People From Paris

"Galois... accomplished his task and... few men will... accomplish more. He has conquered the purest kind of immortality. As he wrote to his friends: "I take with me to the grave a conscience free from lie, free from patriot's blood". How many of the conventional heroes of history, how many of the kings, captains and statesmen could say the same?"

"... un auteur ne nuit jamais tant à ses lecteurs que quand il dissimule une difficulté."

"Ne pleure pas, Alfred ! J'ai besoin de tout mon courage pour mourir à vingt ans !"

"When genius... explodes suddenly, at the beginning and not at the end of life, or when we are at a loss to explain its... genesis, we can but feel that we are in the sacred presence of something vastly superior to talent. ...[I]t is not necessary to introduce any mystical idea, but it is one's duty to acknowledge the mystery. ...Galois' fateful existence helps one to understand Lowell's saying: "Talent is that which is in a man's power, genius is that in whose power man is." If Galois had been simply a mathematician of considerable ability, his life would have been far less tragic, for he could have used his mathematical talent for his own advancement and happiness; instead... the furor of mathematics—as one of his teachers said—possessed him and he had no alternative but absolute surrender to his destiny."

"Although the Greek use of s in the case of greatest common measure was well known in the Middle Ages, the modern theory of the subject may be said to have begun with Bombelli (1572). ...The first great memoir on the subject was Euler's De fractionibus continuis (1737), and in this work the foundation for the modern theory was laid. Among other interesting cases Euler developed e as a continued fraction, thus: e = 2 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{4 + \cfrac{1}{1+...}}}}} Of the later contributors to the theory, special mention should be made of Lagrange (1767) and Galois."

"How to console oneself for having exhausted in one month the greatest source of happiness which is in man — of having exhausted it without happiness, without hope, being certain that one has drained it for life? Oh! come and preach peace after that! Come and ask men who suffer to take pity upon what is! Pity, never! Hatred, that is all. He who does not feel it deeply, this hatred of the present, cannot really have in him the love of the future. ...I like to doubt your cruel prophecy when you say that I shall not work any more. But I admit that it is not without likelihood. To be a savant, I should need to be that alone. My heart has revolted against my head. I do not add as you. do: It is a pity."

"Nous avons transcrit textuellement la démonstration que nous avons donnée de ce lemme dans un mémoire présenté en 1830. Nous y joignons 154 IV The First Memoir comme document historique le note suivante qu’a cru ^devoir^ y apposer M. Poisson. On jugera. Note de l’auteur [We have faithfully transcribed the proof of this lemma that we have given in a memoir presented in 1830. We append as a historical document the following note which Mr Poisson believed he should add. Posterity will judge. Note by the author]"

"In his last year at the college, 1828-1829... [his] teacher of mathematics... wrote of him: "This student has a marked superiority over all his school-mates. ...He works only at the highest parts of mathematics." ...[O]ther teachers were less indulgent. For physics and chemistry, the note often repeated was: "Very absent-minded, no work whatever.""

"The final lines are not mine: they come from an experiment on soft matter, after Boudin… An English translation might run like this:"

"The modern in general is commonly said to date from Abel and Galois. The latter's posthumous (1846) memoir... established the theory... To him is due the discovery that to each equation there corresponds a group of substitutions (the "group of the equation") in which are reflected its essential characteristics. Galois' early death left without sufficient demonstration several important propositions, a gap which has since been filled."

"There are six major episodes to be observed, four of which will be described... The four are the definition by Gauss, E. E. Kummer.., and Dedekind of s; the restoration of the in s by Dedekind’s introduction of ideals; the definitive work of Galois on the solution of algebraic equations by radicals, and the theory of finite groups and the modern theory of fields that followed; the partial application of arithmetical concepts to certain linear algebras by R. Lipschitz.., A. Hurwitz.., L. E. Dickson.., Emmy Noether.., and others. All of these developments are closely interrelated."

"Le vrai point d'honneur [d'un scientifique] n'est pas d'être toujours dans le vrai. Il est d'oser, de proposer des idées neuves, et ensuite de les vérifier."

"Dès le commencement de ce siècle, l'algorithme avait atteint un degré de complication tel que tout progrès était devenu impossible par ce moyen, sans l'élégance que les géomètres modernes ont su imprimer à leurs recherches et au moyen de laquelle l'esprit saisit promptement et d'un seul coup un grand nombre d'opérations. Il est évident que l'élégance si vantée et à si juste titre n'a pas d'autre but. Du fait bien constaté que les efforts des géomètres les plus avancés ont pour objet l'élégance on peut donc conclure avec certitude qu'il devient de plus en plus nécessaire d'embrasser plusieurs opérations à la fois, parce que l'esprit n'a plus le temps de s'arrêter aux détails. ... Sauter à pieds joints sur les calculs, grouper les opérations, les classer suivant leurs difficultés et non suivant leurs formes; telle est, suivant moi, la mission des géomètres futurs; telle est la voie où je suis entré dans cet ouvrage."

"The duel took place on the 30th in the early morning, and he was grievously wounded by a shot in the abdomen. He was found by a peasant who transported him at 9:30 to the Hôpital Cochin. His younger brother... came and stayed with him, and as he was crying, Evariste tried to console him, saying: "Do not cry. I need all my courage to die at twenty." ...[H]e refused the assistance of a priest. ...[H]e breathed his last ...the following morning."

"A dense film of a conventional surfactant is quite impermeable. On the other hand, a dense film of Janus grains always has some interstices between the grains, and allows for chemical exchange between the two sides; "the skin can breathe". This may possibly be of some practical interest."

"To support himself Galois announced that he would give a private course of higher algebra... [A] new copy of his second lost memoir... communicated... to the Academie... was returned to him by Poisson, four months later, as being incomprehensible. Galois was partly responsible... for he had taken no pains to explain himself clearly. This was the last straw. ...[H]e plunged himself entirely into the political turmoil. ...He is said to have exclaimed: "If a corpse were needed to stir the people up, I would give mine.""

"May 9, 1831, at the end of a political banquet, being intoxicated—not with wine but with the ardent conversation of an evening—he proposed a sarcastic toast to the King. He held his glass and an open knife in one hand and said simply: "To Louis Philippe!" Of course he was soon arrested and sent to Ste. Pélagie. The lawyer persuaded him to maintain that he had actually said: "To Louis-Philippe, if he betray," and many witnesses affirmed... His attitude before the tribunal was ironical and provoking, yet the jury rendered a verdict of not proven and he was acquitted. On the following Fourteenth of July, the government... [had] him arrested as a preventive measure. He was given six months' imprisonment on the technical charge of carrying arms and wearing a military uniform, but he remained in Ste. Pélagie only until... sent to a convalescent home... A dreadful epidemic of was then raging in Paris, and Galois' transfer had been determined by the poor state of his health."

"The publication in the "Gazette des Ecoles" of a letter of Galois... which... scornfully criticised the director's tergiversations was... the last of many offenses. On Dec. 9, he was invited to leave the school, and his expulsion was ratified by the Royal Council on Jan. 3, 1831."

"He was now a prisoner on parole and took advantage... to carry on an intrigue with a woman... probably not... reputable ("une coquette de bas etage," says Raspail) ."

"[H]e was probably pressed by his friend, [Auguste] Chevalier, to join the Saint-Simonists, but he declined, and preferred to join... the "Societe des amis du peuple"."

"It was... another man who reentered the Ecole Normale in the autumn of 1830. ...The revolution had opened to him a fresh source of disillusion..."

"[G]enius possessing... a mere boy, a fragile little body divided within itself by disproportionate forces, an undeveloped mind crushed mercilessly between the exaltation of scientific discovery and the exaltation of sentiment."

"[U]nder a more liberal guise, the same oppression, the same favoritism, the same corruption soon took place under Louis-Philippe as under Charles X."

"His French biographer very clearly explains his attitude:There was in him a hardly disguised contempt for whosoever did not bow spontaneously and immediately before his superiority, a rebellion against a judgment which his conscience challenged beforehand and a sort of unhealthy pleasure in leading it further astray and in turning it entirely against himself. Indeed, it is frequently observed that those people who believe that they have most to complain of persecution could hardly do without it and, if need be, will provoke it. To pass oneself off for a fool is another way and not the least savory, of making fools of others."

"In the... ensuing year, he sent three more papers to mathematical journals and a new memoir to the Academie. The permanent secretary, Fourier, took it home with him, but died before having examined it, and... [it] was not retrieved... Thus his second memoir was lost like the former."

"In 1829 he entered the Ecole Normale... then passing through the most languid period of its existence. ...[T]here too, the main student body inclined toward liberalism, though their convictions were very weak and passive as compared with the... Polytechnique... Evariste suffered doubly, for his political desires were checked and his mathematical ability remained unrecognized."

"He considered himself a victim of a... social organization which... sacrifices genius to mediocrity, and... he cursed the hated regime of oppression which... precipitated his father's death and against which the storm was gathering."

"On July 2... 1829, his father had been driven to commit suicide... This terrible blow, following many smaller miseries, left a very deep mark... His hatred of injustice became the more violent... his father's death incensed him, and developed his tendency to see injustice and baseness everywhere."

"Evariste was in the possession of his general principles by the beginning of 1830... at the age of eighteen, and that he... knew their importance. ...[H]e did not trouble himself to write his memoirs with sufficient clearness and to give the explanations... necessary because his thoughts were... novel. ...Instead ...Galois enveloped his thought in ...secrecy by his efforts to attain ...conciseness, that coquetry of mathematicians."

"Four days later two men challenged him to a duel. ...According to Evariste's younger brother the duel was not fair. Evariste, weak as he was, had to deal with two ruffians hired to murder him."

"was... the highest mathematical school in France and... also a daughter of the Revolution who had remained faithful... The young Polytechnicians were the natural leaders of every political rebellion; liberalism was for them a matter of traditional duty. ...[T]hus twice sacred to Galois, and his failure to be accepted was a double misfortune."

"Benjamin Franklin performed a beautiful experiment using surfactants; on a pond at Clapham Common, he poured a small amount of oleic acid, a natural surfactant which tends to form a dense film at the water-air interface. He measured the volume required to cover all the pond. Knowing the area, he then knew the height of the film, something like three nanometers in our current units. This was to my knowledge the first measurement of the size of molecules. In our days, when we are spoilt with exceedingly complex toys, such as nuclear reactors or synchrotron sources, I particularly like to describe experiments of this Franklin style to my students. Surfactants allow us to protect a water surface, and to generate these beautiful soap bubbles, which are the delight of our children."

"[A]t the age of sixteen he believed that he had found a method of solving general equations of the fifth degree. ...[B]efore succeeding in proving the impossibility of such resolution, Abel had made the same mistake."

"Galois was already trying to... enter the ... as early as 1828 — but failed. This failure was very bitter to him... he considered it as unfair... [but] his extra knowledge could not compensate for his deficiencies... The next year he published his first paper, and sent his first communication to the Academie des Sciences... lost through Cauchy's negligence. This embittered Galois even more. A second failure to enter Polytechnique seemed to... climax... his misfortune..."

"The last letter addressed to... Auguste Chevalier, was a... scientific testament. Its seven pages, hastily written... contain a summary of the discoveries which he had been unable to develop. This statement is so concise and... full that its significance could be understood only gradually... It proves the depth of his insight, for it anticipates discoveries of a much later date. At the end of the letter, after requesting his friend to publish it and to ask Jacobi or Gauss to pronounce upon it, he added: "After that, I hope some people will find it profitable to unravel this mess. Je t'embrasse avec effusion." ...[T]he greatest mathematicians of the century have found it very profitable ...to clear up Galois' ideas."

"Galois... had nothing of the topologist about him..."

"Although... structural theories of a major division of analysis originated in the late nineteenth century, they are more in the spirit of the general analysis of the twentieth. Their primary objectives are to discover what can be done rather than to do it, and to give criteria for what cannot be done. ...[A]s in Abel’s proof that the general quintic is not solvable by radicals, a demonstration of impossibility definitely disposed of what might seem a reasonable problem. Once more the methodology of Abel and Galois made an outstanding contribution to the development of mathematics. In this connection it is interesting to recall Lie’s opinion that the pattern of nineteenth century mathematics was laid out by four men, Gauss, Cauchy, Abel, and Galois."

"With this highly abstract definition of a space in mind, we return once more to Klein’s program and its successors. It is interesting to observe the abstract identity between the following description of spacial structure and structure as described in connection with modern algebra, and further to note once more that the basic concepts originated with Galois. Two spaces are called equivalent or (simply) isomorphic if there is a one-one correspondence between the objects in the spaces which establishes a one-one correspondence between all the properties constituting the structures of the respective spaces. When this is applied to two spaces which are the same, there is thus defined what is called an of the space. It follows... from these definitions that all the automorphisms of a given space form a group."

"Congruences were responsible for one theory of far more than merely arithmetical interest. The notation for a congruence suggests the introduction of appropriate 'imaginaries' to supply the congruence with roots equal in number to the degree of the congruence when there is a deficiency of real roots. As in the corresponding algebraic problem, it is not obvious that imaginaries can be introduced consistently. That they can, was first proved in 1830 by Galois, who invented the required 'numbers,' since called Galois imaginaries, for the solution of any irreducible congruence F(x) = 0 mod p , where p is prime. He thus obtained a generalization of Fermat’s theorem, and laid the foundation of the theory of s. ...Galois was eighteen when he invented his imaginaries."

"Probably almost anyone who has ever seriously attempted to solve differential equations by the will appreciate the labor inherent in any such heroic project as Wilczynski’s and agree with Galois that, whatever the nature of its unchallenged merits, the theory of groups does not afford a practicable method for solving equations. Galois of course was speaking of s, but his opinion, in the judgment of experts in the Lie theory, carries over to differential equations. Beyond a not very advanced stage of complexity, the calculations become prohibitive to even the most persevering obstinacy."

"Gauss, the penniless son of a day laborer, was educated by society as represented by the Duke of Brunswick. Today he would be educated at public expense... An Abel, no doubt, would be sent by the municipal health authorities to a sanitarium, where he might recover. A Galois... would find himself at outs with respectability, or in the protective custody of the police on some trumped-up charge... or in a concentration camp. For there is but little evidence that teachers are less helpless in the disturbing presence of a mind of the very highest intelligence than they were... or that the guardians of law and order are less nervous than they were when they sentenced Galois to... jail on a legal technicality. Aesop’s fable of the peacock and the crows has an element of permanence... you are different from us; get out or be plucked."

"[A]s noted by Lie in the grand summary (1893) of his lifework, the nineteenth century’s greatest effort in formal algebra—as distinguished from the more abstract, structural algebra originating with Galois—was absorbed in analysis."

"[A]ny mathematician today must be impressed by the apparent permanence of the ideas introduced by Abel... and Galois.., and the profound difference between their approach to mathematics and that of their predecessors including, in some respects, Gauss... To these young men, perhaps more than to any other two mathematicians, can be traced the pursuit of generality which distinguishes the mathematics of the recent period, beginning with Gauss in 1801, from that of the middle period. They initiated... the deliberate search for inclusive methods and comprehensive theories. Their forerunners in the middle period were Descartes with his general method in geometry; Newton and Leibniz with the differential and integral calculus created to attack the mathematics of continuity by a uniform procedure; and Lagrange, with his universal method in mechanics. Their contemporary in recent mathematics was Gauss, who in his arithmetic sought to unify much of the uncorrelated work of the leading arithmeticians from Fermat to Euler, Lagrange, and Legendre. Both Abel and Galois acknowledged their indebtedness to the theory of cyclotomy created by Gauss; and although they went far beyond him in their own algebra (Abel in analysis also), it is at least conceivable that neither Abel nor Galois would have chosen the road he followed had it not been for the hints in the Gaussian theory of binomial equations."

"Both Abel and Galois died long before their time, Abel at the age of twenty-seven from tuberculosis induced by poverty, Galois at twenty-one of a pistol shot received in a meaningless duel. When Abel’s genius was recognized, he was subsidized by friends and the Norwegian government. By nature he was genial and optimistic. Galois spent a considerable part of his five or six productive years in a hopeless fight against the stupidities and malicious jealousy of teachers and the smug indifference of academicians. Not at first quarrelsome or perverse, he became both."

"Galois, who was Lie's idol, indirectly inspired the application of continuous groups to differential equations. In a letter of 1874 to A. Mayer, Lie observes that "In the theory of algebraic equations before Galois only these questions were proposed: Is an equation solvable by radicals, and how is it to be solved? Since Galois, among other questions proposed is this: How is an equation to be solved by radicals in the simplest way possible? I believe the time is come to make a similar progress in differential equations.""

"Whoever, if anybody, was responsible for the colossal waste represented by these two premature deaths, it seems probable that mathematics was needlessly deprived of the natural successsors of Gauss. What Abel and Galois might have accomplished in a normal lifetime cannot be even conjectured. ...Early maturity and sustained productivity are the rule, not the exception, for the greatest mathematicians. It may be true that the most original ideas come early; but it takes time to work them out."

"The simple of any two groups was defined in connection with the postulates for a group. Galois considered simply isomorphic groups as the same group which, abstractly, they are."

"The earliest discussion from the standpoint of groups of the (modular) equations arising in the division of s was by Galois."

"The work of Galois and his successors showed that the nature, or explicit definition, of the roots of an algebraic equation is reflected in the structure of the group of the equation for the field of its coefficients. This group can be determined nontentatively in a finite number of steps, although, as Galois himself emphasized, his theory is not intended to be a practical method for solving equations. But, as stated by Hilbert, the Galois theory and the theory of algebraic numbers have their common root in that of algebraic fields. The last was initiated by Galois, developed by Dedekind and Kronecker in the mid-nineteenth century, refined and extended in the late nineteenth century by Hilbert and others, and finally, in the twentieth century, given a new direction by the work of Steinitz in 1910, and in that of E. Noether and her school since 1920."

"Even before Galois coined the term 'group,' A. L. Cauchy... made (1815) extensive investigations in what are now called s, and discovered some of the simpler basic theorems."