La vie d'Evariste Galois

SECOND CAHIER OF THE FIFTH SERIES

Foreword by Jules Tannery

The mathematical works of Galois form a volume of sixty-one pages; (1) the author died at twenty, killed in a duel. On the eve of his death, he wrote to his friend Auguste Chevalier a letter in which his principal discoveries are summarized, the certain results that “had been in his head for a year,” and in which are indicated, in a single stroke, the ideas that were fermenting within him: “…but I do not have time, and my ideas are not yet well developed on this terrain, which is immense.” It concludes with these words: “After that, there will be, I hope, people who will find it to their advantage to decipher this mess.”

This disdainful phrase is too harsh; but it is true that those who rediscovered or clarified Galois’s thought, and who developed its consequences, have also been the greatest mathematicians of the nineteenth century.

As long as there are mathematicians on earth, the name of Galois will be illustrious; it will remain attached to the most beautiful discoveries of the last century; the few pages he left behind will be read by a small number of

(1) They were collected by Liouville, who published them in 1846 in his Journal of Pure and Applied Mathematics. The Mathematical Society of France issued a definitive edition, preceded by an Introduction by Mr. Emile Picard.

scholars, alone capable of understanding their meaning or grasping their scope.

They relate to the most elevated and most abstract parts of Algebra and Analysis; but the thought is so profound that it most often transcends its object, and Galois’s doctrines have penetrated into almost all parts of mathematics, which they dominate today.

Works like those of Galois are regarded as useless by short-sighted philosophers who see in science only its immediate applications: these applications are possible only because we know better the world in which we live; mathematics alone can put order and connection into our knowledge; mathematics itself has an order and a logical connection all its own, which must be discovered by attending to mathematics alone. Those who are capable of this will always be rare.

What can be said of the genius of Galois, which was perhaps unique? How many weeks of his brief and agitated life did this twenty-year-old boy give to science, which owes him so much?

JULES TANNERY

The Life of Évariste Galois

by Paul Dupuy

The first elements of this study were gathered in the course of research on the history of the Ecole Normale. I had planned to use them in writing a short biography as an appendix to the purely scientific study that Mr. Sophus Lie wrote about Galois in the Centenary Volume of the Ecole Normale. Time failed me, less for writing than for completing the research that seemed indispensable to me. I resumed them when the Mathematical Society announced an edition of Galois’s works, hitherto scattered in various scientific journals or collected in Liouville’s Journal, which mathematicians do not always have readily at their disposal. The occasion seemed right to complete my inquiry, which I pushed in every direction, seeking to penetrate Galois’s person as intimately as possible, and to illuminate him from without through an exact knowledge of the times and particular circumstances in which he lived.

I had at first at my disposal a certain number of notes, assuredly very interesting but also very incomplete, published about Galois in various journals, which have until now furnished the elements for articles in biographical or encyclopedic dictionaries. The oldest and principal one is that which his school friend Auguste Chevalier inserted in November 1832 in the Revue encyclopedique of Hippolyte Carnot and Pierre Leroux. The note is preceded by a foreword in which I believe I recognize the hand of Pierre Leroux. Fourteen years later, Liouville, in publishing Galois’s unpublished memoirs, also wrote a notice that added nothing to the one in the Revue encyclopedique. In 1848, the Magasin pittoresque published in its turn a short unsigned biography, but which Mr. Ludovic Lalanne (1) told me was by Flaugergues, (2) and a portrait made from memory by Alfred Galois, Evariste’s brother. In 1849, the Nouvelles Annales de mathematiques gave, following a notice on Mr. Richard, Galois’s advanced mathematics teacher, a short note on Galois himself. Finally, in recent years, the Intermediaire des chercheurs et des curieux pointed out in the Memoirs of Alexandre Dumas pere an interesting passage on the duel in which Galois was mortally wounded.

Such was the body of documents, accessible to everyone, that I could use at first. To check, critique, and complete them, I had to undertake research in various archival repositories. I found material at the archives of the Ecole Normale, those of the Ministry of Public Instruction and the Academy of Paris, the National Archives, and

(1) Mr. Ludovic Lalanne knew Galois at Louis-le-Grand; he had especially heard about him from his brother Leon, who was in advanced mathematics with Galois. (2) Flaugergues, a classmate of Galois at Louis-le-Grand, encountered him again during the last months of 1830 at the Ecole Normale, from which he was expelled like him.

those of Bourg-la-Reine, the Prefecture of the Seine, the Cochin Hospital, and the prisons of the Conciergerie and Sainte-Pelagie. Unfortunately, those of the Prefecture of Police, the court and the tribunal of Paris, where interesting notes and all the proceedings of his legal cases must have been, were burned in 1871. The newspaper collections of the time, preserved either at the National Library or the Arsenal Library, allow only very incomplete substitution for the absence of these documents. However, the Gazette des Tribunaux gave an account of all of Galois’s trials. Other newspapers, like the Tribune, provided me with a certain number of interesting details. Another, the Gazette des Ecoles, supplements the documents preserved at the National Archives and the archives of the Ecole Normale regarding the affair that led to Galois’s expulsion from the Ecole Normale.

Books on the period in which Galois lived are not lacking, but generally provide only very brief indications about him, except for the Memoirs of the Prefect of Police Gisquet and the Letters on the Prisons of Paris by Raspail, which are of little use in adding to his biography. In truth, nothing yet exists that allows us to give a precise account of the history of the republican party in this heroic period of its existence; and I hope that when the papers, memoirs, or correspondence of the party’s principal leaders are given to the public, new information about Galois will be found, particularly about the circumstances in which he met his death, information that might clarify the mystery. I was unable to do so.

I finally addressed myself to Galois’s family, whom a fortunate chance enabled me to find very quickly. The sole survivor among the normaliens who knew Galois at the school, Mr. Benard, had in fact married a first cousin of his schoolmate; through her and through the other surviving relatives she introduced me to,

I was able to gather sometimes direct memories, sometimes traditions preserved in the family for sixty-four years since Galois’s death. On this side, my principal acquisition was the portrait of Galois made from life when he was fifteen or sixteen years old. I owe it to the extreme kindness of his niece, Madame Guinard, daughter of Galois’s elder sister, Madame Chantelot. May I be permitted to address here all my thanks for the zeal with which she assisted me. I owe no less to Mr. and Madame Benard, Colonel Guinard, Mr. Gabriel Demante, former professor at the Faculty of Law of Paris, and his brother, the Abbe Demante, to whom I am indebted for a great number of curious and precise details about Galois’s childhood.

It goes without saying that I endeavored to check all the documents I had in hand against one another. I endeavored to do so without bias, though with ever-increasing sympathy for the brilliant and unfortunate young man who paid with so much suffering for the incredible power of his faculties; I sought above all to explain him, or at least to explain what was explicable in his character and adventures. I always saw him in the midst of the things, people, events, and institutions of his time; an interest of history was thus added for me to an interest of biography. My essential wish is to substitute an accurate portrait of this illustrious mathematician for the vague sketches previously available; but I confess it would also be a keen satisfaction if it were judged that, in recounting the life of Galois, I had been able to cast a curious light on some corners of the Revolution of 1830 and the troubled, vivid years surrounding it.

I

Evariste Galois was born on October 25, 1811, at Bourg-la-Reine, in a house that today bears number 20 on the Grand Rue. Before being painted green and salmon and calling itself, for the Parisian, Villa de Bourg-la-Reine, this house was until recently still a school for young men, whose origins went back beyond the Revolution. It had then been owned by Evariste’s grandfather. Far from suffering from the Revolution, grandfather Galois had owed it, on the contrary, the prosperity of his school: Bourg-la-Reine, having become Bourg-l’Egalite, enjoyed a relative calm at a short distance from Paris; most of the colleges or other boarding schools, almost all run by priests, had disappeared or become suspect: these were so many favorable circumstances from which the Galois school had profited; it had also owed part of its success to the ardent sentiments with which the Galois family had rallied first to the Revolution, then to the order of things that issued from it. While his eldest son, an officer in the Imperial Guard, fought in various parts of Europe, Mr. Galois had yielded his school to his younger son, Nicolas-Gabriel, and the latter, when Evariste was born, had become for a year a true functionary, the head of an institution of the Imperial University.

Nicolas-Gabriel Galois was then thirty-six: he was indeed a man of the eighteenth century, amiable and witty, skilled at rhyming couplets or composing drawing-room comedies; he was at the same time deeply imbued with philosophy. He had welcomed the fall of the monarchy with joy and, even in the decline of the Empire, would still have preferred anything to the return of the old regime. The first Restoration made him the leader of the liberal party at Bourg-la-Reine. During the Hundred Days, the vote of the primary assembly entrusted him with the mayoralty of the village. After Waterloo, he should have returned the position to his predecessor; but the latter had meanwhile been disqualified by bad business dealings and had just left the area: Mr. Galois took advantage of the prefect’s embarrassment to ask to be confirmed or replaced, and, for want of another candidate, he had to be officially reappointed to the function he had never ceased to exercise. (1) He was to keep it until his death, a scrupulous observer, without doubt, of the oath of loyalty he had sworn to the king, but strong enough in the support of his constituents to resist very firmly the omnipotence of the parish priest.

He had married under the Empire a young woman, Adelaide-Marie Demante, whose family, well known at the Faculty of Law of Paris, lived at Bourg-la-Reine,

(1) Archives of the Seine.

almost opposite the Galois house. There too, in modest comfort, traditions of intellectual culture had been preserved for a long time, whose heritage Evariste Galois was to receive from his earliest childhood. His maternal grandfather, Thomas-Francois Demante, was an associate professor at the Faculty of Law of the old University of Paris; the Empire had made him a magistrate, and when Evariste was born he presided over the court of Louviers. He was a passionate Latinist of the old regime: he himself had broken all his children, daughters and sons, to the exercises of the old classical education; he had at the same time given them a solid religious instruction; but on his daughter Adelaide-Marie the imprint of antiquity had been the strongest. Through the apparent monotony of daily translations from the Conciones, the ever-renewed lessons of Roman stoicism had penetrated deeply into the young woman’s soul and given her a virile temper; not that she had ceased to be a Christian; on the contrary, she professed to be one all her life, but without any shade of feminine devotion, bringing the sacred texts alongside those of Cicero and Seneca, and reducing religion almost to the role of an envelope for the principles of morality. With this, an ardent imagination, which further exalted in her the strength of character and gave to her virtues — whether the sense of honor or the forgiveness of injuries — something passionate. (1)

(1) Most of this information about Galois’s father and mother was provided by his family, notably by Mr. Gabriel Demante.

Such was the mother of Evariste Galois. One must know her to understand him well, and one must also know that until the age of twelve he had no other teacher. Regarding this first part of his life, what is known about his mother is about all that is known about him. Since she did not die until 1872, aged 84, it was fairly easy for me to meet people who knew her well and for a long time: they kept a very precise memory of her intelligence, which remained vivid to the end, of her generosity pushed, it seems, to the point of improvidence; I even heard her taxed with eccentricity and oddity, and I thought I should note it, because that too helps explain similar judgments made about her son, who certainly inherited from her the principal traits of his moral character.

It was in 1823 that Galois left his family for college. Two years earlier, a half-scholarship had been granted to him at the college of Reims; (1) but his mother had preferred to keep him near her still, and he left her only to enter the fourth form at Louis-le-Grand as a boarder.

Sensitive as he was, the child must have felt a singular impression in passing from his native village and the paternal house, where life was both serious and cheerful, into that somber dwelling of old Louis-le-Grand, all bristling with iron bars and stirred by passions beneath its prison-like appearance: passion for work and academic triumphs, passion for liberal ideas, passion for memories of the Revolution and the Empire, hatred and contempt for the legitimist reaction. Since 1815, revolts had not ceased there; two headmasters had already worn themselves out in eight years.

(1) Archives of the Ministry of Public Instruction.

At the moment when Galois entered Louis-le-Grand, a new headmaster, Mr. Berthot, had just taken charge and was preparing to govern with a heavy hand. The boarders immediately judged that he had been placed there only to prepare the return of the Jesuits and protested by refusing to sing in chapel; repression was swift: its result was to give the sedition a less negative form and to move it into the study halls; the principal mutineers were then thrown into the street, without even notifying their families, and such was the students’ exasperation that at the Saint Charlemagne feast of 1824 they resolved to keep silent when the headmaster would offer the customary toast to the king. Not only did they remain silent, but when a few professors responded haltingly, their voices, which misfired, were covered by laughter. An abominable scandal! Stricken and furious, Mr. Berthot did not hesitate to expel all students present at the banquet; he thus decapitated his college. (1)

Galois did not yet count among the very first in his class at the end of January 1824, since he was not expelled. Perhaps even the prize and three honorable mentions he obtained at the distribution were due in part to this upheaval; in any case, they were sufficient to attest that the child had done honor to his mother’s lessons and had not lost the habits of regular work she had given him.

(1) Quicherat, History of Sainte-Barbe.

However, when one knows what followed, one must think that what he saw at Louis-le-Grand during this first year of boarding had a decisive influence on his character; it was, no doubt, the first crisis of his childhood. Until then, he had known only in books and maternal conversations the struggles and sacrifices for liberty, the conspiracies against tyranny; and now he had found them immediately realized in this world of the college, new to him, where fear did not succeed in keeping out the breezes of liberty, where the very narrowness of the walls and the severities of the rules gave them more power over young souls intoxicated by the delights of first enthusiasms. His own had been too well prepared not to be immediately taken by what was generous in the spirit of disorder that then reigned in most of the colleges of Paris. It was then, I am convinced, that the feelings that were the faith of his life took root in his heart: he remained until his last day a Louis-le-Grand boy of 1824.

This moral crisis did not, however, slow his work: he even raised his standing in his class and obtained, at the end of the third form, the first prize in Latin verse and three honorable mentions; an honorable mention in Greek translation at the General Competition ranked him among the students the college could count on for the future. It was only in the second form that the first signs of weariness and disgust with schoolwork appeared; he had only four honorable mentions at the lycee. He had probably also been in poor health during the year; to safeguard his health and shore up his successes, the headmaster proposed having him repeat the second form.

The father resisted at first, and at the start of the 1826 school year, Evariste entered the rhetoric class. His work was judged mediocre, his conduct unruly, his mind too young to benefit from the class; in January it was necessary to yield to the headmaster’s insistence: Evariste returned to the second form, in the division of Mr. Saint-Marc-Girardin, and there he found success again, but without making any effort. His behavior appeared most bizarre to his study master: if the subject of an assignment displeased him, he botched it or dispensed with it entirely; as for lessons, no middle ground: either very well learned or not at all; in reality, he devoted only the last two weeks of the year to his class work. This was no doubt enough for his self-esteem, since, in addition to a second prize in Greek translation, he obtained honorable mentions in the four other subjects, and an honorable mention in Greek translation at the General Competition.