Prisoners

"[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"."

"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."

"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..."

"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."

"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."

"[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."

"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."

"[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..."

"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.""

"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."

"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."

"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."

"Younger than Gauss by thirty-four years, and dying twenty-three years before him, Galois... seems more modern... Gauss terminated his investigations on the nature of the solutions of with the binomial equationGalois grasped and solved (1830) the general problem, proving, among other things, necessary and sufficient conditions for the solution by radicals of any algebraic equation. Mathematics after Gauss, and partly during his own lifetime, became more general and more abstract... Interest in special problems sharply declined if there was a general problem including the special instances to be attacked... [i.e.,] mathematics after Gauss turned to the construction of inclusive theories and general methods which, theoretically... implied... detailed solutions of infinities of special problems. In this sense Galois was more modern..."

"He was still a mere boy, yet within these short years he had accomplished enough to prove indubitably that he was one of the greatest mathematicians of all times."

"[T]he soul of Galois will burn on throughout the ages and be a perpetual flame of inspiration."

"It is... not true... that he proved and used the simplicity of s. He did not need to: he was much cleverer than that; his treatment of solubility of equations is... simpler and more elegant than... textbook tradition."

"No existence could be more tragic... and the only one at all comparable... is.. that of... Niels Henrik Abel, who died of consumption at twenty-six in 1829... just when Galois was ready to take the torch from his hand and to run... further. Abel had the inestimable privilege of living six years longer... full years at the time that genius was ripe... of divine inspiration. What would not Galois have given us, if he had been granted six more such years at the climax of his life."

"It is perhaps less well known that [Galois] had also, without any possible doubt, discovered the essentials of the theory of abelian integrals, as Riemann would develop it 25 years later. By what route did he arrive at these conclusions? The fragments of calculations in Analysis found among his papers do not seem to permit much of an answer to that question, but there is room to imagine that he must have been very close to the idea of the Riemann surface associated with an algebraic function, and that such an idea must also be fundamental in his investigations into what he calls the "théorie de l'ambiguïté"."

"Langlands and Grothendieck are both (at least) Giants by any measure, and both were consciously successors of Galois."

"[D]iscoverers of fundamental principles are not generally awarded much recompense. They often die misunderstood and unrewarded. But... the fame... of Galois ...is based upon the unlimited future. He well knew the pregnancy of his thoughts, yet they were even more far-reaching than he could possibly dream of."

"Since my mathematical youth, I have been under the spell of the classical theory of Galois. This charm has forced me to return to it again and again."

"[In the first case, the problem is solving equations, including solving an algebraic equation with an unknown variable. In the second case, the problem is integrating differential equations. Galois was the first to recognize with absolute clarity how extraordinarily important the concepts of substitution group and invariant of a discontinuous group are for dealing with problems of the first kind.] Im ersten Falle hat man das Problem der Auflösung der Gleichungen, unter Anderm der Auflösung einer algebraischen Gleichung mit einer Unbekanuten. Im zweiten Falle hat man das Problem der Integration von Differentialgleichungen. Galois war der erste, der vollkommen klar erkannte, wie ausserordentlich wichtig die Begriffe Substiutionengruppe und Invariante einer discontinuirlichen Gruppe für die Behandlung von Problemen jener ersten Art sind."

"[We will not dwell further on the theory of transformation groups of a linear equation. We believe we have sufficiently demonstrated... the value of this theory, which is simply the natural extension to a question of analysis of the fruitful ideas introduced into algebra by Galois] Nous n'insisterons pas davantage sur la théorie des groupes de transformations d'une équation linéaire. Nous pensons avoir suffisamment montré... l'intérèt de cette théorie, qui n'est que l'extension bien naturelle à une question d'Analyse des idées si fécondes introduites en Algèbre par Galois."

"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]"

"Galois's introduction of imaginary roots of congruences has not only led to an important extension of the theory of numbers, but has given rise to wide generalizations of theorems which had been obtained in subjects like linear congruence groups by applying the ordinary theory of numbers."

"La démonstration de ce lemme n’est pas suffisante; mais il est vrai d’après le No . 100 du mémoire de Lagrange, Berlin, 1771. [The proof of this lemma is insufficient; but it is true according to No . 100 of the memoir by Lagrange, Berlin, 1771.]"

"An excessive desire for conciseness was the cause of this fault which one must try to avoid when writing on the mysterious abstractions of pure Algebra. Clarity is indeed an absolute necessity. ... Galois too often neglected this precept."

"His complete works fill only sixty-one small pages; but a French geometer, publishing a large volume some forty years after Galois' death, declared... it... simply a commentary on... [Galois'] discoveries. Since then, many more consequences have been deduced... and Galois'... ideas have influenced the whole of mathematical philosophy."

"It angered Galois sufficiently that he wrote directly below it:"

"Preserve my memory, since fate has not given me life enough for the country to know my name."

"[This] science is the work of the human mind, which is destined rather to study than to know, to seek the truth rather than to find it."

"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."

"Mais je n’ai pas le temps et mes idées ne sont pas encore bien développées sur ce terrain qui est immense. [But I don't have the time and my ideas are not yet well developed on this immense terrain.]"

"Il parait après cela qu'il n'y a aucun fruit à tirer de la solution que nous proposons. [It seems there is no fruit to be drawn from the solution we offer.]"

"Poisson, reading Galois' First Memoir, found the proof of Lemma III insufficient, and wrote in pencil the following comment."

"It is likely that... [in] a few centuries... [he] will appear... surrounded by the same halo of wonder as... Euclid, Archimedes, Descartes and Newton."

"He had read... books of geometry as easily as a novel... No sooner had he begun to study algebra than he read Lagrange's original memoirs. This extraordinary facility had been at first a revelation... but... it became more difficult for him to curb his own domineering thought and to sacrifice it to the routine of class work. ...By 1827 it had reached a critical point. This might be called the second crisis of his childhood: his scientific initiation. His change of mood was observed by the family. Juvenile gaiety was suddenly replaced by concentration; his manners became stranger every day. A mad desire to march forward along the solitary path... possessed him."

"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."

"... 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 !"

"A person came to make him a visit whilst he was sitting one day with a lady of his family, who retired upon that to another part of the room with her work, and seemed not to attend to the conversation between the Earl and the other person, which turned soon into some dispute upon subjects of religion; after a good deal of that sort of talk, the Earl said at last, "People differ in their discourse and profession about these matters, but men of sense are really but of one religion." Upon which says the lady of a sudden, "Pray, my lord, what religion is that which men of sense agree in?" "Madam," says the Earl, "men of sense never tell it.""

"I wish I could...give you a full notion of the idea which Mr. Locke had of that nobleman's merit. He lost no opportunity of speaking to it, and that in a manner which sufficiently showed he spoke from his heart... In short, Mr. Locke, so long as he lived, remembered with much delight the time he had spent in my Lord Shaftesbury's conversation; and never spoke of his known abilities with esteem only, but even with admiration."

"I will dwell a little longer on his character; for it was of a very extraordinary composition. He began to make a considerable figure very early. … He had a wonderful faculty in speaking to a popular assembly, and could mix both the facetious and the serious way of arguing very agreeably. He had a particular talent to make others trust to his judgment, and depend on it: and he brought over so many to a submission to his opinion, that I never knew any man equal to him in the art of governing parties, and of making himself the head of them. He was, as to religion, a deist at best."

"For close designs and crooked counsels fit, Sagacious, bold, and turbulent of wit, Restless, unfixed in principles and place, In power unpleased, impatient of disgrace; A fiery soul, which, working out its way, Fretted the pigmy-body to decay And o'er informed the tenement of clay. A daring pilot in extremity, Pleased with the danger, when the waves went high, He sought the storms; but, for a calm unfit, Would steer too nigh the sands to boast his wit. Great wits are sure to madness near allied And thin partitions do their bounds divide; Else, why should he, with wealth and honour blest, Refuse his age the needful hours of rest? Punish a body which he could not please, Bankrupt of life, yet prodigal of ease? And all to leave what with his toil he won To that unfeathered two-legged thing, a son, Got, while his soul did huddled notions try, And born a shapeless lump, like anarchy. In friendship false, implacable in hate, Resolved to ruin or to rule the state."

"Shaftesbury had, in reality, no intention of permanently subverting the independence of the individual members or of establishing a dictatorship based on popular support. He was forced to use the people in order to maintain pressure on the King, he had to establish close relations with the radicals, but he did not intend to share power with them. He used the most unscrupulous methods—subsidising perjurers and an inflammatory press, appealing to the masses with a daring and an ability unmatched in the next century and a half, because of the long odds which he faced. The party which he developed might appear to be revolutionary and unprecedented (with the ominous exception of Pym's), Shaftesbury might seem to be a real demagogue, a veritable Tribune of the people, but his ultimate objectives were essentially conservative. His theoretical proposals for the reform of the representative system, involving a drastic reduction in the size of the electorate, would have strengthened the independence of the individual member and established an oligarchy even more secure than that which was to rule in the eighteenth century."

"Not merely were the Whigs forced into total submission and inactivity in the years after 1683, but after 1688 those who called themselves Whigs explicitly repudiated Shaftesbury's example. To them, in retrospect, he appeared to have been a dangerous incendiary, another Pym. They revered Russell and Sidney as martyrs put to death by a tyrant, but they would not acknowledge Shaftesbury as their political ancestor. The frequent changes in his long career pointed to insincerity and opportunism, and his final conversion seemed to have been a tactical change of front rather than a genuine change of heart. It was not an accident that Shaftesbury had to wait so long for a biographer and apologist; he retained too much of the character of the age and circumstances which had produced Pym and Cromwell."

"In closing, I should like to cite a line from William Blake. “To see a world in a grain of sand - - - ” and allude to a possible parallel to see worlds in an electron."

"Borders are always dictated by the strong, never by the weak.… We simply consider it as a legitimate right and interest of the Serb nation to live in one state. This is the beginning and the end.… If we have to fight, by God we are going to fight. I hope that they will not be so crazy as to fight against us. If we do not know how to work properly or run an economy, at least we know how to fight properly."

"I tell you, Izetbegović has earned Sarajevo by not abandoning it. He's one tough guy. It's his."