am Bankett teilnahmen, war Alexandre Dumas. Der zukünftige Schöpfer der drei Musketiere und des Grafen von Monte Cristo Dumas war bereits ein bekannter Dramatiker, nur eine Woche zuvor war eines seiner Stücke zur Theater-Sensation des Jahres geworden. Er war auch ein Artillerist in derselben Nationalgarde-Batterie wie Galois und viele der neunzehn freigesprochenen Republikaner event much later in his MemoirsDumas wrote that “it would have been difficult to find two hundred people more hostile to the government in all of Paris”. Even so, Galois eventually managed to find a way to stand out. As the number of emptied champagne bottles grew and the banqueters began to forego their promise not to make any unapproved toasts, Dumas suddenly became aware that

“[a]n extremely animated scene was taking place fifteen or twenty places down from me. A young man, holdin g his raised glass and an open dagger in the same hand, was striving to make himself heard. It was Évariste Galois, […] one of the most ardent republicans.”

Ardent and, it must be said, drunk. Galois later admitted to a friend that if he’d been sober, he would never have behaved as he did. Dumas could not hear over the immediate roar of the crowd, but he did work out that the words “Louis-Philippe” had been uttered, that Galois’s open dagger was unambiguous, and that there were limits to his own radicalism. Catching each other’s eye, Dumas and his neighbour hopped out a window and skedaddled—which, from a banquet on the ground floor, was admittedly easier than it might have been.

What Galois had done was to openly call for regicide—a crime so unthinkable that the criminal code classified it as being as unnatural as parricide. Dumas knew well that plays could be banned simply for alluding to it. This action was too radical even for the radicals: republicans did not want their movement reminding the nation of the constant guillotining that had marked the 1789 Revolution. But Galois’s fiery outburst was soon all over the newspapers, causing considerable embarrassment. The morning after the banquet, Galois was arrested at his mother’s house.

Galois’s supporters latched onto the fact that the threat had technically been conditional—“To Louis-Philippe, should he betray [his oath to uphold the constitution]”—with the crowd’s noise burying the second half. Astonishingly, at his own trial on 15 June 1831, Galois did not take advantage of this escape rope. Instead, he doubled down in every possible way. Not only did he not blame drunkenness, he insisted that he had intended what he’d said, conditionals be damned. In open court, with a gobsmacked audience looking on, Galois confirmed that it had been an assassination threat, that he had not just been expressing his personal opinion, and that he was attempting to goad others into making attempts on the king’s life. Indeed, Galois went on, in his opinion Louis-Philippe probably already had betrayed his oath. Around this point, the judge cut the interrogation short—either to keep a lid on Galois’s outrageous sentiments, or simply to stop him from digging himself even more deeply into a hole.

King Louis-Philippe I" title="King Louis-Philippe I" style="width:340px;"/></div><figcaption style=King Louis-Philippe I

Surprisingly, Galois was acquitted. Dumas’s explanation was that the jurors either agreed with Galois, or simply thought he was beyond sanity. Most other sources agree that the judge and jury took pity on him because he was so young—still only 19.

Against the odds, Galois had slipped out of the political noose. Not only that, but the publicity dislodged the apparent impasse at the Academy of Sciences. On the day his trial started, the newspaper Le Globe published a lengthy letter detailing Galois’s experiences with the Academy. Specifically, the publication argued that Poisson—who had requested the new paper from Galois in the first place had really dropped the ball. Perhaps encouraged to hurry, Poisson and his co-referee submitted their report a few weeks after Galois’s acquittal, on 4 July 1831, then presented it publicly to the Academy a week later.

A months-long wait for a response to a manuscript was not actually anything unusual. The Academy was swamped with submissions. Mathematical historian Caroline Ehrhardt reports that two-thirds of the papers submitted to the Academy never received a report at all—and of those that did, few got more than a few terse sentences. Galois’s complaint of neglect in March was baseless. Fortunately, the Academy did not dismiss him out-of-hand for his impudence—which is just as well, for this manuscript contained what astrophysicist/author Mario Livio later called “one of the most imaginative breakthroughs in the history of algebra”—the core of his invention of group theory.

To their credit, the reviewers produced a lengthy, thorough, and detailed report on Galois’s manuscript. It was clear that they saw and appreciated the links between Galois’s ideas and Abel’s. But their evaluation was not what Galois had hoped for: overall, Poisson and Lacroix confessed that they were baffled. The math itself was not the problem; rather, they were not always able to follow the argumentation.

“[…] we have made every effort to understand Mr. Galois’s demonstration. His reasoning is neither clear enough not developed enough for us to have been able to judge its exactness […] The author announces that this proposition […] is part of a general theory liable of many other applications. It is often the case that the different parts of a theory, by mutually illuminating one another, are easier to grasp when taken together rather than in isolation. We can therefore wait for the author to have published his work in its entirety before coming to a definitive opinion […]”

Several contextual factors worked against Galois. He still did not have training in the conventions of writing out mathematical discoveries—a skill he could have picked up at the Polytechnique. Added to this, his work was pure mathematics; this would have resonated with the absent Cauchy, but for the other members of the Academy at the time, what was important was applied mathematics. Plus, algebra was still thought of as a tool rather than a subfield of mathematics in and of itself; Poisson and Lacroix would have been judging Galois’s manuscript through the lens of mathematical analysis—what we now call differential calculus—and looking for its practical potential. Galois’s ideas likely struck them as a pointless excursion into terra incognita. With the benefit of hindsight, Mario Livio slams the reviewers or reacting so lukewarmly to Galois’s manuscript. However, in modern academic parlance, the referees’ report is much more “revise and resubmit” than it is an outright rejection. More than one modern mathematician has admitted they would likely have made the same decision in response to the manuscript as it was.

None of this counted for Galois. Furious, he became convinced once and for all that the Academy was out to get him. And as usual, mathematical disappointment dovetailed with political trouble. Only days later, on 14 July 1831, the police set out very early for Galois’s home. Knowing him to be a troublemaker, they were hoping to nab him in a pre-emptive roundup of republican rowdies known to have plans for commemorating the fall of the Bastille. No Évariste Galois was found—because he had already left home that morning. The police eventually caught up with him and his friend Ernest Duchâtelet. The young men were going to an illegal march, with Galois illegally wearing his National Guard uniform—not to mention carrying a rifle, a couple of pistols, and a dagger. Placed under arrest, the pair complied calmly enough that the officers didn’t think to confiscate their prisoners’ rifles until they were halfway to the police station.

Sainte-Pélagie prison" title="Sainte-Pélagie prison" style="width:476px;"/></div><figcaption style=Sainte-Pélagie prison

Galois was thrown in jail to await trial, but his stay in Sainte-Pélagie prison got off to an eventful start. As the anniversary of the July Revolution approached, the republican prisoners got more and more excited. On 28 July 1831, they happily spent the day yelling anti-royalist slogans, insulting passers-by, and throwing things out of windows. Towards evening, a loaf of bread hit a woman on the head so hard that she was left bruised and bleeding (a testament to the sort of food the inmates were given). Half an hour later, one of Galois’s cellmates began to caterwaul La Marseillaise as he undressed for bed in front of the window. Outside, an exasperated local citize n sick of the ruckus let off a round of buckshot that hit the man clear in the face. Inside, panic erupted. The guards on the scene managed to regain control only by throwing three of the inmates—including both Galois and Monsieur La Marseillaise—into the dungeon. This action itself nearly caused a riot, and rumours flew that the shot had been fired by a prison employee at the governor’s order.

Galois was only in the dungeon for three days, but it was three months before he came to trial. This time around, Galois backed down and feebly claimed that he hadn’t realised that wearing his uniform was illegal. The judge was unconvinced. Probably feeling that Galois’s previous acquittal had failed to teach him a much-needed lesson, he sentenced the young man to a further six months in jail. This sentence was disproportionately long, especially considering that Duchâtelet—who had sketched a guillotine on his cell wall and added verses that threatened Louis-Philippe—would have to do only three.

Imprisonment did not suit Galois. Still, the dampness and discipline of Sainte-Pélagie was probably more comfortable than boarding school had been. Prisoners could receive visitors every Thursday and Sunday, chat among themselves, walk about. Nevertheless, Galois soon fell into despair. Some of his co-inmates began referring to him as “an old man of twenty”. His sister Nathalie, who visited constantly, thought the same, finding him as hollow-eyed as someone 30 years older. Galois had already proven at the banquet in May that he could drink beyond his limits—an ability that he now had cause to demonstrate repeatedly. Fellow inmate François-Vincent Raspail, the president of the banned Société des amis du peupledescribes how prisoners taunted Galois by calling him a water-drinker. In response, Galois began downing entire bottles of brandy in one go, with predictable consequences. His mental state deteriorated, reopening his grief over the death of his father. He may even have attempted suicide, prevented only by Raspail’s intervention.

But Galois also spent a great deal of time in prison pacing and doing mathematics in his head. He even worked out a plan to bypass the Academy and publish two manuscripts privately with the help of his friend Auguste Chevalier. One of these papers—”On the Conditions for Solubility of Equations by Radicals”—was a revision of the paper that the Academy had balked at; the other was a new work on “Primitive Equations Solvable by Radicals”, which Galois (now particularly adept at anything having to do with radicals) seems to have undertaken starting in the summer of 1830. Before he got out of prison, he had put considerable energy into drafting a preface to the two works. This was a five-page manifesto dripping with vitriol and heavy-handed sarcasm, along with self-congratulatory sentiments about his own independence from the Academy and “how lowly I esteem my adversaries”. Among other things, he was still stewing over his lost paper.

“I must mention how manuscripts most often end up lost in the folders of Messrs. the members of the Institute, though truly I can’t conceive of such carelessness on the part of men who have the death of Abel on their conscience.”

He snarkily outlines how he could have clogged his manuscript with useless details to make it easier to understand and appease his reviewers’ request for additional information. He witheringly suggests that the examiners in charge of testing candidates for the Polytechnique deserve to be members of the Academy, given that “they certainly have no place in posterity”. He presents himself as a martyr figure, “knowingly exposing myself to the mockery of dunces”, and even obliquely lambastes the Academy for favouring applied over pure mathematics. Clearly, prison had done nothing to temper either Galois’s simmering rage or his self-esteem—nor, for that matter, his politics.

In spite of this, Galois was released from prison in March, before he had completed either his manuscripts or his sentence. The authorities sent him to a small halfway house run by a man named Denis Faultrier to recover his broken health. The transfer dramatically changed things. Back in prison, Galois had told Raspail that the one thing he truly lacked was someone he could love “with his heart alone”. Now that lack was about to be rectified—but it would not end well.

&#39;Évariste&#39; superimposed on &#39;Stéphanie&#39; at top left, and E and S monograms bottom left" title="&#39;Évariste&#39; superimposed on &#39;Stéphanie&#39; at top left, and E and S monograms bottom left" style="width:476px;"/></div><figcaption style=&#39;Évariste&#39; superimposed on &#39;Stéphanie&#39; at top left, and E and S monograms bottom left

After his transfer, Galois became acquainted with a woman named Stéphanie, and reacted to her rather as he had to his first encounter with mathematics. It turned out that Galois fell in love as wildly as he did everything else. Even as he went over old papers, presumably still working on his pair of manuscripts, he doodled Stéphanie’s name in the margins. On one page, he superimposed the names “Évariste” and “Stéphanie”; on another, he sketched out some rather elegant monograms combining the initials “E” and “S”. Clearly, Galois had it bad.

The object of his fixation was almost certainly Stéphanie Félicité Poterine du Motel, a cousin of Denis Faultrier’s. Nearly 20 years old, she did not seem to reciprocate Galois’s affections. On 14 May 1832, she wrote to hi m:

“Let us please make an end of this matter. I do not have sufficient wit to follow a correspondence of this type, but I will try to have enough to converse with you as I did before anything happened… no longer think about things that could not exist and that never will exist.”

Another excerpt was even more crushing:

“…be persuaded, Sir, it would doubtless never have been more; you are assuming wrongly and your regrets are ill-founded.”

She ruled out even the possibility of a friendship, telling Galois that he was wrong to believe that men and women could ever be true friends.

A heartbroken Galois swirled downwards into an emotional whirlpool. His friend Auguste Chevalier grew alarmed, thinking that Galois was revelling in his own misery; Chevalier accused him of “being drunk on the putrefied muck of a rotten world infecting [his] heart, [his] mind, and [his] hands.” In a consoling letter, Chevalier apparently urged him, not for the first time, to seek refuge in religion. On 25 May 1832, Galois wrote back.

“[H]ow can one destroy the traces of emotions as violent as those I have passed through? How can one console oneself for having exhausted in one month the greatest source of bliss available to man, to have exhausted it without bliss, without hope, certain one has drained it dry for life? […] for your part, you feel obliged to do your best to convert me. But it is my duty to warn you, as I’ve done a hundred times, that your efforts are in vain. […] I’m disenchanted with everything, even the love of glory.”

Still, this all-encompassing existential loathing did not stop Galois from looking forward to reuniting with Chevalier in a few days. But the trip never happened. What actually happened in the next four days is unknown, but the night of the 29th found Galois desperately dashing off additional letters instead of sleeping. One was addressed to a pair of his republican friends:

“I have been challenged [to a duel] by two patriots…it was impossible for me to refuse.”

Officially, duelling was illegal at the time, but Galois was right. Like most of the rest of Europe, plus the New World (hello Mr. Hamilton, Mr. Burr), France was in the grip of a duelling frenzy. If anything, the historical reality makes the way the Three Musketeers duel at the drop of a hankie seem downright restrained. More than 200 people died in duels in France between 1826 and 1834 alone. Alexandre Dumas knew this firsthand: he had won his first duel when he was 22 (though his trousers had fallen down in the process). Politicians blew out the brains of other politicians, journalists from opposing sides of the political divide killed each other with pistol or sword, authors and literary critics could be found on the fighting fields, young men took aim at each other over women, professional killers went around challenging people for jostling them in the street. In Bordeaux, a duelling club was founded whose members swore to only ever fight to the death: the association lasted three years, until a newcomer systematically killed all twelve surviving members. In 1837, two law professors took up swords over whether a certain passage in the 6th-century Digest of Justinian should end with a colon or a semicolon. (Professor Semicolon won by sticking three inches of steel into Professor Colon’s arm.) The poet Lamartine, who ended up in a duel in 1825 after one of his poems included a mildly uncomplimentary line about Italy, summed up the ethos by remarking that “It takes more courage to refuse one duel than to fight ten”.

In this atmosphere, there was never any question as to whether Galois would accept the challenge: any young man refusing would forever be branded a coward, and a fellow of above-average hotheadedness was unlikely to consider the possibility anyway. But Galois did not think he was fighting a duel for a particularly glorious cause. In a letter he addressed to “all republicans”, he wrote:

A page from Galois&#39;s memoir/article, with "I don&#39;t have time" scrawled in the margin" title="A page from Galois&#39;s memoir/article, with "I don&#39;t have time" scrawled in the margin" style="width:350px;"/></div><figcaption style=A page from Galois&#39;s memoir/article, with "I don&#39;t have time" scrawled in the margin

“I die the victim of a shameless flirt and her two dupes. My life is being extinguished in a miserable bit of gossip. Oh! Why die for something so small—die for something so contemptible!”

He closed by describing himself in Latin: “Nitens lux, horrenda procella, tenebris æternis involuta”, which translates to “a brilliant light, swallowed by an awful tempest, wrapped in eternal darkness.” Id est, the turducken of melodrama.

As Samuel Johnson said, the certainty of being hanged in the morning concentrates the mind wonderfully. The possibility of being shot did precisely the same to Galois. A few days earlier, he had been doubting his ability to ever do math again; now he began a letter to Auguste Chevalier. “My dear Friend,” he wrote, “I’ve done several new things in analysis.” Then, in small, neat handwriting, over seven pages, he frantically set about getting down as many of his original ideas as he could. These provided enough material, Galois claimed, for three manuscripts. The first was the one the Academy had shrugged at; Galois insisted he stood by it, with only a small number of corrections. He then sketched out the most important parts of the other two manuscripts that he envisioned. Only at the end of the letter did some sense of finality come into play:

“You know, my dear Auguste, that these are not the only topics that I have explored […] But I don’t have time and my ideas are not yet very well developed in this immense area.”

He concluded by asking Chevalier to publish the entire text of the letter itself in the Revue encyclopédiqueand to publicly ask mathematicians Carl Gustav Jacob Jacobi or Carl Friedrich Gauss, or both, “to give their opinion, not on the truth, but on the importance of t h ese theorems”. He also defended himself against the risk of again being accused of cursory mathematical argumentation:

“In my life, I’ve often taken the risk of advancing propositions about which I wasn’t certain. But everything I’ve written here has been in my mind for almost a year, and it is too much in my interest not to make mistakes for anyone to suspect that I’ve here formulated theorems for which I don’t have complete demonstrations.”

And yet, on the night of the 29th, Galois also revisited the copy of the manuscript that the Academy had returned to him. At one point, he scrawled in a margin, “There is something to be completed in this proof. I don’t have time”—a poignant admission that reviewers’ calls for greater clarity had not been entirely undue.

At the scheduled time, on the outskirts of Paris, with a handful of witnesses, Galois and his challenger chose their pistols. It is possible that only one of them was loaded—an uncommon but not unknown option that added an element of Russian roulette to the mix. The two men walked to twenty-five paces, turned to face one another, and shot. Hit in the abdomen, Galois dropped to the ground. Someone—possibly an onlooker, possibly a passerby—transported him to the nearby Cochin Hospital.

Conscious, but badly wounded, Galois lay in a room with four or five other patients. The only family member to have heard about the duel was his younger brother Alfred, who raced to the hospital and soon became despondent. Galois’s wound was not only severe, but also oddly positioned. It was as if he had not tried to minimise his chances of being shot; he might not even have turned to the side as he faced his foe as was customary. The injury was extensive, and already infected. The hospital surgeon and both brothers all knew that there was little to be done. Alfred was in tears, but Galois stoically instructed his brother, “Don’t cry. I need all of my courage to die at the age of twenty.” He characteristically rejected an attempt to have a priest attend to him, then, at 10:00 a.m. on 31 May 1832, Évariste Galois died in his brother’s arms.

Pistol duel in the XIX century by Bauce and Rouget" title="Pistol duel in the XIX century by Bauce and Rouget" style="width:476px;"/></div><figcaption style=Pistol duel in the XIX century by Bauce and Rouget

Legend has long held that a large crowd attended Galois’s funeral that Saturday. The Prefect of Police claimed much later in his memoirs that some two to three thousand people were there. However, on the day itself, the same man’s report to the Minister of the Interior stated that the actual number was around 150, mostly other republicans. They set off at 11:30 a.m. on 2 June to accompany Évariste Galois from Cochin Hospital to the Montparnasse graveyard. After some good old fiery political speeches, Galois’s body became the 18th of 21 to be placed in a common grave—after which the attendees pas sed the hat around to pay for the funeral costs.

Perhaps more than 150 had intended to be there. Paris was (again) on the verge of insurrection, and the republicans were looking for an excuse to gather and start a riot. Galois’s funeral might have fit the bill perfectly—except that the day before saw the death of General Jean Maximilien Lamarque. Lamarque was a prominent advocate for the wretched, miserable poor—who, despite what certain modern stage productions suggest, were not inclined to be tuneful about their state. The republicans quickly realised that Lamarque’s funeral would be the higher-profile event, and rescheduled the rebellion. Thus, Évariste Galois was once again shoved aside and left out, even in death. His duel deprived him by just a few days of the opportunity to die a glorious republican martyr in the ill-fated 1832 June Rebellion. His friends of the Société des amis du peuple would appear thinly disguised in one of the best-selling novels of all time, but there are no mathematicians among the students in Victor Hugo’s Les Misérables. Adding insult to fatal injury, among those who fought heroically on the June barricades, and survived, was a certain Étienne-François Pecheux d’Herbenville—who most likely fired the bullet that prevented Galois from ever getting to join a revolution.

The identity of Galois’s opponent has long been one of the many mysteries surrounding the duel. Alexandre Dumas specifically named him as d’Herbenville, but Dumas is not infallible, and other evidence seemed to suggest otherwise. Two days after the duel, a newspaper from the town of Lyon, Le Précurseurreported on the duel, describing the victor as “one of [Galois’s] old friends, like him a very young man, like him a member of the Société des amis du peuple, and who had … also been a figure in a political trial”. The article attributed the duel to a romantic argument, and identified Galois’s foe by the initials “L.D.”

The Précurseur article is full of small inaccuracies, but even taking these into account, it is tricky to make their depiction match Dumas’s identification. On the one hand, d’Herbenville—a charming young man who liked to wrap his cartridges in pink silk paper—was indeed a republican. In fact, he was one of the nineteen republicans whose acquittal was celebrated with the notorious banquet that ended with Galois’s drunken oath and Dumas’s autodefenestration. But beyond this there was no evidence of any connection between the two, let alone of old friendship. Given that Galois did not have many friends, other candidates have been sought. Mario Livio, among others, has argued for Galois’s friend and fellow prisoner Ernest Duchâtelet, who has the advantage of both being a friend and having at least one of the right initials.

Recent discoveries, however, make it almost certain that Dumas was right after all. One key piece of evidence is a copy of France’s 1791 revolutionary constitution. Among its owners was a Swiss medical student named Larguier, who wrote on it, “This manuscript was given to me by Gallois killed in a duel by Pécheux d’Herbinville”. Meanwhile, Olivier Courcelle’s research has discovered that d’Herbenville’s names were rarely spelt the same way twice in the press. Among the variations were forms such as “Lepescheux” and “Dherbinville”—these would give us the “L. D.” reported by Le Précurseur. Most tellingly of all, recent close analysis of Galois’s manuscripts has shown that, though crossed out, d’Herbenville’s name appears in Galois’s papers, proving that the two were indeed connected somehow. How, and as of when, is unknown, but Courcelle has discovered that d’Herbenville took classes at Louis-le-Grand at the same time that Galois resided at the school. Moreover, d’Herbenville studied mathematics at school before turning his attention elsewhere, and eventually became an engineer. In d’Herbenville, Galois would have found a radical republican he could talk shop with.

But knowing the identity of Galois’s killer does not solve the greater mystery of why the duel was fought at all, nor explain its oddities. The confusion over who took Galois to the hospital suggests that he went into the duel without any witnesses of his own—a baffling choice given that one of the witnesses’ duties was to ensure prompt medical care for the wounded. Between this and Galois’s uncommon injury, several commentators have suggested that Galois went into the duel intending to die. One theory even holds that he was sacrificing himself for his political cause. But this is difficult to reconcile with his open letter to “all republicans”, in which he begs “my friends the patriots not to reproach me for dying for something other than the country”. Instead, he attributes the death he foresees for himself to “a miserable bit of gossip”. Moreover, one of Galois’s last-minute letters suggests that the challenger(s) “charged me on my honour not to inform any patriot” (italics in original). Galois’s devastated brother Alfred, meanwhile, thought Galois had been shot by secret police agents acting on behalf of the king. This is profoundly unlikely. Even if Louis-Philippe’s regime had been prone to assassinating people—which it does not appear to have been—Galois was simply not important enough to warrant eliminating, particularly in such a convoluted manner.

A French postage stamp commemorating Évariste Galois" title="A French postage stamp commemorating Évariste Galois" style="width:476px;"/></div><figcaption style=A French postage stamp commemorating Évariste Galois

More likely, it was simply a ‘matter of honour’—and the generally accepted explanation is that the honour in question was Stéphanie du Motel’s. Most commentators identify the ‘shameless flirt’ of Galois’s letter as Stéphanie, though whether she deserved that appellation is entirely unknowable—as is whether the challengers were in any way ‘dupes’. Quite possibly, Galois had simply importuned her so much that she had no option but to ask for assistance. Certainly, Galois indicates that he insulted his challengers, and did so to their faces: he wrote in his open letter that he “repent[s] having spoken a fateful truth to men who were so poorly prepared to hear it coolly”. With Galois’s temper, it seems unlikely that any of it was done coolly. As research continues into Galois’s papers and what he crossed out where, answers may yet come to light. Recent findings have confirmed that there was a police report about the duel; perhaps it will be found in an archive somewhere.

Alfred Galois and Auguste Chevalier were left with the weighty task of doing justice to Évariste Galois’s mathematical legacy, and it took time. It was not until 1843 that Galois’s luck changed, when his work reached French mathematician (and member of the Academy) Joseph Liouville. As he worked through Galois’s papers, Liouville noted their brevity and relative opacity, but realised that what Galois had proposed years before was both mathematically rigorous and far ahead of its time. In 1846, Liouville published Galois’s “ingenious and profound” work in his own internationally renowned journal, and thus it reached the wider mathematical community at last. Within 15 years, what came to be known as Galois theory was being taught in algebra classes. Historian Amir Alexander says that Galois “had become an iconic figure of the field, a revered martyr to mathematics”.

It is a curious fact that what seems at first glance to be nothing but “pure” mathematics often later turns out to have important applied uses. While the Academy of Sciences could not see the practical side at the time, Galois’s new ways of approaching symmetries, permutations, and groups turned out to apply to, well, basically everything. Subtle symmetry appears to play a profound, central role in the laws of physics as we understand them. It applies just as well to the miniscule (particle physics) as to the humongous (cosmology), and is scattered throughout just about everything in nature that shows organised behaviour. One can only imagine what Galois might have been able to contribute with more than just a few years of research.

The story of Évariste Galois—a revolutionary in every sense—has become something of a legend in the last 150 years, not least because of the dual figure he presents as mathematical visionary and political lightning-rod. Early obituaries all focused on him as a republican. As early as 1846, however, Liouville could dismiss Galois’s political activities as nothing more than “a pity”, and for several decades this was the common verdict. Neither of these is the full story. Galois’s mathematical thought and his political thinking are deeply intertwined. In one of his draft papers, an equation that cannot be broken down further leads him to write the word “Indivisible”, and beneath that, “Indivisibility of the republic”, followed on a new line by “Liberty, equality, fraternity, or death”. Among the scrawls on the same page are the words “Une femme” (“a woman”)—and, deeply scrawled out and now visible only with specialised equipment, the name of Pecheux d’Herbenville.

Galois closed his final mathematical statement with the bitter hope that after Jacobi and Gauss had given their opinion on the importance of his theorems, “there will be, I hope, some people who will find it to their advantage to decipher all this mess.”

There were, and they did.

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