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UNIVERSITAS
No. 11, May 1998
German mathematics in France. The role of Charles Hermite
by Tom Archibald

Universitas

No. 13 - Dec 2000 (PDF)

No. 12 - Sep 1999

No. 11 - May 1998

No. 10 - Jun 1997

Introduction

Explaining the attainment of dominance by German pure-mathematical values, even enunciating what they are, is a complex task. In the German-French context following the Franco-Prussian War, we might be tempted to make the analogy with the post-World-War-II ascendancy of American cultural styles in, for example, Germany. Yet, looking at France in the 1870s and 1880s, we do not see German dominance in other cultural realms such as literature, art, and music, indeed far from it. Thus the specific situation in mathematics calls for some specific factors in that area which give a unique importance to the value of understanding and appreciating German work, and which facilitate its assimilation and transmission.

In this brief note I will suggest that one of these factors is Charles Hermite (1822-1901). This is not a particularly original observation; more original is the notion that Hermite's political opinions and cultural values, particularly as expressed in his correspondence with Paul Du Bois-Reymond and Gösta Mittag-Leffler, give us a means to link general historical developments with the content of mathematical activity.


Motives of Hermite in promoting German mathematical values in France

The urgency that Hermite felt to engage in his role as interpreter of German work was intimately linked to his view of social conditions in France following the Franco-Prussian War. In particular, his efforts were connected on the one hand to his ideal of an international quasi-aristocratic elite of scientists and savants living in gentlemanly harmony, and on the other to his concerns about the consequences for France of continued repudiation of the Germans.

The fact that many mathematicians at the end of the nineteenth century saw themselves almost as aristocrats is easy to forget. Recent studies, in particular those of Christophe Charle, provide a sociological picture of this group, and make it clear that indeed successful professors are among the haute bourgeoisie (Charle 1994, Ringer 1992, Zerner 1991). Charle's clear identification of the aristocratic aspirations of most of the professoriate in the realm of lettres, together with Zerner's reflection that a busy senior professor could earn up to 50,000 Francs annually at this period (a considerable sum) indeed place the professoriate among an elite that was more than intellectual, despite the liberal associations of academic protest in the Dreyfus affair and other public debates of the time.

Charles Hermite (1822-1901) was born in a small town in Lorraine. He was a hereditary anti-radical, his grandfather's fortune having been destroyed by the revolution and his great-uncle guillotined. He had a rather unorthodox career, and was largely self-taught in mathematics. His affinity for German mathematical work dates from his earliest years, when he successfully extended some important theorems of Jacobi on elliptic functions. This was the foundation of his career, achieved at the early age of 21. Curiously, he never learned to read German - Jacobi's papers were in Latin and available in French summaries. Unusually for France of the 1840s, he did not work in applied areas at all, and was ignorant of the basics of physics. He thus may be associated with the Jacobian ideal of pure mathematics, and was doubtless aware of Jacobi's statement in response to Fourier that mathematics was undertaken not for application but for the honour of the human spirit. I will not give details of his career, but I will mention that he was appointed to the Académie des Sciences in 1856 and held various teaching positions, notably at the Ecole polytechnique and at the Collège de France.

Of key importance for his influence in the transmission of German mathematics are Hermite's dynastic connections, in what Martin Zerner has called the Kingdom of Mathusia, ruled rather surprisingly by Joseph Bertrand. Bertrand's mathematical work is now little recalled, but his unique position of power in French mathematics and science in the 1870s and 1880s was guaranteed by his role as one of the two perpetual secretaries of the Académie des Sciences. As Bertrand's brother-in-law, Hermite had unique access to Bertrand at times, though their relations were sufficiently difficult that at times they did not speak. Hermite was father-in-law to the mathematician Emile Picard, and was likewise related by marriage to another important young mathematician of the period, Paul Appel. Indeed his extended family also has links to J. M. C. Duhamel and Olinde Rodrigues (in earlier generations) and Emile Borel (later). One final biographical detail of importance. Hermite had been very closely touched by the Franco-Prussian War. His home town, where many of his relatives still lived, had been occupied by Prussian officers during the siege of Metz. A portion of his family was in territory which was ceded to Germany at the conclusion of hostilities (Lampe 1916, Letter 4), and he felt the changes that had occurred very deeply.

As for the first of the two correspondents, Paul Du Bois-Reymond (1831-1889) was born in Berlin, and studied there. His father was a self-made Prussian diplomat from Neuchatel in Switzerland (a Prussian fiefdom until the mid-nineteenth century) who ensured his fluency in both French and German. Paul's brother Emil, the physiologist, was at the top of Berlin scientific world in 1870, and was a well-known and rather chauvinist critic of the French at the time of the war (E. Du Bois-Reymond 1870). During the period of the correspondence Du Bois-Reymond was an ordinary professor at Tübingen, then moved in 1885 to the Technische Hochschule Charlottenberg, back in Berlin. He had close ties to Weierstrass but had difficult relations with many members of the Weierstrass school, most notably H. A. Schwarz. His correspondence with Hermite revolves around mathematical questions in analysis, mostly concerning the convergence of trigonometric series and improper integrals. However Hermite's candid discussion of his life as a professor and his efforts to promote Du Bois-Reymond's work in France give us a valuable picture of his role as a vector of German ideas in France. As Du Bois-Reymond's portion is lost, except for one letter, his responses must be inferred. The letters from Hermite are published in (Lampe, 1916)

The letters from Mittag-Leffler have likewise been transcribed (Dugac, 1984). Mittag-Leffler, certainly one of the most intriguing figures in international mathematics of the late nineteenth century, was a pivotal figure in the transmission of information between Paris and Berlin. Visiting Paris on a post-doctoral travel grant in 1874, he studied elliptic functions with Hermite; then, acting on Hermite's advice, he went to Berlin to attend the lectures of Weierstrass and Kronecker. He subsequently held positions in Helsinki (then Helsingfors) and in Stockholm, where he founded the journal Acta Mathematica with royal support. He kept in very regular contact with both Paris and Berlin, both via travel and in correspondence, and Acta became an important vehicle for the transmission of mathematical information from one country to another.


Hermite's critique of radicalism

As we have mentioned, then, Hermite was hereditarily conservative. In his case this meant that he favoured the restoration of the monarchy, and the retention and indeed expansion of religious involvement in education. He was strongly opposed to democratic measures of various kinds, as is amply attested in the letters. He is particularly caustic about universal suffrage, which he found deeply wrong-headed.

In a letter to Mittag-Leffler of 1883 he describes his horror at picking up his voters' card at the Mairie:

I savoured the impression of contact with the sovereign people, Our Lords the labourers, tailors, junk-shop merchants (brocanteurs), wine merchants, tripe merchants, etc, who demanded their cards while harassing the employees. (Dugac 1984, 195).

Beyond a mere quasi-aristocratic distaste for the common people, however, Hermite shows a genuine fear of the consequences of radical republicanism. He frequently mentions the possibility of German armed intervention in an unstable or radical French state, and went so far as to investigate acquiring a property in Brittany for the express purpose of having a place to go in the event of another invasion.

This needs a little explanation. The aftermath of the war had left a complicated political situation in France, in which the positions of Hermite were those of a substantial minority on the right. The notion of rapprochement with and even appeasement toward Germany was also a common right-wing view, while the left was much more belligerent towards the Germans. The right was however insufficiently well-organised, divided between Bourbonists and Orleanists, for example, and the superior leadership of the left led to its dominance, especially from about 1879. For Hermite, this had important implications, not just on general political grounds. The left favoured and indeed carried out much educational reform, a key element of which focussed attention on the Sorbonne and the Grandes Ecoles because of the proposal to eliminate members of religious orders from teaching.

This focus had various consequences. Members of some orders, notably the Jesuits, were indeed banned from teaching at all levels. On the other hand, the question of the content of education and its appropriateness for republican life came up in a variety of ways. In these discussions the question of foreign influence had an ambiguous status. While Germany was generally seen as bad, the organisation of higher education there had various elements that were politically correct for at least some of the left, for example the distribution of universities across the country and the well-developed system of higher technical schools. As well, the obvious rise of German science and technology compared to the relative stagnation of French efforts was apparent. Thus an interest in German models was expressed by people on both the right and the left.

This atmosphere was one in which Hermite could pursue his efforts at the integration of German mathematics into the research community and the curriculum, though not without obstacles. Likewise, it afforded various opportunities for integration of the mathematical communities in the two countries.


Franco-German relations and how to promote them

Hermite's expressed view of international relations was essentially this, that the scientific confraternity transcends national boundaries. Writing to Du Bois-Reymond in 1875, he asks:

Do you believe that just because we belong to countries with different languages, between which the war has made a bloody trench, that there is no longer any resemblance in the work and the professional duties, between those who have devoted their entire lives to the same studies? At the end of a tiring lecture... I take great pleasure in conversing with a colleague from Germany, and deploring just as energetically as he would the impossibility of uninterrupted reflection. (Lampe 1916, Letter 3).

And further

The analogy of our situations creates a natural and legitimate sympathy, of which, above and against the resentments of politics and the war, I make myself the medium.

As for the current view of Germany from the French scientific community:

Certain signs announce that a real and general peace will be concluded, to restore to science its most precious privilege, to create a bond of esteem and personal affection between all those who dedicate themselves to it.

Hermite insists here, and frequently throughout the course of the correspondence, that the hopes and fears he expresses are not uniquely his own. In fact he insists so much that we may wonder a bit whether he is really correct. He frequently mentions, however, the agreement of his colleague Bouquet, and also sometimes of Bertrand.

Hermite's admiration of German mathematics dated to the beginning of his career. His admiration of the institutional setting, academic life, and cultural values was strongly reinforced by a visit to Göttingen in 1877 for the centenary of Gauss's birth. Here Hermite was deeply moved, as he explains to Du Bois-Reymond, by the fact that such a large celebration would be given in honour of a man whose life's work had been dedicated to science alone. Such honours would be impossible in France, he felt. Subsequent to this, his comments comparing France unfavourably to Germany seem to become more frequent, particularly as the French political situation becomes more unstable and more republican.

We now turn to look at Hermite's methods for promotion of German mathematics. These take two forms: attempts to get official French recognition for German mathematical achievements, and attempts to promote the teaching and learning of current German mathematics.

Academy membership

Already around 1875 Hermite's efforts in creating this "real and general peace" in the scholarly community, may be seen in the appointment of Borchardt as corresponding member of the Académie des Sciences (Lampe 1916, Letter 4). One of the standard methods of recognition of foreign savants, a corresponding membership conferred publication rights in the Comptes Rendus, an important platform internationally. For the corresponding member, it was also a form of recognition which could prove useful in a variety of ways, such as for professional advancement, and could thus be both a material and a symbolic reward. The extent to which Hermite valued such appointments himself is shown by his own response to admission to the Swedish Academy. He states to Mittag-Leffler how delighted he is, how pleased Mme. Hermite is that "they are now Swedish", and looks forward to receiving the diploma so that he can display it in his home.

Borchardt, the man who had succeeded August Leopold Crelle in 1856 as the editor of Crelle's Journal für die reine und angewandte Mathematik, joined a number of his compatriots who had been nominated before the war: among mathematicians, Kummer (as foreign associate), Weierstrass and Kronecker (as corresponding members). Borchardt was both a colleague and a friend of Hermite, whom he had met during a trip to Paris in 1846 (Dugac 1984, Letter 21). The two had worked on closely related subjects, and Hermite had frequently published papers in Borchardt's journal. Borchardt, an Academy member in Berlin, was also at the heart of the Berlin mathematical establishment. The editorial board of his journal included Weierstrass, Kronecker, and Helmholtz.

Getting someone into the academy, either as a corresponding member or as a regular member, was a complex task. In the case of corresponding members, the secrétaire perpetuel - in this case Hermite's brother-in-law Bertrand - had a good deal to do with starting the process that would lead to the position being filled. A committee of members from related domains meets in secret to rank nominees, following which a vote is taken in the academy as a whole, in awareness of the ranking but not necessarily in obedience to it. There is thus room for much behind-the-scenes manoeuvring at all stages of this process. The result in this case was positive. Borchardt had good relations in France and since the war had remained in correspondence with Hermite, Bertrand and Chasles. It was not a simple matter, however: he failed to obtain a majority in the first round, and in the second round received 29 votes, with 23 opposed and 4 abstaining.

Bad feeling towards Germany, in the Academy, the scientific community, and the community at large, was certainly not a myth at this time, despite Borchardt's election. The republican parties of the left, increasingly powerful in the mid-1870s and ascendant to government in 1879, contained a strongly anti-German component, with a rhetoric which harked back to the days of the revolution and leaders who had bitterly opposed surrender. Since these parties had a close eye on the educational system advancing a pro-German cause in the context of their leadership was not without risk of reprisal. There was nevertheless an element of the left which shared the rather widely-held view that French science in particular had fallen behind that of the Germans, and that the higher educational system was particularly weak in this regard. Pasteur was a famous proponent of this view, already in the 1860s, and it is taken up as a theme in criticizing the educational system, especially its more conservative elements, and especially its religious elements.

In the mathematical community anti-German feeling displayed itself through a variety of mostly rather minor acts. Seen from the German standpoint, namely Borchardt's, these included the failure to cite German work on which French work was based (Jordan, Lipschitz corr. 17 Dec. 1885); pettily critical reviews in Darboux's Bulletin (by M. Levy, of a paper by Borchardt); resignation from membership in scientific societies (Jordan again); and outright plagiarism. Borchardt draws a specific analogy between the ordinary plunder of war and this kind of plunder in the scientific sphere, citing in particular a work of Laurent which he felt had been stolen directly from Eduard Heine.


Medals

Another form of distinction that could be accorded to Germans consists in the award of prizes and medals. For the established French scientist, recognition by the state came in one principal form: admission to a national order, accompanied by a medal. In France, the different degrees of the Legion of Honour were the standard awards to meritorious servants of the public good.

A medal is a form of recognition accorded not merely to the savant, but to the military man, public servant, or private citizen. It therefore permits comparisons with others in the different walks of life whose efforts may be thus recognized. It may be worn, or displayed in a prominent place in the home, and is thus a portable and easily understood expression of distinction. A medal in an aristocratic society may carry with it the sense of inclusion in the aristocracy, a feeling that one is quasi-royal. It is an honour to the family as well as to the recipient. Some academics were inveterate collectors of such honours - Mittag-Leffler is one example. An anecdote still told recounts how he appeared at an official function in a fine jacket, covered with medals all of whose ribbons matched its blue colour. When asked how it happened that the ribbons should all match, Mittag-Leffler supposedly gave the obvious answer: "These are the blue ones."

Of course, to get a medal usually requires a nomination, in addition to activity which is understood to merit it. In France, one could nominate oneself, or you could arrange to have a third party nominate you. Hermite felt such behaviour was not appropriate, feeling presumably that true merit would be recognized and rewarded appropriately. He therefore refused to ask for himself, and as a result had only a lowly position in the Legion of Honour. He attributed this also to his religious activity (going to church with the family) and to his politics (anti-republican), probably with some justice.

Hermite thus knew well the interest of academics in receiving such honours. Unhappy about his own lack of advancement, in April of 1882 he seized a political opportunity in order to seek awards for foreign colleagues: first Weierstrass, then Kronecker. The procedure was to ask one of the secrétaires perpetuels, in this case the chemist Dumas, to approach the president of the Conseil des Ministres, concerning the possibility of an award. The decision appears to have been essentially up to this individual, who at the time was Charles de Saulces de Freycinet (1828-1923), a polytechnicien and engineer. Freycinet had sought Hermite's support for a position as a free member of the Academy, and Hermite grabbed his chance, writing at once to Dumas requesting an award for Weierstrass. Almost at once (Dugac 1984, April 18, 1882) he realized that Kronecker would be jealous, and learning that Weierstrass would receive an award he quickly sought one for Kronecker as well.

Hermite had to provide the basis for the honour. For Weierstrass the arguments were interesting, since they insist on the importance of his work for the French State. For example Hermite cites the fact that Weierstrass's work was so fundamental that it was being taught to students at the Ecole Polytechnique. For Kronecker he was obliged to fall back on his own prestige as the main basis for the award, since his argument was in essence that Kronecker had worked in the same area as Hermite and that they had got very similar results using related but different methods simultaneously. These results, were very famous at the time they were produced (in the mid-1850s), at least in scientific circles, since they concerned a long-open problem which had defied generations of researchers: the solution of equations of the fifth degree (impossible algebraically), achieved by Hermite and Kronecker using elliptic functions. These medals followed closely similar (actually slightly better) awards to Helmholtz and Kirchhoff. These appear to have been the first Germans to be so honoured since the war.


Bringing German mathematics to French students: a thankless task

If medals and academy membership were important in establishing Franco-German links at the highest professional levels, much more important in terms of long-term developments were the efforts of Hermite to make the work of German researchers known in France, as well as to publicize French work in Germany. This took several forms. One direction involved teaching mathematics which would allow students access to German work. The other direction is oriented toward research, and includes encouraging German publication in French journals, arranging for translation of work he thought important, and ensuring that German mathematicians were well-aware of the efforts of the French. I will first discuss teaching.

Here we should very briefly note marked differences in the systems in Germany and in France. In Germany, the professor typically lectured on research, which was necessarily somewhat current. By contrast, in France there was a greater emphasis on lower-level pedagogical lecturing, and in some institutions (notably the Ecole Polytechnique) committees kept a close watch on curriculum. This led to a textbook tradition, and published French mathematical cours (lecture courses) often had long lives, running many editions over decades with only minor modifications. In Germany such books were a late development, and indeed French books were much used there by students who likewise paid tutors to help them get the basics. Hermite's conviction about the superiority of the German system was reinforced once again by his 1877 visit to Göttingen, where he saw with admiration and amazement the professors and the students sitting, drinking beer together and singing. This may have encouraged him in subsequent efforts to follow a more German pattern in his lecturing by the inclusion of current work.

Hermite frequently complained to Du Bois-Reymond about the amount of effort he devoted to teaching. Into his analysis course at the Faculté des sciences beginning in 1882 he incorporated a good deal of recent German work, notably due to Weierstrass on functions of a complex variable (primary factorization). These results were brand-new. They had appeared for the first time in 1874 in Weierstrass's Berlin lectures, and Hermite had learned about them only by virtue of his attendance at the Gauss Memorial symposium in Göttingen in 1877. These results were published only in 1878 and were to some degree completed by Mittag-Leffler's work of 1880 through 1882. Thus the students were lucky enough (in Hermite's view) to be getting German-style lectures on the most recent work. In fact these results are introduced quite early in the course, since they permitted certain economies in the treatment and allowed him to devote more time to his favourite topic, elliptic functions. It should be noted that these theorems are not that easy, the more so because the treatment uses Weierstrassian arguments which were unknown to all but a handful of professional mathematicians in France (and indeed to relatively few people in Germany). Thus students could not rely on the traditional method of paying a tutor to get them through the difficult spots.

Hermite learned from his colleague Biehler, then director of Preparatory studies at the College Stanislas (a post-secondary institution preparing students for the entry competitions of the grandes écoles) that most of the students couldn't follow him. It seems rather characteristic of Hermite that this was a real surprise - indeed his letters of this period to Du Bois-Reymond and Mittag-Leffler up to this point frequently note with what enthusiasm the students are hearing of these theorems. His first response was that of a highly responsible pedagogue. He asked for Bouquet (whose students at the Ecole Normale Supérieure came to the lectures) to arrange for notes to be taken and then lithographed and then sold at a bookstore for around 0.25 francs a lecture (a then-common system in France).

However negative student reaction was not turned aside by these efforts, and achieved expression in a modern form. In July 1881 the republican government had passed legislation declaring the freedom of the press. In June 1882, just at the end of the university year, a new journal called Le Passant had the idea that they should include some critical investigations of the educational community, and commissioned a writer (A. Dunoiset) to make reports. He began with the science faculty, a particularly attractive target. Staffed by what he termed "grizzled Methusalahs" deviously standing in the way of promised reforms, the faculty had serious space and equipment problems. His complaints have ring remarkably reminiscent of student complaints today: for example, the chemist Wurtz "most distinguished scientist among the chemists is, you guessed it, the most detestable of teachers". Indeed the question of why researchers were needed to teach first and second year courses was brought up repeatedly by the critic.

Mathematics fared a little better at the hands of Le Passant, but Hermite was singled out on two grounds: his own teaching and his role in obtaining a position for his son-in-law Picard. Picard "would still be in Toulouse if he weren't the son-in-law of Hermite" (indeed, unknown to Le Passant Hermite got him the job in Toulouse as well) and his nomination to the chair of Kinematics was "a real scandal". Hermite himself was criticised as follows:

You come to learn how to integrate and M. Hermite presumes that you have been integrating all your life. His course is not a course, it is a dazzling conversation, broken up into bits, with interminable digressions and meanderings all across Europe [a reference to the frequent inclusion-

-of discussions of recent foreign research] ... he is detested by the students."

His references to religion and a belief in providence likewise come in for criticism.

The inclusion of foreign work didn't fail to figure in the list of complaints.

Hermite's response was shock. In letters to both Du Bois-Reymond and to Mittag-Leffler he expresses several times dismay that his students really might detest him. However, the shock to his dignity at being treated this way in public was clearly as much in his mind as his concern for his students. His ultimate diagnosis: freedom of the press is an evil:

Without doubt, Sir, you are liberal and subscribe to the freedom of the press. Alas the use of this liberty under the government of the republic disgusts me profoundly.


German mathematics and the research community

To turn to Hermite's multifarious efforts at improving research communication, we may note that this communication before 1870 was not in a good state. For a French academic to read German was atypical, particularly in the older generations that dominated the French university community. Furthermore, the library situation in Paris (and a fortiori everyplace else in France) was not good, so that in general one would actually have to subscribe personally to many journals in order to see them. This was expensive, though it was the route followed for example by Darboux. Translations of a few articles did appear in Liouville's journal, though this was erratic, and a German practice earlier in the century of trying to announce their key results in letters (to Liouville or others in France) intended for publication had waned as the French mathematical community weakened and the German one attained independence and maturity. There were no regular international meetings and no exchanges as such.

The situation began to change in 1870 with the appearance of Darboux's journal, which has been studied by Helene Gispert (1987). Darboux presented summaries, or at least titles, of papers in most of the international journals, and presented translations of some articles of critical importance, a noteworthy example being Riemann's thesis on trigonometric series. However there were various problems: for one thing, reliable translators were in short supply. For another, the mathematics in some of the papers was incomprehensible even when in French - not only the notoriously difficult Riemann, but also the work of Weierstrass and his students.

Hermite undertook efforts to have longer work translated. For example, in the Mittag-Leffler correspondence we have detailed accounts of his efforts to translate fundamental papers of Weierstrass and Cantor, and in the Du Bois-Reymond correspondence a book by Du Bois-Reymond. His chosen translators were initially his best students. For example, Emile Picard, at that time only a potential son-in-law, was drafted to translate the Weierstrass paper, and Poincaré did Cantor. Such papers could be published in the Annales of the Ecole Normale Supérieure, run by Pasteur, or they could appear in Acta Mathematica, Mittag-Leffler's new journal, once that was begun. Such efforts at translation were not without their problems. Poincaré in particular complained about Cantor "mixing in philosophy with the mathematics". It wouldn't be so bad, he said, if he just had philosophical sections of the work, which we could then just skip. Instead it is all mixed up in the proofs.

Such efforts of translation were obviously very important in assisting the young mathematicians engaged in them to assimilate German concepts and German language. At times they were immediately able to produce research in the same vein - Picard is an example. Indeed expository accounts of German work became common thesis topics at the Sorbonne during this period. Students were encouraged to go beyond the original, but understanding was all that was required for the doctorate. German work that was treated in this way included work by Schwarz, Hölder, Riemann and Fuchs (Gispert, 1991).

Also useful were efforts to have German mathematicians publish in French journals. Darboux's journal was clearly a pioneer in this regard (Riemann, Lipschitz - particularly relevant to French since it improved on Cauchy) but Hermite also attempted to get things in the Comptes Rendus, much more prestigious, widely available and universally read. The letters contain various accounts of Hermite's efforts to place German research in the Comptes Rendus. One of his first letters to Du Bois-Reymond explains the system to him: it must be no more than three pages, and be forwarded by an academician. Hermite then served as a vehicle for work by Du Bois-Reymond and several of his students (Lampe 1916, Letter 3). Since Du Bois-Reymond was essentially a native French speaker there is no obstacle to publication, which had to be in French.

Hermite did face obstacles here though, namely the redoubtable brother-in-law Bertrand, who as secretary decided what went into the Comptes Rendus and what did not. Bertrand's opposition to the newer mathematics is nicely shown here, and is not restricted to German work. For example, Bertrand set up obstacles for the appearance in Comptes Rendus to a note of Darboux inspired by Riemann (CR, 95, 1882, 69-72). According to Hermite, Bertrand seems to have formed the systematic idea to make an obstacle to the numerous publications of young mathematicians ostensibly because he wants things that are ripened longer, more complete, more profound. In reality, we may speculate that apart from his lack of grasp of the details of these works, he may well not have understood their interest - it was certainly a question that Hermite posed from time to time in the correspondence, especially regarding Cantor's work. At one point Hermite feared that Bertrand would go so far as to exclude mathematical work from the Comptes Rendus.


Concluding remarks

While Hermite stresses international cooperation, there is also a strong feeling of rivalry with the Germans, particularly among the younger men who are his students. And it is likewise clear that Hermite did not like to be the target of criticism coming from Germany, for example by Schwarz, who criticised the definition of surface area given in his lectures (as did Peano). This rivalry was felt in both directions, and Weierstrass (Dugac 1984, June 19 1882) was very impressed by the current French crop of young mathematicians, including Poincaré, Picard, and Appell. Apparently speaking to an assembled group he stated,

We are going to have to pull together like the devil, gentlemen, if Paris is not to become once again the mathematical capital.

Hermite's program was enormously successful, and though it was certainly not his alone, his leadership was central to the development of a mastery of German concepts, displayed all the more clearly in the next generation, when individuals such as Fréchet and Borel were to take real-variable theory in important new directions.

Yet to return to the theme enunciated at the beginning, that of the international brotherhood of science and its transcendence of petty politics, such rivalries, though they existed, were to Hermite far from insurmountable. The friendships he enjoyed with Kronecker and Borchardt were for him close friendships, and even his rivalry with Schwarz could be set aside more or less permanently in the atmosphere of civility associated with a visit to the home, where Schwarz could dine well and admire Hermite's dsiplayed medals and diplomas. These expressions of civility - dining en famille, meeting at a spa during the vacations, experiencing the fraternity of mathematics at a scientific meeting - were more important to Hermite than rivalry with those he felt deeply to be people of his own kind.

Selected References

Charle, Christophe, 1995. "Intellectuels, Bildungsbürgertum et professions au XIXe siècle: essai de bilan historiographique comparé (France, Allemagne)". Actes de la recherche en sciences sociales 106-107, 85-95.

Charle, Christophe, 1994. La république des universitaires, 1870-1940. Paris, Seuil.

Du Bois-Reymond, Emil. 1870. "Über den deutschen Krieg". Rectoral address given Aug. 3 1870 at the University of Berlin. Translated in Revue des cours littéraires de la France et de l'étranger, 7, 17 Sept. 1870, pp. 658-668.

Dugac, Pierre, 1984. "Lettres de Charles Hermite à Gösta Mittag-Leffler (1874-1883)". Cahiers du Séminaire d'histoire des mathématiques 5, 49-285.

Gispert, Hélene, 1987. `La correspondence de G. Darboux avec J. Houël: chronique d'un rédacteur (déc. 1869 - nov. 1871)". Cahiers du Séminaire d'histoire des mathématiques, 8, 1987, pp. 67-202.

Gispert, Hélène. 1991. "La France mathématique". In Cahiers d'histoire et de philosophie des sciences.

Hermite, Charles, 1882. Cours d'analyse de la Faculté des Sciences, ed. Andoyer. Paris, Hermann. Lithographed notes.

Lampe, Emil, 1916. "Briefe von Ch. Hermite zu P. du Bois-Reymond aus den Jahren 1875-1888". Archiv der Math. und Phys. (3) 24, 193-220, 289-310.

Ringer, Fritz, 1992. Fields of Knowledge: French Academic Culture in Comparative Perspective, 1890-1920. Cambridge and Paris: Cambridge University Press and Editions de la Maison des Sciences de l'Homme.

Zerner, Martin, 1991. "Le regne de Joseph Bertrand (1874-1900)." in Cahiers d'histoire et de philosophie des sciences.

Acknowledgements

I wish to thank the International Centre for the History of Universities and Science, University of Bologna, for the invitation to present this paper in March 1998. Thanks are also due to the Centre de recherche en histoire des sciences et techniques at the Cité des Sciences et de l'Industrie, La Villette, Paris, which has hosted me as an associate reseracher (CNRS) for the academic year 1997-1998; and to Acadia University for according me a sabbatical.-

Address: Tom Archibald, Department of Mathematics, Acadia University, Wolfville, Nova Scotia, BOP 1X0, Canada.

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