The theory of equations and the training of French mathematicians (1794–1830)

Since the 1980s, scholars have shown that the practices and skills central to doing science are difficult to master (Collins Reference Collins1974; Pickering Reference Pickering1995, 1–37, 113–156). Although focused on scholarly science, these studies suggest how important education is for establishing a scientific culture (Belhoste Reference Belhoste1998; Schubring Reference Schubring2001; Warwick Reference Warwick2003, 1–48, 227–285; Garcià-Belmar, Bertomeu-Sánchez, and Bensaude-Vincent 2005; Olesko Reference Olesko and Kaiser2005). Looking at the teaching of mathematics is particularly relevant because Galois was still a student when he sent his memoir to the Academy in 1830. Showing the links between the training and the scientific practices in this specific time and place will allow us to understand how Galois's mathematical world was ruled. In using the phrase “Galois's time,” I mean to suggest that educational processes are long-lasting ones. The French education system was profoundly transformed during the Revolution, in its global structure as well as in its programs. In spite of successive reforms about the general scientific training in high schools between 1808 and 1830, the program of the mathematical courses for students who wanted to make a scientific career remained mostly unchanged from the first years of the century to the 1830s.

Early in the nineteenth century, mathematicians did not differentiate between algebra and the theory of equations, which itself was divided between algebraic solution and numerical solution. The former was aimed at determining a sequence of operations in order to obtain the exact values of the roots to an equation from unspecified coefficients; only the four algebraic operations and square roots were allowed in this algorithm. The latter consisted in finding approximate values of the roots to an equation whose coefficients were numerically known; in that case, one could use analytical calculus, and in particular the notions of limits and derivative function. Thus, the two branches of the theory of equations used different mathematical skills. In particular, numerical solution employed calculus methods that, being beyond algebra, required more advanced mathematical knowledge.

Both skills were usually mastered by members of the French mathematical milieu in 1830. We can gain insights into the specifics of the training the members of the Parisian mathematical community had received by looking at the institution around which this community revolved, namely, the École polytechnique. Created in 1794, this school was devoted to training state engineers and commissioned officers; but because it was the only place to learn advanced mathematics, it rapidly became the main venue of choice for aspiring mathematicians (Belhoste Reference Belhoste, Bottazzini and Dahan2001; Belhoste Reference Belhoste2003, 73–104). The topics selected for the entrance examination to the École and the curricula of the mathematical courses taught there remained largely unchanged until 1830. Thus, at the beginning of the nineteenth century, aspiring mathematicians were expected to master the algebraic solution of equations from degrees two to four, polynomial calculation, cyclotomic polynomials, imaginary numbers and some methods of numerical solution (“Programme d'admission à l'École Polytechnique,” Le Moniteur, n° 179, 9 Ventôse an VIII (February 28th, 1800), in Belhoste Reference Belhoste1995, 73–77; Gilain Reference Gilain1989, 98–132). Moreover, the books usually used by students, the Éléments d'algèbre à l'usage de l'École centrale des Quatre-Nations (Lacroix Reference Lacroix1799), and the Compléments des éléments d'algèbre à l'usage de l'École centrale des Quatre-Nations (Lacroix Reference Lacroix1800), both by Sylvestre-François Lacroix, included the recent developments of the field (Lamandé Reference Lamandé2004).Footnote ⁴ For instance, Lacroix quoted the mathematical courses Laplace gave at the École Normale in 1795, which were quite innovative, and, in the third edition of 1804, he took some results from Gauss's Disquisitiones Arithmeticae, which had not been translated into French at that time. Lacroix's textbooks stand out for their longevity, but comparing the successive editions reveals that this course hardly evolved after 1804: the corpus of the theory of equations was fixed for almost 30 years until Abel and Galois undertook to renew it.Footnote ⁵ Hence the stability of the corpus allows us to assert that anyone who intended doing mathematics in 1830 had to master all the algebra expounded in these textbooks, whatever their first training had been, and whatever generation they belonged to. In Galois's times, the students’ algebraic knowledge was very similar to their elders’.

Analyzing the standard mathematical textbooks helps us to paint this picture with a finer brush by showing the practical skills attached to the theory of equations. These books not only inform us about the kind of knowledge that was to be passed on, but also elucidate the contours of each sub-discipline, the way knowledge was introduced and used in examples and applications. They embody cognitive and methodological preferences, which will be adopted by the later generations in the course of their training, and which will be considered as the right way to ask questions and to solve problems (Schubring Reference Schubring1987). Thus opening Lacroix's textbooks allows us to see Galois's paper through the eyes of its readers.

Lacroix's Élements d'algèbre and their Compléments covered algebra, from the rules of formal calculation to the general solution of equations. In the 1800s, they were fairly up-to-date textbooks, both in their content and in the author's theoretical (rather than empirical) way of introducing the mathematical concepts, and they were not overtaken yet in the 1830s. Nevertheless, at the beginning of the nineteenth century, this “modern” algebra wasn't an abstract one. To Lacroix, algebra was associated with the solution of problems, which means that what was needed was to effectively find the roots of the studied equation. In fact, Lacroix presented an extensive compilation of results linked to the practical solution of specified equations by numerical methods in the first book, and, in the second one, he maintained that the general solution wasn't efficient: “If one has to give up finding the solutions to an unspecified equation by a limited number of algebraic operations on its coefficients, one has to be all the less sorry that the form of these expressions makes the numerical application always very long and sometimes impossible” (Lacroix Reference Lacroix1800, 61).Footnote ⁶

We can clarify this by considering the structure of the Compléments d'algèbre. Lacroix began with two chapters on the general theory of equations, taken almost totally from Lagrange. But this coherent theory was only a small part of the book. All the other chapters were methodological rather than theoretical: Lacroix provided simpler ways to solve the equations of degree three and four, gave techniques for reducing the degree, methods to know if the roots were real or imaginary numbers, or if they were positive or negative, formulas to calculate radicals quickly. Even with few numerical examples in this advanced textbook, the general purpose was to teach the students how to apply the theoretical frame to applications, and how to make the necessary calculations to solve problems. Knowing different theoretical results was insufficient; one had to be able to implement a wide range of concrete processes. The algebraic disciplinary habitusFootnote ⁷ appears in Lacroix's work as much as a theory as a calculation practice, the primary objective being to be able to obtain numerical results when they would be needed. In a letter he wrote between 1804 and 1810, Lacroix explained that he went “into each analytical discovery in relation to how promising it is for the physico-mathematical sciences” (Lamandé Reference Lamandé2004).

Nevertheless, one could argue that the goal of these textbooks was above all to instill in students an habitus in tune with the working practices of the engineers they were to become. The goal was not to train would-be scholars. Indeed, most mathematicians of 1830 were former students of the École polytechnique, but most former polytechniciens were state engineers, that is to say men who used mathematics only to solve concrete situations as they practiced their profession. In this context, the high theoretical level of the mathematical training was above all a way to provide a systematic method to solve practical problems.Footnote ⁸ Let us now go and look more carefully at the working practices of mathematical scholars to see to what degree they were filled with this disciplinary habitus too.

The theory of equations in scholarly practices

The predominance that Lacroix's textbooks gave to concrete problem-solving can be explained by the theoretical framework in which they were located. Indeed, Lacroix's reference book for this question was then the Traité de la resolution des équations numériques de tous les degrés, first published by Joseph-Louis Lagrange in 1797. But Lagrange's book was not simply used for teaching: contemporary mathematicians’ statements certified its authority within the community. This book was famous and held up as an example in many reports and papers of the Académie des sciences dealing with equations for the years 1795–1835 (Procès-verbaux des séances de l'Académie des sciences, vol. 2, 668–669; vol. 5, 214–216, 265–267, 292–295; Fourier [Reference Fourier1820] 1888–1890, 309; Fourier [Reference Fourier1827] 1888–1890, 129; Sturm Reference Sturm1835, 273). In Fourier's words, Lagrange's book must have been “well-known by every geometer” (Fourier Reference Fourier1831, 12). Moreover, it was considered to be the best possible theoretical synthesis at that time. In 1808, Poinsot awarded it the title of the most complete and thorough algebra treatise that one could wish for, because of “the author's gift for simplifying and extending the doctrines at the same time” (Poinsot Reference Poinsot1808, 344). Clearly, Lagrange's research was the keystone of scholarly algebra at the beginning of the nineteenth century (Grattan Guinness Reference Grattan-Guinness1990, vol. 1, 187–269). Any new algebraic research then had to find its way in relation to this treatise, either to follow it or to oppose it.

Yet, some historians of mathematics have ranked Lagrange among the precursors of group theory: the use he made of permutations of roots to develop his research on algebraic equations has earned him a respected position in the traditional account of the history of algebra, where he comes, together with Gauss, right before Abel and Galois (Kiernan Reference Kiernan1971; Dahan Reference Dahan and Walusinski1983; Wussing Reference Wussing1984, 71–75; Hirano Reference Hirano1995). This assessment seems anachronistic, for it relies on a structural point of view with which Lagrange was not familiar.Footnote ⁹ But since the tools Galois used – which, according to these historians, are supposed to have made his endeavor so different – were directly inspired from Lagrange's method of resolvents, Lacroix and Poisson could not have been disconcerted by it. On the other hand, both academicians were aware of the problems that Lagrange's approach had raised: the method of resolvents did not help to solve equations of degree more than four, and the new numerical methods that Lagrange had developed couldn't be applied in a practical way. Moreover, Lagrange didn't manage to make the theory of equations agree with his own wish to avoid what he called the “custom” of using curves: for this theory made it necessary to use calculus, it fell short of the methodological purity that this mathematician was looking for (Sinaceur Reference Sinaceur1991, 71–76).

Finally, Lagrange himself thought that his research ended in semi-failure for what he called the “théorie générale des équations” [general theory of equations]: “Even if one succeeded in solving on a general way equations of fifth degree or more, one would only have algebraic formulas. They would be valuable in themselves but not very useful to solve given equations effectively and numerically. As a consequence, one wouldn't be exempted from coming back to arithmetical methods” (Lagrange [Reference Lagrange1808] 1867–1892, 15–16).Footnote ¹⁰

Still, the Lagrangian agenda generated the two main strands of research on equations at the beginning of the nineteenth century. First, solving an equation algebraically meant finding an explicit formula that may enable mathematicians to express the roots in a finite form from the coefficients; progress could come from simpler methods, but not from determining whether such a formula did exist or not, which was what Galois did. Second, the aim of the theory of equations was above all practical; the method was only valuable if it ended by an approximate value of the roots – even if it had to cross the borders of algebra and use analytical tools for that. Applicability was much more important than methodological purity or the cleverness of the argument.

Yet, Galois's heterodoxy on the calculation point is striking. Indeed, Galois provided written arguments, when common practice was to show calculations whose accuracy was the only evidence of the exactness of the proof. This habit, very common in the usual textbooks and treatises that constituted the core of the scholarly tradition vis-à-vis equations, was also manifest in everyday practice of scholars working on this theory.Footnote ¹¹ We can understand this habit by looking at the reports the Academy made on papers dealing with the theory of equations, which usually started with a summary of these works. In 1805, when Rochat proposed a new method (otherwise mistaken!) to solve the third degree equation, the academician Legendre explained that he had done it with “a fairly specious analysis” and a “clever” method of calculation (Procès-verbaux des séances de l'Académie des sciences, vol. 3, 21 Brumaire, year XIII, 151). In the same fashion, in 1813, Poisson wrote a report about Dubourguet's geometrical method to determine the number and the sign of the real roots of some kinds of equations. This method consisted, first, in substituting two new equations of two unknown quantities constructed by means of curves for each initial equation. Then Dubourguet examined these curves to determine the number of their intersections, and therefore the number of real roots of the initial equation (Procès-verbaux des séances de l'Académie des sciences, vol. 5, 17 May 1813, 211–212). A paper on the same subject, one that used parabolas, was submitted by Salvage in 1820; but Cauchy criticized its method for being much more complicated than the ones already known (Procès-verbaux des séances de l'Académie des sciences, vol. 7, 10 January 1820, 5). As for Budan de Boilaurent's method of numerical solution, in 1806, the referee Legendre explained that it was based on a “process to transform a given equation of an unknown quantity x into a sequence of equations whose unknown quantities would be x-1, x-2, x-3,” and added that this process was efficient because it made the transformations easier, and avoided multiplying the coefficients of the given equation by those of the powers of the binomial (Procès-verbaux des séances de l'Académie des sciences, vol. 3, 8 Brumaire, year XII, 21–23).Footnote ¹² In 1823, another paper on this topic was presented by Ranson. It also relied exclusively on calculation, which the author claimed to be simpler but proved to be false (Procès-verbaux des séances de l'Académie des sciences, vol. 8, 8 December 1823, 591).

Thus, work on the theory of equations at the beginning of the nineteenth century usually consisted of exhibiting a method of calculation of roots. The value of this method derived from its effectiveness compared to the ones already known. Finally, geometers were following Lagrange's advice. This was what Poisson and Lacroix looked to find in Galois's paper; it was what geometers usually did, and it was what Lagrange himself had defined as the only legitimate method. And this certainly was where their first criticism originated. Of course, they understood that Galois was dealing with another kind of problem; but for them, the task following Lagrange's research was to provide a practical method to solve some particular equation within the precisely bordered scholarly tradition of algebra. It was not to compete with Lagrange or to try to encompass the whole problem with a brand new theory.

Galois, however, thought otherwise. The new mathematical idea of group of permutations of roots that he used and the notion of adjunction he took from Gauss and Abel required that he argued on objects that were not numbers and that, consequently, could not use traditional algebraic calculation. But this change in outlook made the application of Galois's theorems problematical. The priority had to be inverted: necessarily, theory had to take precedence over applications in dealing with new objects that were not based in ad-hoc methods that would have made them fit applications. The practical techniques were not yet developed to apply Galois's ideas. Instead, Galois offered the Académie des sciences nothing less than a new way to look at the problem. He would define it some months later, in a text that would become very famous, as the “analysis of analysis” (Galois [Reference Galois1962] 1997, 3–11). So Galois defined his own mathematical practice as “analysis,” and not algebra. But what was analysis?

Mathematical analysis

Nowadays, “analysis” refers to a specific part of mathematics, the study of functions with differential and integral calculus. The Greek classical tradition saw in it a special way of reasoning: opposed to synthesis, analysis refers to arguments based on the decomposition of the problem to be solved (see Otte and Panza 1997, 260–271, for a more general perspective on this point). At the beginning of the nineteenth century, “analysis” was nearly synonymous with “mathematics”: significantly, all the mathematical works (and a large part of the physical ones) submitted to the French Academy were then filed in the “mathematical analysis” chapter of the published accounts of the meetings. But this doesn't mean that analysis was just a nineteenth-century word for mathematics “in general,” as an a-historical category. On the contrary, as we will see, research in mathematics in the 1830s implied a specific practice, a practice which was called “analysis” embodying mathematical theory and skills, ways of thinking, and a concrete approach to mathematical questions.Footnote ¹³

The mathematical body of knowledge on which analysis rested is well known. Inherited from Leibniz and Newton, its technical tools were developed during the eighteenth and nineteenth centuries in such founding treatises as Euler's Introductio in analysin infinitorum (Euler Reference Euler1748), Lagrange's Théorie des fonctions analytiques (Lagrange Reference Lagrange1797) and Cauchy's Cours d'analyse (Cauchy Reference Cauchy1821). The body of knowledge associated with analysis at the beginning of the nineteenth century was what we call today differential and integral calculus (Jahnke Reference Jahnke and Jahnke2003). Moreover, the development of these mathematical concepts implied a change in mathematical practices. Analysis was essentially a method of calculation that relied upon functional relations, a method for which no visual representation was needed; hence finite or infinitesimal quantities could be used, but not curves nor geometrical methods (Belhoste Reference Belhoste2003, 232–235).

The traditional “Greek” sense of analysis as a general method of investigation for solving problems with the help of decomposition into simpler steps, was still used by mathematicians. For instance, Lacroix and Laplace, both analysts in the modern sense, defined it in their lectures as the method of splitting a problem into simpler ones for solution: “In analysis . . ., if one supposes that the question is solved, one can encompass all the subject and one can get to the solution by making a different formulation of the problem or by making it go through different forms” (Lacroix Reference Lacroix1805, 207–208).Footnote ¹⁴

But mathematical analysis at the beginning of the nineteenth century was much more than either a new mathematical tool or a traditional method. Calculus was only the technical side of mathematical analysis. When Laplace was writing his Système du monde (Laplace Reference Laplace1799) or when Fourier was writing his Théorie analytique de la chaleur (Fourier Reference Fourier1822), they were actually doing mathematical analysis. At the beginning of the nineteenth century, mathematical analysis had become a major scientific discipline whose scope was much larger than differential and integral calculus (Rashed Reference Rashed and Rashed1988). In particular, one of the facets of analytic orthodoxy was the possibility to come and go from general theory to calculation-based applications.

This point of view is revealed in the text that Dupin read at the public annual session of the Académie des sciences, on 28 December 1835, which was entitled “Coup d'œil sur quelques progrès des sciences mathématiques en France depuis 1830” [A glance at the progress of mathematical sciences in France until 1830] (Dupin Reference Dupin1835). The mathematical science that Dupin described embodied Legendre's book on elliptic functions as well as Poisson's research on astronomy or the project to survey the country using the triangulation method. Among the prominent works published between 1830 and 1835, Dupin quoted Fourier's posthumous book on equations, and took this opportunity to pay tribute to the former Perpetual Secretary's works. Thus Dupin explained what made them so valuable: “Every theoretical discovery is illustrated by significant applications. Moreover, thanks to the general analysis methods that he invented for this purpose, new applications, which couldn't be done before, have become possible” (ibid., 565).Footnote ¹⁵

Dupin did not differentiate between Fourier's work on equations and his famous book on the theory of heat. Algebra and mathematical physics weren't different scientific disciplines (see Goldstein, Schappacher, and Schwermer 2007, 3–129, for a historical reflection on this concept). They were parts of the same one – analysis. Moreover, according to Dupin, the legitimacy of analysis could be explained by strictly internal factors: it derived from its capacity to account for physical phenomena, in respect of both theory and numerical application; that is to say from its efficiency and its generality. But efficiency and generality are not ontological properties of a mathematical theory. They are values that people, in one given place and at one given time, attributed to this mathematical theory. In Galois's times, the scientific legitimacy of analysis went hand in hand with a social one.

These were values strongly linked to the structure of the French scientific community. Taught at the École polytechnique, whose former students, like Dupin, ruled official mathematical science, analysis had become an habitus, shared by all the actors of the mathematical community. These men agreed on a corpus of knowledge and a system of representations it was related to; they shared epistemological and cognitive views about the doing of science (Dhombres and Julia Reference Dhombres and Julia1993). The analytical skills they had learned at the École defined their positions and their roles in the scientific world: being a polytechnicien meant having symbolic capital, whose guarantee was precisely these analytical skills.Footnote ¹⁶ So, in the 1830s, mathematicians who, thanks to their institutional positions, were able to assess the value of any new research shared a view of mathematics characterized by a permanent dialogue between theory and applications and by the primacy of analysis, which was then seen both as a corpus of knowledge and as a universal method of invention.Footnote ¹⁷ In the case of Galois's work, these referees were Cauchy, Poinsot, Fourier, Poisson, and Lacroix.Footnote ¹⁸ Each of these academicians, of course, had his own mathematical preferences; but reviewing their publications, we can see that their agenda was not guided by the distinctive features of a subject but by the uniformity of problems and methods, circulating from one field to another thanks to the generality of viewpoint that analysis afforded.

In this context, it seemed necessary for anyone looking to undertake mathematical research to master all sides of the analytical skill, even if this research was not what one would call a “typical” analytical one. In particular, algebra was not an autonomous discipline with its own methodology or reasoning. The theory of equations was part of “algebraic analysis,” which also encompassed real and complex functions, convergent and divergent series, and the expansion of rational functions (Cauchy Reference Cauchy1821, ij). Just like physics or cartography, it was a field of mathematical science for which the whole analytical habitus should be applied. For instance, while reporting on Fourier's posthumous book Analyse des équations déterminées, which dealt with numerical solution of equations and was, as such, an algebra book,Footnote ¹⁹ Dupin stressed that it was based on the same methodology as all Fourier's research and in particular that “it was characterized by and impregnated with methods of integration” (Dupin Reference Dupin1835, 564–565). In fact, for Fourier, “the principles [of algebraic analysis] belonged to general analysis” and his theory “was not limited to algebraic equations” but “could solve every determined equation” (Fourier Reference Fourier1831, 3).

The circulation of problems and methods among the different domains to which analysis applied also had institutional consequences, which can be seen in the academic commissions appointed to examine the submitted memoirs. A close examination of the exchanges that often occurred among the different sections shows that, for everyday working practices, the only real division within the Academy was between mathematical and natural sciences. Specialists in physics, astronomy, mechanics, and geometry actually worked together to referee mathematical papers (Chabot Reference Chabot, Brian and Demeulenaere-Douyère2002).

So Galois's readers used thinking patterns united by the practice of analysis, and they looked for Galois to do the same. One key to understanding the reaction to Galois's work, then, is to determine the way his work fitted with the prevailing mental equipment (patterns of thought): Galois wanted to make “the analysis of analysis,” but one may wonder whether he was still practicing analysis as his readers understood it.Footnote ²⁰

The analytical practice of Algebra: Cauchy (Reference Cauchy1815) vs Galois (Reference Galois1830)

To understand how algebraic research relied, in a practical way, on the nineteenth-century French mathematical habitus, and then what Galois's readers expected to find in his paper, we will compare this paper to another. But as every mathematical research is unique by definition, we will first explain what this means. First, we need a paper that could have been seen by the academicians as one that resembles Galois's with respect to the author's position in the community and to its subject. As we have seen, the algebraic theory of equation wasn't considered an autonomous field of research but part of a larger one, the theory of equation; therefore, it won't be necessary to pick up a paper that would be on exactly the same topic as Galois's. Moreover, to make the comparison, we need to consider a work which was approved by the Academy. Consider for example Cauchy's paper “Sur la determination du nombre de racines réelles dans les equations algébriques” [On the determination of the number of real roots of algebraic equations] (Cauchy Reference Cauchy1815), which fills all these conditions: it deals with equations, and it was sent and approved in 1813, when its author had just finished his studies. Moreover, in both cases, Poisson was asked to referee and he pointed out the same weakness in Cauchy's as in Galois's memoir: it was difficult to make effective calculation. Second, we would like to add that we will not focus our attention on the mathematical contents of these papers (which are uncommensurable), but rather on their global argumentative structure: the aim is to analyze how the two authors write their proofs to convince their readers of its interest.

Cauchy's paper sought to “translate into analysis” a result that the Abbé de Gua had reached in 1741 by geometrical methods involving the study of curves. As Cauchy's proof was all about “analysis,” the use of derivative functions was essential. Working on an equation the number of real roots of which was to be determined, Cauchy began to show that the problem could be reduced to the solution of the derivative equation, then to the solution to the same kind of problem on two equations of lower degrees. Thus, the problem was gradually broken down into the solution of a larger number of simpler equations (Sinaceur Reference Sinaceur1991, 112–116). Nevertheless, this theoretical part only took up 10 pages out of 88. Some were devoted to provide further practical explanations about this result including taking account of special cases. Lastly, the larger part of the dissertation (62 pages) was entitled “développements analytiques” [analytical developments], and sought to apply the former results to various problems and particular cases, such as equations of known degree.

Hence, Cauchy's paper showed the features we have identified with the analytical processes at the beginning of the nineteenth century. The dissertation began with a general theory whose proof was based on a decomposition of the problem to solve on the one hand, and, on the theoretical tools provided by differential calculus, on the other hand. Then Cauchy adjusted the theory to make it useful before finally looking back at its applications, with the introduction of many examples. All the patterns of the analytical habitus were present in Cauchy's algebraic practice, precisely as expected for this kind of subject. Despite some objections about the applicability of the method, after Poisson's report, the text was approved by the Academy (Procès-verbaux des séances de l'Académie des sciences, vol. 5, 24 May 1813, 214–216).

We can now understand what Poisson and Lacroix meant when they blamed Galois's work for being neither clear enough nor developed enough. These two elements were deeply linked, derived from the attention paid to and the place taken by the applications of the theoretical result. For these academicians, the clarity of a proof necessarily proceeded from analytical reasoning and was sustained by the presence of applications developed in the text. They criticized Galois's work for lacking this clarity because they didn't recognize the mathematics they were themselves practicing in it. In fact, as Galois explained in the preliminary discourse of his dissertation:

Galois preceded his work with these lines because he realized perfectly the importance mathematicians lent to these questions, and that it was a weakness in his research. He anticipated criticism, explaining that adding a hypothesis on the equation to solve made the calculation doable. This argument played the same role as the second part of Cauchy's dissertation: Galois wanted to show that general theorems could become operative for some particular cases, such as equations of prime degree, as long as they were suitably modified. The way Galois wrote his paper was in fact close to the two first parts of Cauchy's paper. It began with a general theory, called “principles,” whose proof was based on a kind of analytical reduction process.Footnote ²² Like Cauchy, Galois tried to show how his theory could work when one added general hypothesis in a second part explicitly called “application”; and, for both Galois and Cauchy, effective calculation was not possible at that stage.

Nevertheless, these papers differed on two points. First, Cauchy devoted much of his memoir to this work of clarification (part 2), for which he gave many applications; Galois only dealt with one application, the equations of prime-number degree. Second, Cauchy persuades the reader that calculation was not an insurmountable problem, because he actually showed a large number of concrete examples for which the theory worked (part 3). On the contrary, the equation of fifth degree was the only example in Galois's paper, and Galois gave the associated group without even discussing whether it was soluble or not. In this context, Galois's preliminary discourse looked like a sleight of hand – a mere declaration of principle.

Thus, if we compare Galois's dissertation with Cauchy's, both submitted to the Académie des sciences, and if we measure them up against the analytical standards of their time, we see that only the latter seems complete. This is neither because of the novelty of the ideas, nor because of the incompletion of some proofs. In fact, the clarity of a mathematical text was not a matter of theoretical rigor in the 1830s. A mathematical text could only seem clear to a “polytechnique-trained” reader if he could find concrete applications of the general theory within it.

The texts that Galois wrote to explain and justify his research clearly showed that he was aware of this requirement: he recognized the question of application. All the comments he made about his work can be read as attempts to legitimate by epistemological arguments an objective fact that he had to deal with, while being sure that it was actually a heterodox one: calculations could not be made. Yet he recognized how far his research broke new ground. Thus, he decided to stress the significance of his discoveries, showing that they were not “outside” the mathematical values and questions of his time. In fact, the “analysis of analysis” Galois wanted to introduce took applications into account, but changed their object: “The highest calculations carried out up to now are considered as particular cases with which it was useful and necessary to deal, but which it will be disastrous not to give up for broader research” (ibid., 11).Footnote ²³

Galois thus wanted to take mathematical research to higher abstraction so that the abstract objects that belonged to the classical analytical field could become applications of this new analytical field. The link between the traditional mathematical objects and the new analysis that Galois promoted would be exactly the same as the link between numerical applications and the traditional analysis. Our mathematician remained within the theory-application dialectic typical of the beginning of the nineteenth century, but hoped to take it to a further level of abstraction. In fact, Galois didn't break with the mathematical practice of his time; he just proposed to broaden it in order to overcome the limits of the ordinary analytical method. In this he was not alone; others felt that a period was ending. For example, a famous German scholar residing in France, Lejeune-Dirichlet, also suggested that it was time to “substitute ideas for calculations.”Footnote ²⁴

However, this viewpoint still had no place in the systems of representations of the older generation of French mathematicians. We need to take into account the generational break between Galois and the members of the Academy to understand why our mathematician's works were rejected. In the twentieth century, Galois was recognized as a visionary genius for the epistemological dimension people saw in his thought, but he was only a young student for his contemporaries. Of course, the Academy didn't accept his dissertation, but that didn't mean Galois was definitively and irrevocably banished from the mathematical community. Indeed, Lacroix and Poisson ended their report with words of encouragement, inviting Galois to continue his research. They found that Galois was a student with good prospects. But it was a long way from up-and-coming student to accomplished mathematician. A single research work wasn't enough, and extra-mathematical facts also played a part. To explain why Galois's work was not recognized as valuable, we now have to examine his case within the institutional dynamics of the Académie des sciences.

French mathematics in the 1830s: a big community or a small world?

In response to Poisson's report, Galois wrote several critical texts against the Academy and, more generally, against the Parisian scientific milieu (ibid., 3–27). Reading them, one can imagine that his experience reflected some personal hostility from the Academy. Indeed, these texts have brought the historical debate around to the contacts Galois had with its members. But we must remember the “self-fashioning nature” of these texts: seeking to interpret his own life, Galois painted himself as a persecuted young geometer (see Terral Reference Terral2006 for a historiographical note about the notion of self-fashioning). This brings us back again to the more general question of social history and the socialization of French scientists in the 1830s.

At the beginning of the nineteenth century, the French scientific community was much larger than it had been before the Revolution. The new advanced military schools and the Napoleonic lycées provided a common training, while new journals, such as the Annales de Gergonne, the Bulletin de Férussac, or the Journal de l'École polytechnique, made publishing research easier and faster. For the first time, a generation of well trained young men could claim the status of professional scientist. No longer the exclusive privilege of academicians, the status “professional scientist” was relatively widely available to increasing numbers of teachers due to the reform of the educational system. Because of this growth of the community, Galois could hope to earn his living in the field of science and to practice mathematics outside the Academy (Dhombres and Dhombres Reference Dhombres and Dhombres1989, 171–198; Karady Reference Karady, Fox and Weisz1980). For example, Joseph Gergonne, famous for founding the Annales de mathématiques pures et appliquées [Annals of pure and applied mathematics] and editing it from 1810 to 1831, started as a mathematics teacher before becoming a professor of Astronomy then chief education officer at the University of Montpellier (Dhombres Reference Dhombres, Ausejo and Hormigon1993; Gerini Reference Gerini and Condette2008; Otero Reference Otero1997, 10–14; Stigler Reference Stigler1976). Despite the success of his journal and the many articles he published, Gergonne never became a full member of the Academy, and was only elected as Corresponding Member in 1830. Arbogast, Servois, and the Français brothers were other teachers who influenced scholarly debates at the beginning of the nineteenth century without being integrated into the academic circle (Grattan-Guinness Reference Grattan-Guinness1990, vol. 1, 211–219; Friedelmeyer Reference Friedelmeyer1994, 254–264). Galois, who had published his first papers in mathematical journals (Galois Reference Galois1829a; Galois Reference Galois1829b; Galois Reference Galois1830a; Galois Reference Galois1830b; Galois Reference Galois1830c), could have followed their path if he had not been excluded from the École préparatoire – the French school where teachers-to-be were trained, and which would be called again the École Normale Supérieure from 1830 – because he had publicly criticized its director's actions during the July Revolution (Dupuy Reference Dupuy1896, 222–232).Footnote ²⁵

Thus, the scientific field offered Galois a “space of possibilities” much larger than what was labeled by the Academy of Sciences. Nevertheless, Galois's student position was at the bottom of the scale of positions within this field – in Bourdieu's words, Galois was in a “dominated” position. Still, Galois thought that he deserved more than a teacher's job in province and for this, the highest scientific rewards, he needed the Academy of Sciences whose members were in a “dominating” position within the scientific field through an institution whose role in evaluating research meant that it still embodied the scientific temperament (Crosland Reference Crosland1992, 242–278; Demeulenaere-Douyère Reference Demeulenaere-Douyère, Brian and Demeulenaere-Douyère1996). As Galois's friend Raspail wrote, it was the “Temple of Glory” for whoever wanted to have his work accepted as scientifically valid (Raspail Reference Raspail1837, VIII). Being recommended by the Academy was much better than being published in a scientific journal. Moreover, some members of the Academy provided influential advice to the Ministère de l'Instruction Publique: they chose the candidates for teacher jobs in higher education and must have chosen most of the Parisian high school teachers of mathematics.Footnote ²⁶

In 1830, Raspail violently criticized the fact that all the power was concentrated among a few within the French scientific field: “Everything is focused in Paris. . . . This is the place where one appoints, where one puzzles, where one obtains, where one teaches, where one publishes. Then, the coteries get together within the same surrounding walls, at the gates of the same power, on the seats of the same academies” (Raspail Reference Raspail1830, 150).Footnote ²⁷ This account may go too far, but it illuminates an important fact. Even if Galois's world had expanded, there remained a small group, namely the academicians, who could decide who would make a “great” career and who would not. At this point, we again face the question of the implicit rules governing the Academy of Sciences in the 1830s.

The men who lent their support to the beginning and progress of young aspiring mathematicians like Galois were the six members of the mathematics section of the Academy, to whom one can add some members elected in other sections but whose skills and work fit the scientific agenda. Between 1828 and 1830, these academicians were Ampère, Biot, Cauchy, Dupin, Fourier, Lacroix, Legendre, Navier, Poinsot, Poisson, and Prony.Footnote ²⁸ The situation changed in 1830: Arago became Perpetual Secretary instead of Fourier, who had died, and Cauchy, a royalist, went into exile after the July Revolution. In theory, all the scientists of this small group had the same status at the Academy, except Fourier, who was the Perpetual Secretary. But in fact, symbolic capital was not evenly distributed. Thus, if we look at the positions of the academicians who were called to report on Galois's work, this will help us to follow the evolutions of Galois's own position.

The two referees for the first paper were Poinsot and Cauchy. From 1820 to 1831, both served almost exclusively on committees dealing with papers on what one would call today “abstract” mathematics. While Poinsot was not very active in research or in evaluating research, Cauchy was the academician who devoted by far the most time to research, writing no less than an average of six new papers a year. But both remained aloof from the academic community (Bertrand Reference Bertrand1872; Belhoste Reference Belhoste1991, 50–60). For example, they were never asked to give their opinion regarding prizes or lists of candidates to a position. Historian René Taton showed that Cauchy helped Galois and read his work (Taton Reference Taton1971). This may have helped the young mathematician to improve his work and, as every one at the Academy recognized Cauchy's great scientific skill, this must have lent credibility to Galois's work. Still, Cauchy's help was of limited value to Galois, because he wasn't in a position to assign jobs. Saigey's necrology of Abel which was deliberately polemical, confirms this: “I must say, may it lessen the good opinion the academy formed about M. Abel's merit, that he had great regard for this academician's [Cauchy's] works, and went as far as giving him the top place among geometers” (Saigey Reference Saigey1829, 319).Footnote ²⁹ Then, Fourier was the referee for the paper Galois had sent for the Grand Prix de mathématiques of 1830. We don't know if he actually read it, but the fact that the Perpetual Secretary had agreed to referee it may prove that Galois had status at the Academy.

Finally, Poisson and Lacroix were appointed for the third paper. And, this time, a report was written. Whatever the conclusion of this report, the preparation of a report highlights Galois's growing stature in the scientific field. First, Lacroix and Poisson were respected members of the Academy. They were often asked to referee papers, especially in mathematics. At Galois's time, Lacroix was an old man, but his authority was unchallenged. Poisson loomed ever larger in the Academy and was becoming Laplace's heir (Costabel Reference Costabel, Costabel, Dugac and Métivier1981; Arnold Reference Arnold1983; Arnold Reference Arnold1984; Libri Reference Libri1840). After Fourier's death in 1827 and Cauchy's exile in 1830, there is no doubt that he was France's leading mathematician. Unlike Cauchy, Lacroix and Poisson had the privilege of playing important roles in awarding prizes and allocating jobs.Footnote ³⁰ Moreover, they also had duties within the educational field, so they could provide somebody with a job through other ways than the traditional academic one. That both men were in touch with Galois reveals how seriously his work was taken (Taton Reference Taton and Walusinski1983; Taton Reference Taton1971).

Second, it is quite sure that both men had heard about Galois. Even in Paris, there were very few advanced students of mathematics. Galois was one of the twelve students in the science section of the École préparatoire (Ehrhardt Reference Ehrhardt2008), and had to attend the courses at the Faculté des Sciences, where Lacroix taught calculus. Moreover, this school was the object of governmental attention: the Ministère de l'Instruction, where Poisson worked, was carefully watching the professors-to-be (Zwerling Reference Zwerling1990, 56–59).

Third, even a negative report shows that the members of the Academy respected Galois's scientific undertaking. While the Academy was supposed to report on each paper submitted to it, it did so for less than 50 per cent of the papers submitted between 1820 and 1835, and in most cases these were favorable reports. Faced with an increasing number of submissions, the academicians only wrote on those papers they believed worth the time to be spent on them. Lacroix and Poisson were particularly selective: they wrote on fewer than 20 per cent of the works they refereed (Chabot Reference Chabot1999, 260–324). Clearly, the Academy was not rebutting Galois. Rather, it was inviting him to carry on with his work.

Notably, Poisson mentioned in his report that Galois's project was worthy of interest and that he should actually expand on his work as much as improve it qualitatively. Coming from a man who spent an entire career improving the theories of his time instead of creating new ones – a man who knew to what extent each new mathematical object remained improvable – this could hardly be interpreted as a rebuke. Moreover, the assessment Poisson made of Galois's work was exactly the opposite of that he made about the theory of equations in general. Indeed, he gave his opinion on the interest of some mathematical questions in notes he had taken for his personal works: Poisson believed that it was unlikely that any progress would be made in the field of numerical or algebraic equations (Libri Reference Libri1840, 432).Footnote ³¹ In deciding not to express these doubts in the public report, Poisson might have sought to preserve Galois's budding scientific capital.

All in all, it seems that the mathematicians of the Academy shared Sophie Germain's opinion, to wit, that Galois was “a student who, in spite of his impertinence, seemed to have superior gifts” (Henry Reference Henry1879, 631–632). Tellingly, in the cover letter he sent to apply to the École préparatoire, Galois explained that he was “encouraged by people at the top of the learned world to attempt to launch into a mathematical career” (Archives Nationales, F/17/4176). All the candidates wrote such letters, but Galois was the only one to emphasize scientific recommendations. Galois's lamentations notwithstanding, he had received significant support from the mathematical community, which suggests that we must delve into the social process whereby a young scientist came to be accepted by the community as one of its own. Just how did one become a mathematician in the early nineteenth century?

Entering the world of mathematics

Now that we have shown that Galois wasn't especially neglected by the Academy of Sciences, we are able to analyze his over-reaction to the possibilities that the structure of the scientific field gave him – trying to understand why he felt persecuted. Moreover, the historical focus on the Galois affair tends to make one forget that Galois wasn't the only young mathematician in the 1830s, and that others obtained full recognition from the Academy. Comparing Galois's career-making tactics to those of other young mathematicians at the time is thus crucial to understanding his short and unhappy career.Footnote ³²

We have seen that the Academy of Sciences was so overwhelmed by the growing number of submissions that its members only wrote reports on half the papers they reviewed. In this context, helping young aspiring talents was no longer a priority, and others besides Galois complained. In a letter to Lacroix of 1814 – four years before his election – Dupin wrote that he had no news from Poinsot, Carnot, and Lamé, who were asked to referee one of his papers six months earlier; Dupin felt that his work was treated “with disdain, almost with contempt,” which was “cruel” from the Academy (Bibliothèque de l'Institut de France, Ms 2396, Fol. 78).Footnote ³³ Other contemporaneous accounts that underline this point were published – Libri, for one, mentioned in his necrology of Abel, “the nonchalance of modern geometers, of which each of them has been the victim when he was a young scientist, and because of which the papers of young mathematicians were almost never read” (Libri Reference Libri and Michaud1843, 59).Footnote ³⁴ Saigey went even further, claiming that “so few scientists can actually read, even superficially, the work that is submitted to them,” that “favoritism, or mere luck, decided the fate of a paper” (Saigey Reference Saigey1829). Waiting for months for academicians to review one's work, or being ignored completely, were normal, so common that Galois must have been aware of the fact. Moreover, the need to be recommended for career-making was not specific to the 1820s and 1830s. It was the norm under the Ancient Regime – Lacroix owed his career to Monge, for example. And this practice had not been abolished during the Revolution – Biot owed his career to Lacroix, Poisson owed his to Laplace (Frankel Reference Frankel1978; Belhoste Reference Belhoste and Dhombres1992; Hahn Reference Hahn2005, 154–155).

What had changed was that young scientists felt that their situation was unfair, and dared to express it. French education changed faster than the Academy and by the 1830s there were many people well trained in mathematics who believed in the meritocracy that the examination system of the École polytechnique promoted.Footnote ³⁵ Against them were a small number of mature geometers – academicians who worked in mathematics were on average 57 years old – who could not give jobs to all. Career-making then became a competitive process whose rules differed from those of the educational system.Footnote ³⁶

Moreover, a reference letter hardly sufficed to have a paper published. It was common for a young scientist to see his first submission turned down, and this certainly did not prevent the author from making a name for himself. Joseph Liouville, later himself a member of the Academy, submitted an early paper on electro-dynamics that was rejected (but he was encouraged to carry on) (Procés verbaux des séances de l'Académie des sciences, vol. 9, 15 September 1828, 119). It is equally significant that none of Liouville's next four submissions received a report (Procès verbaux des séances de l'Académie des sciences, vol. 9, 1 December 1818, 151; 20 April 1829, 231; 29 June 1829, 274; 2 November 1830, 528). After he was awarded the Mathematics prize in 1827 with Colladon, Charles Sturm, another young man on his way to mathematical fame, sent five articles to the Academy between 1828 and 1831 (Procès verbaux des séances de l'Académie des sciences, vol. 9, 25 May 1829, 253; 1 June 1829, 257; 8 June 1829, 259; 3 August 1829, 283; 19 October 1829, 286). All five were rejected. Indeed, his theorem on the numerical solution of equations shows that a result could have an important and immediate impact on the French mathematical community without having been approved by the Academy: after a summary was published without proof in the Bulletin de Férussac (Sturm Reference Sturm1829), a full version appeared in two different textbooks in 1832 and 1833 (Choquet and Mayer Reference Choquet and Mayer1832, 383–396; Lefébure de Fourcy Reference Lefébure de Fourcy1833, 483–493; Sinaceur Reference Sinaceur1991, 35–36).

Like Sturm, others responded to the Academy's lack of interest in their research by publishing parts of their work in a specialized journal. Publishing in extra-academic journals helped to expand the reach of the French scientific community and gave visibility to new scholars. But the papers they published were not the same as the academic ones. They were in general shorter – the Bulletin de Férussac, for example, only published summaries of current research. The authors didn't need to present a general theory – in the Annales de Gergonne, they often published short notes on particular problems. Thus, the working investment one had to make to publish in these journals was less important than the one for an academic submission. Still there were rewards. The publishing process was faster in these than in academic journals; one didn't have to wait several years for work to be read by the other members of the community. For all these practical reasons, the articles in mathematical journals, even less prestigious than the Mémoires de l'Académie, were not second-rate publications. Even academicians often used this way to publish their research.Footnote ³⁷ But this was also an easy way to publish one's work, in particular because of the shortness of the paper. As the chief editor of the Bulletin de Férussac, Sturm was in a privileged position to do that. The first part of his article on electrodynamics – which had been refereed by the Academy – was published in the Annales de Physique et de chimie [Annals of physics and chemistry], and a summary was included in the Bulletin de Férussac (Liouville Reference Liouville1829). Liouville also took this route; a part of the paper on heat theory that he had sent to the Academy to enter the contest for the 1830 mathematics prize also came in the Annales de Gergonne (Liouville Reference Liouville1830). As for Galois, his work was published in the Bulletin de Férussac and in the Annales de Gergonne (Galois Reference Galois1829a; Galois Reference Galois1829b; Galois Reference Galois1830a; Galois Reference Galois1830b; Galois Reference Galois1830c). Clearly, a young geometer bent on making a name for himself could do so by using those publications.

As a result it seems doubtful that Galois suffered from an inferior position in the mathematical world – rather, his trajectory was very similar to those of the young mathematicians of his generation, including Sturm and Liouville, for example. Galois enjoyed the same amount of support, his work on equation was refereed by the Academy, and his articles were published in the same journals. Finally, he used the same strategy as his competitors to respond to his period's “career freeze.” What made his own path different, however, was his untimely death at twenty years old, and the whiff of scandal that soon enveloped his name.

One may even argue that the Academy was not an institution from which any young scientists could really expect much respect. It required patience and perseverance to be used in applying oneself to a wide range of scientific questions. Looking at the first years in the careers of young scientists, one is struck by the fact that they did not try to cast themselves as experts in one mathematical field. During the first two years of his career, for example, Liouville worked on electrodynamics, the theory of heat, and calculus. Sturm, on the other hand, manifested an interest for differential equations, the theory of heat, and numerical equations. As for Corancez, one of the members of Napoleon's expedition to Egypt, he submitted three papers on the solution of equations to the Academy, one of which was actually refereed, while another was published in the Recueil des savants étrangers (Procès verbaux des séances de l'Académie des sciences, vol. 4, 371, 377, and 459; vol. 5, 221, 244, and 459). But tellingly, when he decided to apply for the appointment that Laplace was leaving, he sent a paper about differential equations connected to the movement of water in vases (Procès verbaux des séances de l'Académie des sciences, vol. 9, 52; Biographie universelle ancienne et moderne, vol. 9, 169–171). Or consider the case of Lamé, who did unsuccessful research on geometry and number theory before presenting the work he had carried out with Clapeyron on the stability of archways and the equilibrium of solid bodies, as well as a paper on the application of the theory of heat to polyhedra. Eventually, the Academy was to award Lamé an appointment at the École polytechnique (Bertrand Reference Bertrand1879).

To be sure, the line separating pure and applied mathematics at the time was much fuzzier than today, and all these scholars worked within one field, namely, as we have seen, mathematical analysis. Nevertheless, working on a wide range of subjects was more than pure curiosity, it was a strategy adapted to the rules of the Academy of Sciences. The Academy's implicit rules encouraged this because each time the Academy received a paper it asked an expert on the topic at hand to referee it. Hence, choosing a subject was a way to attract the attention of the academicians. Someone working on electricity could expect to catch the eye of Ampère, while others trying their luck at theory of heat hoped that Fourier would notice them. Someone working on pure mathematics could hope that Cauchy or Poisson would read his dissertation. Thus, by tackling a variety of topics, young geometers followed two rules of the scientific fields. The first one is related to the values attributed to a scientific work at that time: this wide range of subjects enabled aspiring geometers to prove the breadth of their skills in analysis and to earn the respect of academicians. It was a way to prove that the way they made mathematics was the right one. The second one is related to the institutional rule of the elections at the Academy. For every job, a list of candidates was drawn up by the six members of the section in which the job had opened. Then, all the 66 academicians attending the meeting took part in the final vote. In this context the strategy of combining several research interests, from theory of heat to the theory of equations, could pay off, because one could maximize his chances to get some support, inside and outside the Geometry section.

Thus, from April 1827 to October 1829, Sturm submitted no less than eight papers to the Academy: Arago, Cauchy, Dulong, Fourier, Fresnel, Gay Lussac, Poinsot, Poisson, Navier, and Thénard were asked to read them. A few years later, in March 1833, Sturm was selected to enter the election to the Academy. Liouville, who was also selected in March 1833, had sent six papers between 1828 and 1832, attracting the attention of Ampère, Arago, Cauchy, Dulong, Fourier, Lacroix, Maurice, Poisson, and Navier. Neither won election. Instead, by an overwhelming majority, the academicians chose a well known mathematician, an author of several books, who happened to be the Academic correspondent member in Italy – Guillaume Libri (Procès verbaux des séances de l'Académie des sciences, vol. 10, 11 March and 18 March 1833, 226–227; Maccioni Ruju and Mostert 1995, 86–93).

Seen from this perspective, Galois's weaknesses seem obvious – his work focused on algebra, which limited the number of academicians who would become familiar with him. But beginning with algebra wouldn't have prevented Galois from becoming a famous mathematician, as long as he would have been able to show he could do something else too. Indeed, Galois had already started diversifying his research and worked on the theory of elliptic functions. This was a good choice from a career-making point of view. Abel had just earned the Grand Prix of mathematics for his research on this topic, causing Saigey to write that this theory was “a field outside of which neither scientific progress nor personal benefit was likely” (Saigey Reference Saigey1829). In other words, Galois planned to work on the trendiest mathematical subject of his times.

All in all, there was nothing unusual or uncommon in the reception that the Academy accorded Galois. Indeed, this social and institutional analysis shows that the Academy dealt with Galois much as it dealt with all young geometers. Both Galois's connections in the mathematical community and his publishing choices confirm that he was poised to become an established mathematician by 1831, when he sent his last paper on the algebraic solution to equations to the Academy. The simple fact that Poisson accepted the task of writing a report on it after having read the letter of April 1831 – in which Galois excoriated the Grand Prix jury and claimed that it was unfair – demonstrates the Academy's benevolence towards him (Galois [Reference Galois1962] 1997, 465; Procès verbaux des séances de l'Académie des sciences, vol. 9, 4 April 1831, 597). Sophie Germain's testimony also supports this: in a letter she wrote to Libri on 18 April 1831, she explained that Galois had fallen prey to a “curse cast on all things mathematical.” Indeed, Germain went on to compare the decision of the École préparatoire to expel Galois to Cauchy's exile and Fourier's death (Henry Reference Henry1879, 631–632). What Galois needed, but did not enjoy, was more time to expand on his research.

The legend of Galois has been mostly constructed upon what Galois himself wrote about this affair. Galois sought to bequeath notes that would underwrite a narrative about a failure of mythical proportions. But by using other archival sources, particularly those of the Académie des sciences, we can reconstruct Galois's own position in the mathematical milieu, and thus differentiate between the aspiring mathematician that he really was and the cursed genius he was to become in collective memory.