by E. T. Bell
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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’ 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 timeserving vacillations of the director and the wishy-washy loyalty of the students with their exact opposites at the Polytechnique. 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.
Footloose 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,” but he did not state how long it had taken him to reach his remarkable conclusion. 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.”
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 pocket knife 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 the great Alexander 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.
A clever lawyer, with the help of Galois’ loyal friends, devised an ingenious defence, 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 would not claim the saving clause.
During the trial Galois’ 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 the forces 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 semiprivate banquet. On this nice point of law hung the liberty of Galois. But it was evident 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.
He did not keep his freedom long. 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. But they were hard put to it to find a legal accusation under which he could be brought to trial. True, he had been armed to the teeth when arrested, but he had not resisted arrest. Galois was no fool. Should they accuse him of plotting against the government? Too strong; it wouldn’t go; no jury would convict. Ah! 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. Why not? “Let justice prevail though the heavens fall.”
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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 somber 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 Galois 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 and only love affair. In this, as in everything else, he was unfortunate. Some worthless girl (“quelque coquette de bas étage”) initiated him. Galois took it violently and was disgusted with love, with himself, and with his girl. To his devoted friend Auguste Chevalier he wrote, “Your letter, full of apostolic unction, has brought me a little peace. But how obliterate the mark of emotions as violent as those which I have experienced? . . . On re-reading your letter, I note a phrase in which you accuse me of being inebriated by the putrefied slime of a rotten world which has defiled my heart, my head, and my hands. . . . Inebriation! I am disillusioned of everything, even love and fame. How can a world which I detest defile me?” This is dated May 25, 1832. Four days later he was at liberty. He had planned to go into the country to rest and meditate.
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What happened on May 29th is not definitely known. Extracts from two letters suggest what is usually accepted as the truth: Galois had run foul of political enemies immediately after his release. These “patriots” were always spoiling for a fight, and it fell to the unfortunate Galois’ lot to accommodate them in an affair of “honor.” In a “Letter to All Republicans,” dated 29 May, 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
E. GALOIS.”
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These were the last words he wrote. All night, before writing these letters, he had spent the fleeting hours feverishly dashing off 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. He had found, once and for all, the true solution of a riddle which had tormented mathematicians for centuries: under what conditions can an equation be solved? But this was only one thing of many. In this great work, Galois used the theory of groups (see chapter on Cauchy) with brilliant success. Galois was indeed one of the great pioneers in this abstract theory, today of fundamental importance in all mathematics.
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In addition to this distracted letter Galois entrusted his scientific executor with some of the manuscripts which had been intended for the Academy of Sciences. Fourteen years later, in 1846, Joseph Liouville edited some of the manuscripts for the Journal de Mathématiques pures et appliquées. Liouville, himself a distinguished and original mathematician, and editor of the great Journal, writes as follows in his introduction:
“The principal work of Évariste Galois has as its object the conditions of solvability of equations by radicals. The author lays the foundations of a general theory which he applies in detail to equations whose degree is a prime number. At the age of sixteen, and while a student at the college of Louis-le-Grand . . . Galois occupied himself with this difficult subject.” Liouville then states that the referees at the Academy had rejected Galois’ memoirs on account of their obscurity. He continues: “An exaggerated desire for conciseness was the cause of this defect which one should strive above all else to avoid when treating the abstract and mysterious matters of pure Algebra. Clarity is, indeed, all the more necessary when one essays to lead the reader farther from the beaten path and into wilder territory. As Descartes said, ’When transcendental questions are under discussion be transcendentally clear.’ Too often Galois neglected this precept; and we can understand how illustrious mathematicians may have judged it proper to try, by the harshness of their sage advice, to turn a beginner, full of genius but inexperienced, back on the right road. The author they censured was before them, ardent, active; he could profit by their advice.
“But now everything is changed. Galois is no more! Let us not indulge in useless criticisms; let us leave the defects there and look at the merits.” Continuing, Liouville tells how he studied the manuscripts, and singles out one perfect gem for special mention.
“My zeal was well rewarded, and I experienced an intense pleasure at the moment when, having filled in some slight gaps, I saw the complete correctness of the method by which Galois proves, in particular, this beautiful theorem: In order that an irreducible equation of prime degree be solvable by radicals it is necessary and sufficient that all its roots be rational functions of any two of them”II
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Galois addressed his will to his faithful friend Auguste Chevalier, to whom the world owes its preservation. “My dear friend,” he began, “I have made some new discoveries in analysis.” He then proceeds to outline such as he has time for. They were epoch-making. He concludes: “Ask Jacobi or Gauss publicly to give their 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. E. Galois.”
Confiding Galois! Jacobi was generous; what would Gauss have said? What did he say of Abel? What did he omit to say of Cauchy, or of Lobatchewsky? For all his bitter experience Galois was still a hopeful boy.
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 five 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. Perhaps he remembered his father. His young brother, the only one of his family who had been warned, 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 today there remains no trace of the grave of Évariste Galois. His enduring monument is his collected works. They fill sixty pages.
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I. That is, so far as actually published work is concerned up to the present (1936). Euler undoubtedly will surpass Cayley in bulk when the full edition of his works is finally printed.
II. The significance of this theorem will be clear if the reader will glance through the extracts from Abel in Chapter 17.
CHAPTER TWENTY ONE
Invariant Twins
CAYLEY AND SYLVESTER
The theory of Invariants sprang into existence under the strong hand of Cayley, but that it emerged finally a complete work of art, for the admiration of future generations of mathematicians, was largely owing to the flashes of inspiration with which Sylvester’s intellect illuminated it.—P. A. MACMAHON
“IT IS DIFFICULT to give an idea of the vast extent of modern mathematics. The word ’extent’ is not the right one: I mean extent crowded with beautiful detail—not an extent of mere uniformity such as an objectless plain, but of a tract of beautiful country seen at first in the distance, but which will bear to be rambled through and studied in every detail of hillside and valley, stream, rock, wood, and flower. But, as for every thing else, so for a mathematical theory—beauty can be perceived but not explai
ned.”
These words from Cayley’s presidential address in 1883 to the British Association for the Advancement of Science might well be applied to his own colossal output. For prolific inventiveness Euler, Cauchy, and Cayley are in a class by themselves, with Poincaré (who died younger than any of the others) a far second. This applies only to the bulk of these men’s work; its quality is another matter, to be judged partly by the frequency with which the ideas originated by these giants recur in mathematical research, partly by mere personal opinion, and partly by national prejudice.
Cayley’s remarks about the vast extent of modern mathematics suggest that we confine our attention to some of those features of his own work which introduced distinctly new and far-reaching ideas. The work on which his greatest fame rests is in the theory of invariants and what grew naturally out of that vast theory of which he, brilliantly sustained by his friend Sylvester, was the originator and unsurpassed developer. The concept of invariance is of great importance for modern physics, particularly in the theory of relativity, but this is not its chief claim to attention. Physical theories are notoriously subject to revision and rejection; the theory of invariance as a permanent addition to pure mathematical thought appears to rest on firmer ground.