Men of Mathematics

by E. T. Bell

  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, the place looked like a prison, and it was. The France of 1823 still remembered the Revolution. It was a time of plots and counterplots, of riots 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. His studies, owing to his mother’s excellent instruction in the classics, went very well and he won prizes. But he had also gained something more lasting than any prize, the stubborn conviction, right or wrong, that neither fear nor the utmost severity of discipline can extinguish the sense of justice and fair dealing in young minds experiencing their first unselfish devotion. This his fellow students had taught him by their courage. Galois never forgot their example. He was too young not to be embittered.

  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 teachers 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. Mathematics was taught more or less as an aside to the serious business of digesting the classics, and the pupils of various grades and assorted ages took the elementary mathematical course at the convenience of their other studies.

  It was during this year of acute boredom that Galois began mathematics in the regular school course. The splendid geometry of Legendre came his way. It is said that 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 when we consider what sort of mind Galois had. Here 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. Ignoring the meticulous pettifogging of his teacher, Galois 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—the memoirs on the numerical solution of equations, the theory of analytical functions, and the calculus of functions. His class work in mathematics was mediocre: the traditional course was trivial to a mathematical genius and not necessary for the mastering of real mathematics.

  Galois’ 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.

  With this first realization of his tremendous power, Galois’ 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. All very reprehensible, no doubt, but they might have used their eyes. The boy had discovered mathematics and he was already being driven by his daemon. 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 and 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.” The last does not refer to vice, because Galois had no viciousness in him. It is merely a strong word to describe the heinous inability of a mathematical genius of the first rank to squander his intellect on the futilities of rhetoric as expounded by pedants.

  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.” But the pedagogue redeemed 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.”

  At the age of sixteen Galois made a curious mistake. Unaware that Abel at the beginning of his career had convinced himself that he had done the impossible and had solved the general equation of the fifth degree, Galois repeated the error. For a time—a very short time, however—he believed that he had done what cannot be done. This is merely one of several extraordinary similarities in the careers of Abel and Galois.

  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, founded during the French Revolution (some say by Monge), to give civil and military engineers the best scientific and mathematical education available anywhere in the world, made a double appeal to the ambitious Galois. At the Polytechnique his mathematical talent would be recognized and encouraged to the utmost. And his craving for liberty and freedom of utterance 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’ boyish eyes, were no race of puling rhetoricians like the browbeaten nonentities at Louis-le-Grand, but a consecrated band of young patriots. Events were presently to prove him at least partly right in his estimate.

  Galois failed in the examinations. He 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. Nearly a quarter of a century later Terquem, editor of the Nouvelles Annales de Mathématiques, the mathematical journal devoted to the interests of candidates for the Polytechnique and Normale schools, reminded his readers that the controversy was not yet dead. Commenting on the failure of Galois and on the inscrutable decrees of the examiners in another instance, Terquem remarks, “A candidate of superior intelligence is lost with an examiner of inferior intelligence. Hic ego barbarus sum quia non intelligor illis [Because they don’t understand me, I am a barbarian.] . . . Examinations are mysteries before which I bow. Like the mysteries of theology, the reason must admit them with humility, without seeking to understand them.” As for Galois, the failure was almost the finishing touch. It drove him in upon himself and embittered him for life.

  In 1828 Galois was seventeen. It was his great year. For the first time he met a man who had the capacity to understand his genius, Louis-Paul-Émile 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 advanced lectures on geometry at the Sorbonne in his spare time and kept himself abreast of the progress of living mathematicians to pass it on to his pupils. Timid and unambitious on his own account, he threw all his talent on the side of his pupils. The man who would not go a step out of his way to advance his own interests counted no sacrifice too great where the future of one of his students was at stake. In his zeal to advance mathematics through the work of abler men he forgot himself completely, although his scientific friends urged him to write, and to his inspired teaching more than one outstanding French mathematician of the nineteenth century has paid grateful tribute: Leverrier, codiscoverer with Adams by pure mathematical analysis of the planet Neptune; Serret, a geometer of repute and author of a classic on higher algebra in which he gave the first systematic exposition of Galois’ theory of equations; Hermite, master algebraist and arithmetician of the first rank; and last, Galois.

  Richard recognized instantly what had fallen into his hands—“the Abel of France.” The original solutions to difficult problems which Galois handed in were proudly explained to the class, with just praise for the young author, and Richard shouted from the housetops that this extraordinary pupil should be admitted to the Polytechnique 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. On the first of March, 1829, Galois published his first paper, on continued fractions. This contains no hint of the great things he had done, but it served to announce him to his fellow students as no mere scholar but an inventive mathematician.

  The leading French mathematician of the time was Cauchy. In fertility of invention Cauchy has been equalled by but few; and as we have seen, the mass of his collected works is exceeded in bulk only by the outputs of Euler and Cayley,I the most prolific mathematicians of 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. To Cauchy’s carelessness mathematics is indebted for two of the major disasters in its history: the neglect of Galois and the shabby treatment of Abel. For the latter Cauchy was only partly to blame, but for the inexcusable laxity in Galois’ case Cauchy alone is responsible.

  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, but he forgot. To put the finishing touch to 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 stupid 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. The authorities gave him no peace to harvest the rich field of his discoveries, but pestered him to distraction with petty tasks and goaded him to open revolt by their everlasting 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 did not know it.

  Two further disasters in his eighteenth year put the last touches to Galois’ character. He presented himself a second time for the entrance examinations at the Polytechnique. 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 Polytechnique were closed forever against him.

  That examination has become a legend. Galois’ habit of working almost entirely in his head put him at a serious disadvantage before a blackboard. Chalk and erasers embarrassed him—till he found a proper use for one of them. During the oral part of the examination one of the inquisitors ventured to argue a mathematical difficulty with Galois. The man was both wrong and obstinate. Seeing all his hopes and 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. It was a hit.

  The final touch was the tragic death of Galois’ 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 well-known 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’ 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 which he hated, he could see no good in anything.

  After his second failure at the Polytechnique, Galois returned to school to prepare for a teaching career. The school now had a new director, a timeserving, 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 tragic influence on Galois’ last years.

  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 expr
essing his ideas, but he is intelligent and shows a remarkable spirit of research. He has communicated to me some new results in applied analysis.” In literature: “This is the only student who has 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, at the age of nineteen, 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 had 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 still the blue ribbon in mathematical research; only the foremost mathematicians of the day could sensibly compete. Experts agree that Galois’ 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 grew, and he flung himself into politics on the side of republicanism, then a forbidden radicalism.