Mathematicians From France

"D'Alembert was always surrounded by controversy. … he was the lightning rod which drew sparks from all the foes of the philosophes. … Unfortunately he carried this... pugnacity into his scientific research and once he had entered a controversy, he argued his cause with vigour and stubbornness. He closed his mind to the possibility that he might be wrong..."

"Historically, Jean d’Alembert precedes Augustin-Louis Cauchy. However, in the context of functional equations, it seems more natural to consider his contributions after Cauchy. Jean d’Alembert was a man of many names. The illegitimate son of an army officer, Louis-Camus Destouches, and a writer, Claudine Guérin de Tencin, he was born in Paris in 1717, while his father was abroad. Shortly after his birth, his mother abandoned him at the church of Saint-Jean-le-Rond. Following tradition, he was named Jean le Rond after the church, and placed in an orphanage. Upon the return of his father, he was removed from the orphanage, and placed with Mme. Rousseau, the wife of a glazier. Although Destouches continued to support his son financially, he chose not to publicly acknowledge his son. In 1738, Jean le Rond entered law school, where he was registered under the name Daremberg. He later changed this name to d’Alembert."

"Rien n'est plus incontestable que l'existence de nos sensations; ..."

"Lagrange did not explicitly recognize groups. Nevertheless, he obtained equivalents for some of the simpler properties of s. For example, one of his results, in modern terminology, states that the order of a of a divides the order of the group. Normal (self-conjugate, invariant) subgroups, basic in the theory of algebraic equations and in that of group structure, were introduced by Galois, who also invented the term 'group.'"

"Both Abel and Galois were indebted to Lagrange in their own, profounder work on s. ...The unique importance of Abel’s proof is that it inspired Galois to seek a deeper source of solvability, which he found in the theorem that an is solvable by radicals its group, for the field of its s, is solvable."

"The Galois theory of equations itself was the concluding episode in about three centuries of effort to penetrate the arithmetical nature of the roots of algebraic equations."

"Galois... made the terminal contribution so far as are concerned, and subsequent reworkings of his initial theory have added nothing basically new to his criteria for solvability by radicals. Even the modernized presentation of the , as in the streamlined model of E. Artin... is a tribute to the mathematical creed of Galois, in its elimination of all superfluous machinery. For this modern release from algebraic calculation, the direct approach of A. E. ‘Emmy’ Noether... in the 1920’s was primarily responsible. Much of her mathematics was in the spirit of Galois. But his methods, sharpened and generalized by his successors, have transcended the problem for which they were invented, and have rejuvenated much of living pure mathematics."

"[C]oncerning the vital residue of the theory of algebraic numbers... it, like the , can be traced to definite, highly special problems. Neither Galois nor the creators of the theory of s set out deliberately to revolutionize a mathematical technique; their comprehensive methods were invented to solve specific problems. Such appears to have been the usual path to abstractness, generality, and increased power. Some difficult problem... is taken as the point of departure without any conscious effort to create a comprehensive theory; repeated failures to achieve a solution by known procedures force the invention of new methods; and finally, the new methods, having been necessitated by a problem which appeared in the historical development, themselves pass into the main stream."

"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."

"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."

"Applications, or developments, of... extensions of number followed two main directions. ...The second, in the arithmetical spirit of Gauss, guided in part by the abstract algebraic outlook of Galois, led to a partial but extensive arithmetization of algebra."

"Of all these influences, two in particular are germane... the development of ; and the infiltration of the ideas of Abel and Galois into algebra as a whole. The of equations was acknowledged by both Dedekind and Kronecker to be the inspiration for their own general and semi-arithmetical approach to algebra. Two of the basic concepts of the Galois theory, domains of rationality, or fields, and groups, were the point of departure. ...Common algebra is the most familiar example of a field."

"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..."

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

"Three new approaches to number, in 1801 and in the 1830's, were to hint at the general concept of mathematical structure and reveal unsuspected horizons... That of 1801 was the concept of congruence, introduced by Gauss [age 24] in... the Disquisitiones arithmeticae... To this and the revolutionary work (1830-2) of E. Galois... in the theory of algebraic equations can be traced the partial execution of L. Kronecker’s... revolutionary program in the 1880's for basing all mathematics on the s."

"We must... indicate the involuntary participation of E. Galois... and N. H. Abel... in the development of Kronecker’s Pythagoreanism. Galois... adhered to no such creed. Nor did Abel. But it was in the attempt to understand and elucidate the of equations, left (1832)... in a... fragmentary and unapproachable condition, that Kronecker acquired some of his skill. Both Kronecker and Dedekind, two of the founders (E. E. Kummer... being a third) of the theory of algebraic numbers, were inspired partly by their scrutiny of the Galois theory to begin their... revolutionary work in algebra and arithmetic. ...Galois and Abel mark the beginning of one modern approach to algebra. The transition from highly finished individual theorems to abstract and widely inclusive theories..."

"The earliest recognitions of fields, but without explicit definition, appear to be in the researches of Abel (1828) and Galois (1830-1) on the solution of equations by radicals. The first formal lectures on the were those of Dedekind to two students in the early 1850's. Kronecker also at that time began his studies on abelian equations. It appears that the concept of a field passed into mathematics through the arithmetical works of Dedekind and Kronecker."

"Gauss took as the subject for his doctor’s dissertation (1799) a proof of the : an algebraic equation has a root of the form a + bi, a , b real. ...After Girard’s conjecture, there had been attempts at proof, including essays by D’Alembert (1746) and Euler (1749). All were faulty, as were the first and fourth attempts (1799) by Gauss. ...[T]he fundamental theorem in its classic form, as proved in the theory of functions of a complex variable, is no longer regarded as belonging to algebra. It is supplanted in modern algebra by a statement which is almost a triviality. The basic ideas of the modern treatment go back to Galois.., Dedekind.., and Kronecker.., not to Gauss."

"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?"

"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."

"It is painful to think that a few rays of generosity from the heart of his elders might have saved this boy or... sweetened his life."

"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."

"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."

"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."

"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."

"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) ."

"[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."

"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 first formulas for the computation of the s of the roots of an equation seem to have been worked out by Newton, although Girard (1629) had given, without proof, a formula for a power of the sum, and Cardan (1545) had made a slight beginning in the theory. In the 18th century Lagrange (1768) and Waring (1770, 1782) made several valuable contributions to the subject, but the first tables, reaching to the tenth degree, appeared in 1809 in the Meyer Hirsch Aufgabensammlung. In Cauchy's celebrated memoir on determinants (1812) the subject began to assume new prominence, and both he and Gauss (1816) made numerous and important additions to the theory. It is, however, since the discoveries by Galois that the subject has become one of great significance."

"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."

"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."

"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.""

"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."

"[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."

"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."

"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."

"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."

"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."

"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."

"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."

"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.""

"[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."

"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."

"[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."

"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."

"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."

"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..."

"[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 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."