Genius and Stupidity
‘Against stupidity the gods themselves strive unvictorious.’—SCHILLER
I
‘WE are poets!’ was the answer of one great mathematician, Leopold Kronecker (1823-1891), when challenged to define the sphere of the mathematician’s activities. In the sense that a poet is a creator, Kronecker was right. And, like poets and musicians, the great mathematicians mature young. It is not unusual for a boy with real mathematics in him to do work of lasting value before he is twenty. In at least one instance such immortality as achievement of the highest order in mathematics can confer was attained by the age of twenty. Évariste Galois reached the front rank well before that age.
Unlike poets, mathematicians of the first rank are fairly immune to accidents of changing taste. Fashions come and go in mathematics as in everything else, but work that is essentially new, simple, and of wide scope lasts for generations. The merely elegant or the complicated quickly passes. What Galois created over a century ago is to-day vividly alive in the re-creation of algebra.
As mathematics continues to expand at an ever-accelerated rate, it becomes both more abstract and more general. Simplicity of concept succeeds complexity of calculation, and masses of blindly manipulated symbols whose significance has been lost sight of give way to general ideas handled with a minimum of symbolism. This direct commerce in general concepts, which is the distinguishing mark of modern pure mathematics, originated in 1831 with Galois. It is at once more difficult and simpler than the older reliance on calculation. Technical facility in torturing formulas is at a discount. Only the ability to think clearly and a certain intellectual maturity are demanded of those who would follow the new way.
II
Where Aldous Huxley got the background for his grim story, The Young Archimedes, I do not know, but anyone acquainted with the history of mathematics can match the tragedy of that rare fiction with fact. To cite only the two most conspicuous examples, Abel (1802-1829) and Galois, both of whom transformed mathematics, had lives as sombre as that of Huxley’s young hero. Perhaps more so; for whereas ‘young Archimedes’ solved his difficulties in suicide, Abel and Galois played their game through to the last card. The record of Galois’s misfortunes might stand even to-day as a sinister monument to all self-assured pedagogues, ruffling patriots, and conceited academicians. Galois was no ‘ineffectual angel,’ but even his magnificent powers were shattered before the massed stupidity aligned against him, and he beat his life out fighting one unconquerable fool after another.
The first eleven years of his life were happy. His parents lived in the little village of Bourg-la-Reine, just outside Paris, where Évariste was born on October 25, 1811. Nicolas-Gabriel Galois, the father of Évariste, was a relic of the eighteenth century, cultivated, intellectual, saturated with philosophy, a passionate hater of royalty and an ardent lover of liberty. During the ‘Hundred Days’ after Napoleon’s escape from Elba, Galois was elected mayor of the village. After Waterloo, he retained his office and served faithfully under the King, backing the villagers against the priest, and delighting social gatherings with the old-fashioned rhymes which he composed himself. These harmless activities were later to prove the amiable man’s undoing. From his father, Evariste acquired the trick of rhyming, and a hatred of tyranny and baseness.
Until the age of twelve, Galois had no teacher but his mother, AdelaideMarie Demante. A woman of strong convictions, somewhat eccentric, with a first-rate classical education and a marked vein of skepticism in her independent nature, she undoubtedly passed on to Évariste some of the qualities which were later to get him into trouble. Adélaïde-Marie took nothing for granted on mere authority. In particular she evaluated Christianity in terms of her classical knowledge. Like her husband, she too hated the smell of tyranny.
As a child, Galois was affectionate and rather serious, although he entered readily enough into the gayety of the recurrent celebrations in his father’s honor, even composing rhymes and dialogues to entertain the guests. All this changed under the first stings of petty persecution and stupid misunderstanding, not by his parents, but by his teachers.
In 1823, at the age of twelve, Galois entered the lycée of Louis-le-Grand in Paris. It was his first school. The place was a dismal horror; barred and grilled, and dominated by a provisor who was more of a political gaoler than a teacher, it looked like a prison, and was. The France of 1823 still remembered the Revolution; it was a time of plots and counterplots and rumors of revolution. All this was echoed in the school. Suspecting the provisor of scheming to bring back the Jesuits, the students struck, refusing to chant in chapel. Without even notifying their parents, the provisor expelled those whom he thought most guilty; they found themselves in the street. Galois was not among them, but it would have been better for him if he had been. Till now tyranny had been a mere word to the boy of twelve; now he saw it in action, and the experience warped one side of his character for life. He was shocked into unappeasable rage. For the rest, his studies, owing to his mother’s excellent instruction in the classics, went well enough to gain him prizes.
The following year marked another crisis in the young boy’s life. Docile interest in literature and the classics gave way to boredom; his mathematical genius was already stirring. His teacher advised that he be demoted. Évariste’s father objected, and the boy continued with his interminable exercises in rhetoric, Latin, and Greek. His work was reported as mediocre, his conduct ‘dissipated,’ and the teachers had their way. Galois was demoted. He was forced to lick up the stale leavings which his genius had rejected. Bored and disgusted, he gave his work perfunctory attention and passed it without effort or interest.
It was during this year of acute boredom that Galois began mathematics. The splendid geometry of Legendre came his way. Two years was the usual time required by even the better mathematicians among the boys to master Legendre; Galois read the geometry from cover to cover as easily as other boys read a pirate yarn. The book aroused his enthusiasm; it was no textbook written by a hack, but a work of art composed by a creative mathematician. A single reading sufficed to reveal the whole structure of elementary geometry in crystal clarity to the fascinated boy. He had mastered it.
His reaction to algebra is illuminating. It disgusted him, and for a very good reason. Here there was no master like Legendre to inspire him. The text in algebra was a schoolbook and nothing more; Galois contemptuously tossed it aside. It lacked, he said, the creator’s touch that only a creative mathematician can give. Having made the acquaintance of one great mathematician through his work, Galois took matters into his own hands and, ignoring the meticulous pettifogging of his teacher, went directly for his algebra to the greatest master of the age, Lagrange. Later he read Abel. The boy of fourteen or fifteen absorbed masterpieces of algebraical analysis addressed to mature professional mathematicians.
His classwork in mathematics was mediocre. His peculiar gift of being able to carry on the most difficult mathematical investigations almost entirely in his head helped him with neither teachers nor examiners. Their insistence upon details which to him were obvious or trivial exasperated him beyond endurance, and he frequently lost his temper. Nevertheless he carried off the prize in the general examination. To the amazement of teachers and students alike, Galois had taken his own kingdom by assault while their backs were turned.
III
With this first realization of his tremendous powers, Galois’s character underwent a profound change. Knowing his kinship to the great masters of algebraical analysis, he felt an immense pride and longed to rush on to the front rank to match his strength with theirs. His family, even his unconventional mother, found him strange. At school he seems to have inspired a curious mixture of fear and anger in the minds of his teachers and fellow students. His teachers were good men and patient, but they were stupid, and to Galois stupidity was the unpardonable sin.
At the beginning of the year they had reported him as ‘ very gentle, full of innocence and good qualities, but .. .’ And they went on to say that ‘there is something strange about him.’ No doubt there was; the boy had unusual brains. A little later they admit that he is not ‘wicked,’ but merely ‘original and queer,’ ‘argumentative,’ and they complain that he delights to tease his comrades. By the end of the year of awakening we learn that ‘his queerness has alienated him from all his companions,’ and his teachers observe ‘something secret in his character.’ Worse, they accuse him of ‘affecting ambition and originality.’ But it is admitted by some that Galois is good in mathematics. His rhetoric teachers indulge in a little classical sarcasm: ‘His cleverness is now a legend that we cannot credit.’ They rail that there is only slovenliness or eccentricity in his assigned tasks — when he deigns to pay any attention to them — and that he goes out of his way to weary his teachers by incessant ‘dissipation.’
One man, to the everlasting credit of his pedagogical insight, declared that Galois was as able in literary studies as he was in mathematics. Galois appears to have been touched by this man’s kindness; he promised to give rhetoric a chance. But his mathematical devil was now fully aroused and raging to get out, and poor Galois fell from grace. In a short time the dissenting teacher joined the majority and made the vote unanimous. Galois, he sadly admitted, was beyond salvation, ‘conceited, with an insufferable affectation of originality.’ The pedagogue, however, did redeem himself by one excellent, exasperated suggestion. Had it been followed, Galois might have lived to eighty. ‘The mathematical madness dominates this boy. I think his parents had better let him take only mathematics. He is wasting his time here, and all he does is to torment his teachers and get into trouble.’
While Galois, at the age of sixteen, was already well started on his career of fundamental discovery, his mathematical teacher, Vernier, kept fussing over him like a hen that has hatched an eaglet and does not know how to keep the unruly creature’s feet on the good dirt of the barnyard. Vernier implored Galois to work systematically. The advice was ignored, and Galois, without preparation, took the competitive examinations for entrance to the École Polytechnique.
This great school, the mother of French mathematicians, fostered by Napoleon to give civil and military engineers the best scientific education available anywhere in the world, made a double appeal to the ambitious Galois. At the Polytechnic his mathematical talents would be recognized and encouraged to the utmost. And his craving for liberty and freedom of expression would be gratified; for were not the virile, audacious young Polytechnicians — among them the future leaders of the army — always a thorn in the side of reactionary schemers who would undo the glorious work of the Revolution and bring back the corrupt priesthood and the divine right of kings? The fearless Polytechnicians, at least in Galois’s boyish eyes, were no race of puling rhetoricians like the browbeaten nonentities at Louis-IeGrand, but a consecrated band of young patriots. He took the examinations and failed.
Galois was not alone in believing his failure the result of a stupid injustice. The comrades he had teased unmercifully were stunned. They believed that Galois had mathematical genius of the highest order and they suspected his examiners of incompetence in their office. As for Galois, the failure was almost the finishing touch. It drove him in upon himself and embittered him for life.
His seventeenth year was his great year. For the first and only time he met a man who had the capacity to understand his genius — Louis-PaulÉmile Richard (1795-1849), teacher of advanced mathematics (mathématiques spéciales) at Louis-le-Grand. Richard was no conventional pedagogue, but a man of talent who followed the lectures on geometry at the Sorbonne in his spare time and kept himself abreast of the progress of living mathematics to pass it on to his pupils. He recognized instantly what had fallen into his hands. The original solutions to difficult problems which Galois handed in were proudly explained to the class, with just praise for their young author, and Richard shouted from the housetops that this extraordinary pupil should be admitted to the Polytechnic without examination. He gave Galois the first prize and wrote in his term report, ‘This pupil has a marked superiority above all his fellow students; he works only at the most advanced parts of mathematics.’ All of which was the literal truth. Galois, at seventeen, was making discoveries of epochal significance in the theory of equations — discoveries whose consequences are not yet exhausted after more than a century.
The leading French mathematician of the time was Cauchy (1789-1857). In fertility of invention Cauchy has been equaled by but few; the mass of his collected works is exceeded in bulk only by the output of Euler (17071783), the most prolific mathematician in history. Whenever the Academy of Sciences wished an authoritative opinion on the merits of a mathematical work submitted for its consideration, it called upon Cauchy. As a rule he was a prompt and just referee; but occasionally he lapsed. Unfortunately, the occasions of his lapses were the most important of all.
Galois had saved the fundamental discoveries he had made up to the age of seventeen for a memoir to be submitted to the Academy. Cauchy promised to present this. He forgot. To crown his ineptitude, he lost the author’s abstract. That was the last Galois ever heard of Cauchy’s generous promise. This was only the first of a series of similar disasters which fanned the thwarted boy’s sullen contempt of academies and academicians into a fierce hate against the whole of the blundering society in which he was condemned to live.
In spite of his demonstrated genius, the harassed boy was not even now left to himself at school. Giving him no peace to harvest the rich field of his discoveries, the authorities pestered him to distraction with petty tasks and goaded him to open revolt by their preachings and punishments. Still they could find nothing in him but conceit and an iron determination to be a mathematician. He already was one, but they were incapable of knowing it.
Two further disasters in his seventeenth year put the last touches to Galois’s character. He presented himself a second time for the entrance examinations at the Polytechnic. Men who were not worthy to sharpen his pencils sat in judgment on him. The result was what might have been anticipated: Galois failed. It was his last chance; the doors of the Polytechnic were closed forever against him.
That examination has become a legend. During the oral part, one of the inquisitors ventured to argue a mathematical difficulty with Galois. The man was both wrong and obstinate. Seeing his whole life as a mathematician and polytechnic champion of democratic liberty slipping away from him, Galois lost all patience. He knew that he had officially failed. In a fit of rage and despair, he hurled the eraser at his tormentor’s face. He had aimed perfectly.
The final touch was the tragic death of Galois’s father. As the mayor of Bourg-la-Reine the elder Galois was a target for the clerical intrigues of the times, especially as he had always championed the villagers against the priest. After the stormy elections of 1827 a resourceful young priest organized a scurrilous campaign against the mayor. Capitalizing the mayor’s wellknown gift for versifying, the ingenious priest composed a set of filthy and stupid verses against a member of the mayor’s family, signed them with Mayor Galois’s name, and circulated them freely among the citizens. The thoroughly decent mayor developed a persecution mania. During his wife’s absence one day he slipped off to Paris and, in an apartment but a stone’s throw from the school where his son sat at his studies, committed suicide. At the funeral serious disorder broke out; stones were hurled by the enraged citizens; a priest was gashed on the forehead. Galois saw his father’s coffin lowered into the grave in the midst of an unseemly riot. Thereafter, suspecting everywhere the injustice he hated, he could see no good in anything.
IV
After his second failure at the Polytechnic, Galois returned to school to prepare for a teaching career. The school now had a new director, a time-serving, somewhat cowardly stoolpigeon for the royalists and clerics. This man’s shilly-shally temporizing in the political upheaval which was presently to shake France to its foundations had a fatal influence on Galois’s future.
Still persecuted and maliciously misunderstood by his preceptors, Galois prepared himself for the final examinations. The comments of his examiners are interesting. In mathematics and physics he got ‘very good.’ The final oral examination drew the following comments: ‘This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research. He communicated to me some new results in applied analysis.’ In literature: ‘This is the only student who answered me poorly; he knows absolutely nothing. I was told that this student has an extraordinary capacity for mathematics. This astonishes me greatly; for, after his examination, I believed him to have but little intelligence. He succeeded in hiding such as he had from me. If this pupil is really what he has seemed to me to be, I seriously doubt whether he will ever make a good teacher.’ To which Galois, remembering some of his own good teachers, might have replied, ‘God forbid!’
In February 1830, in his nineteenth year, Galois was definitely admitted to university standing. Again his sure knowledge of his own transcendent ability was reflected in a withering contempt for his plodding teachers, and he continued to work in solitude on his own ideas. During this year he composed three papers in which he broke new ground. These papers contain some of his great work on the theory of algebraic equations. It was far in advance of anything that had been done, and Galois hopefully submitted it all (with further results) in a memoir to the Academy of Sciences, in competition for the Grand Prize in mathematics. This prize was the blue ribbon in mathematical research; only the foremost mathematicians of the day could sensibly compete. Experts agree that Galois’s memoir was more than worthy of the prize; it was work of the highest originality. As Galois said with perfect justice, ‘I have carried out researches which will halt many savants in theirs.’
The manuscript reached the Secretary safely. The Secretary took it home with him for examination, but died before he had time to look at it. When his papers were searched after his death, no trace of the manuscript was found, and that was the last Galois ever heard of it. He can scarcely be blamed for ascribing his misfortunes to something less uncertain than blind chance. After Cauchy’s lapse, a repetition of the same sort of thing looked too providential to be a mere accident. ‘ Genius,’ he said, ‘is condemned by a malicious social organization to an eternal denial of justice in favor of fawning mediocrity.’ His hatred flamed, and he flung himself into politics on the side of republicanism, then a forbidden radicalism.
The first shots of the revolution of 1830 filled Galois with joy. He tried to lead his fellow students into the fray, but they hung back, and the temporizing director put them on their honor not to quit the school. Galois refused to pledge his word, and the director begged him to stay in till the following day. In his speech the director displayed a singular lack of tact and a total absence of common sense. Enraged, Galois tried to escape during the night, but the wall was too high for him. Thereafter, all through ‘the glorious three days,’ while the heroic young Polytechnicians were out in the streets making history, the director prudently kept his charges under lock and key. Whichever way the cat should jump, the director was prepared to jump with it. The revolt successfully accomplished, the astute director very generously placed his pupils at the disposal of the temporary government. This put the finishing touch to Galois’s political creed. During the vacation he shocked his family and boyhood friends with his fierce championship of the rights of the masses.
The last months of 1830 were as turbulent as is usual after a thorough political stir-up. The dregs sank to the bottom, the scum rose to the top, and, suspended between the two, the moderate element of the population hung in indecision. Galois, back at college, contrasted the time-serving vacillations of the director and the wishywashy loyalty of the students with their exact opposites at the Polytechnic. Unable to endure the humiliation of inaction longer, he wrote a blistering letter to the Gazette des Écoles, in which he let both students and director have what he thought was their due. The students could have saved him; but they lacked backbone, and Galois was expelled. Incensed, Galois wrote a second letter to the Gazette, addressed to the students. ‘I ask nothing of you for myself,’ he wrote; ‘but speak out for your honor and according to your conscience.’ The letter was unanswered, for the apparent reason that those to whom Galois appealed had neither honor nor conscience.
Foot-loose now, Galois announced a private class in higher algebra, to meet once a week. Here he was at nineteen, a creative mathematician of the very first rank, peddling lessons to no takers. The course was to have included ‘a new theory of imaginaries [what is now known as the theory of “Galois imaginaries,” of great importance in algebra and the theory of numbers]; the theory of the solution of equations by radicals, and the theory of numbers and elliptic functions treated by pure algebra ’ — all his own work.
Finding no students, Galois temporarily abandoned mathematics and joined the artillery of the national guard, two of whose four battalions were composed almost wholly of the liberal group calling themselves ‘Friends of the People.’ He had not yet given up mathematics entirely. In one last desperate effort to gain recognition, encouraged by Poisson, he had sent a memoir on the general solution of equations — now called the ‘Galois theory’ — to the Academy of Sciences. Poisson, whose name is remembered wherever the mathematical theories of gravitation, electricity, and magnetism are studied, was the referee. He submitted a perfunctory report. The memoir, he said, was ‘incomprehensible.’ This was the last straw. Galois devoted all his energies to revolutionary politics. ‘If a carcase is needed to stir up the people,’ he wrote, ‘I will donate mine.’
V
The ninth of May, 1831, marked the beginning of the end. About two hundred young republicans held a banquet to protest against the royal order disbanding the artillery which Galois had joined. Toasts were drunk to the Revolutions of 1789 and 1793, to Robespierre, and to the Revolution of 1830. The whole atmosphere of the gathering was revolutionary and defiant. Galois rose to propose a toast, his glass in one hand, his open pocketknife in the other: ‘To Louis-Philippe ’ — the King. His companions misunderstood the purpose of the toast and whistled him down. Then they saw the open knife. Interpreting this as a threat against the life of the King, they howled their approval. A friend of Galois, seeing Alexandre Dumas and other notables passing by the open windows, implored Galois to sit down, but the uproar continued. Galois was the hero of the moment, and the artillerists adjourned to the street to celebrate their exuberance by dancing all night. The following day Galois was arrested at his mother’s house and thrown into the prison of Sainte-Pélagie.
With the help of Galois’s loyal friends, a clever lawyer devised an ingenious defense, to the effect that Galois had really said, ‘To Louis-Philippe, if he turns traitor.’ The open knife was easily explained; Galois had been using it to cut his chicken. This was the fact. The saving clause in his toast, according to his friends who swore they had heard it, was drowned by the whistling, and only those close to the speaker caught what was said. Galois himself would not claim the saving clause. During the trial his demeanor was one of haughty contempt for the court and his accusers. Caring nothing for the outcome, he launched into an impassioned tirade against all forms of political injustice. The judge was a human being with children of his own; he warned the accused that he was not helping his own case and sharply silenced him. The prosecution quibbled over the point whether the restaurant where the incident occurred was or was not a public place when used for a semi-private banquet. It was evident, however, that both court and jury were moved by the youth of the accused. After only ten minutes’ deliberation the jury returned a verdict of not guilty. Galois picked up his knife from the evidence table, closed it, slipped it in his pocket, and left the courtroom without a word.
In less than a month, on July 14, 1831, he was arrested again, this time as a precautionary measure. The republicans were about to hold a celebration, and Galois, being a dangerous radical in the eyes of the authorities, was locked up on no charge whatever. The government papers of all France played up this brilliant coup of the police. They now had ‘the dangerous republican, Évariste Galois,’ where he could not possibly start a revolution. After two months of incessant thought, they succeeded in trumping up a charge. When arrested, Galois had been wearing his artillery uniform. But the artillery had been disbanded; therefore Galois was guilty of illegally wearing a uniform. This time they convicted him. A friend, arrested with him, got three months; Galois got six. He was to be incarcerated in Sainte-Pélagie till April 29, 1832. His sister said he looked about fifty years old at the prospect of the sunless days ahead of him.
Discipline in the jail for political prisoners was light, and they were treated with reasonable humanity. The majority spent their waking hours promenading in the courtyard reserved for their use, or boozing in the canteen — the private graft of the governor of the prison. Soon Galois, with his sombre visage, abstemious habits, and perpetual air of intense concentration, became the butt of the jovial swillers. He was concentrating on his mathematics, but he could not help hearing the taunts hurled at him.
‘What! You drink only water? Quit the Republican Party and go back to your mathematics.’ ‘Without wine and women you ’ll never be a man.’ Goaded beyond endurance, he seized a bottle of brandy, not knowing or caring what it was, and drank it down. A decent fellow prisoner took care of him till he recovered. His humiliation when he realized what he had done devastated him.
At last he escaped from what one French writer of the time calls the foulest sewer in Paris. The cholera epidemic of 1832 caused the solicitous authorities to transfer Galois to a hospital on the sixteenth of March. The ‘important political prisoner’ who had threatened the life of Louis-Philippe was too precious to be exposed to the epidemic.
Galois was put on parole, so he had only too many occasions to see outsiders. Thus it happened that he experienced his one love affair. In this, as in everything else, he was unfortunate. Some worthless girl initiated him. Galois took it violently, and was disgusted with love, with himself, and with his girl.
On May 29 he was freed, only to run foul of political enemies immediately after his release. These patriots were always spoiling for a fight, and it fell to the unfortunate Galois’s lot to accommodate two of them in an affair of honor. In a ‘Letter to all republicans,’ dated May 29, 1832, Galois writes: ‘I beg patriots and my friends not to reproach me for dying otherwise than for my country. I die the victim of an infamous coquette. It is in a miserable brawl that my life is extinguished. Oh, why die for so trivial a thing, die for something so despicable! . . . Pardon for those who have killed me— they are of good faith.’ In another letter to two unnamed friends: ‘I have been challenged by two patriots — it was impossible for me to refuse. I beg your pardon for having advised neither of you. But my opponents had put me on my honor not to warn any patriot. Your task is very simple: prove that I fought in spite of myself, that is to say after having exhausted every means of accommodation. . . . Preserve my memory, since fate has not given me life enough for my country to know my name. I die your friend, É. Galois.’
These were the last words he wrote. All night, before writing these letters, he had spent the hurrying hours feverishly scribbling his scientific last will and testament, writing against time to glean a few of the great things in his teeming mind before the death which he foresaw could overtake him. Time after time he broke off to scribble in the margin, ’I have not time; I have not time,’ and passed on to the next frantically scrawled outline. What he wrote in those desperate last hours before the dawn will keep generations of mathematicians busy for hundreds of years.
Galois addressed his will to his devoted friend Auguste Chevalier. He enclosed manuscripts of some of the work intended for the Academy of Sciences. For the preservation of these — Galois’s works — mathematics is indebted to Chevalier. Having outlined such of his unrecorded discoveries as he had time for, Galois concludes: ‘Ask Jacobi or Gauss publicly to give his opinion, not as to the truth, but as to the importance of these theorems. Later there will be, I hope, some people who will find it to their advantage to decipher all this mess. Je t’embrasse avec effusion. É. Galois.’
Neither Gauss nor Jacobi seems to have expressed any opinion.
At a very early hour on the thirtieth of May, 1832, Galois confronted his adversary on the field of honor. The duel was with pistols at twenty-live paces. Galois fell, shot through the intestines. No surgeon was present; he was left lying where he had fallen. At nine o’clock a passing peasant took him to the Cochin hospital. Galois knew he was about to die; before the inevitable peritonitis set in, and while still in the full possession of his faculties, he refused the offices of a priest. His young brother, the only one of his family who had been notified, arrived in tears. Galois tried to comfort him with a show of stoicism. ‘Don’t cry,’ he said; ‘I need all my courage to die at twenty.’
Early in the morning of May 31, 1832, Galois died, being then in the twenty-first year of his age. He was buried in the common ditch of the South Cemetery, so that to-day there remains no trace of the grave of Évariste Galois. His enduring monument is his collected works. They fill sixty pages.