MacTutor
Camille Jordan by Lebesgue
The Notice on the life and works of Camille Jordan (1838-1922) was read by Henri Lebesgue in the session of 4 June 1923 of the Académie des Sciences in Paris. It was published as H Lebesgue, Camille Jordan, Mémoires de l'Académie des sciences de l'Institut de France 58 (1923), 29-66. A shortened version was published in H Lebesgue, Notice sur la vie et les travaux de Camille Jordan (1838-1922), L'Enseignement Mathématique (2) 3 (1) (1957), 81-106. We give below an English translation based on the shortened version.
Notice on the life and works of Camille Jordan (1838-1922), by Henri Lebesgue.
The Geometry Section has just been painfully struck twice. On 22 January 1921, Georges Humbert passed away after a long illness; Camille Jordan lost in him his favourite disciple. Humbert had been his student then his colleague at the École Polytechnique, his colleague in this Academy, his successor at the Collège de France, and he had become his friend. On the eve of the anniversary of a death that had been so sensitive to him, in the night of 20 to 21 January 1922, Camille Jordan passed away in turn, suddenly, without suffering.
Marie-Ennemond-Camille Jordan had just reached the age of 84; he was born on 5 January 1838, in Croix-Rousse, in the most fundamentally Lyonnais neighbourhood of this city of Lyon where the great-uncle whose name he bore had just played such an important political role for more than thirty years.
To speak to you of the great-uncle will already be to speak to you of the great-nephew so great is the spiritual and moral affinity between the members of the Jordan family. We must first note the firmness of their religious convictions, their kindness, their generosity; all nourished by classical literature and great readers of the ancients, they are also willingly attached to the past, in politics as in science; they are moreover with complete good faith and this leads them to be workers of progress. We will constantly see the Jordans being both conservative and innovative.
It is said that at the age of 17, the future orator of the Restoration, Camille Jordan, attended this meeting of the Estates of Dauphiné which took place on 21 July 1788 at his uncle Périer's house, at the Château de Vizille and in which six hundred deputies of the three orders invited the provinces to resist despotism, to refuse the payment of taxes until the convocation of the Estates General. This spectacle must have aroused the combative ardour of Camille Jordan; but, to understand the political line he followed, we must imagine him as deeply religious, if not entirely orthodox Roman Catholic, and obliged in conscience to protest against all excesses. He protested against the civil constitution of the clergy and took a very active part in the uprising of Lyon against the Convention. After 9 Thermidor, on his return from Switzerland and England where he had taken refuge, having become a deputy to the Five Hundred, he was, with Royer Collard, the champion of religious freedoms in favour of which he made a famous plea as rapporteur of worship. It was there that he made the mistake of insisting a little too much on the secondary question of the reestablishment of ringing in churches; he was satirised under the names of Jordan la Cloche, Jordan Bourdon. He also protested against the coup d'état of 18 Fructidor and had to flee to Germany. Returning to France in 1800, political events forced him to make new protests, now directed against too complete returns to absolute power, because the excesses seemed to him to have changed the trend. Thus, in 1802 he wrote his pamphlet: "true meaning of the national vote for the consulate for life". Called to the Council of State by the confidence of Louis XVIII and elected deputy, he would not fear to become, again with Royer Collard, one of the leaders of the opposition against the reaction which followed the assassination of the Duke of Berry. Camille Jordan was therefore conservative and liberal; but he was above all a good and courageous man, who rebelled in the presence of evil and injustice. The breadth of his philosophical views earned him the intellectual esteem of many eminent minds, in France, England and Germany. No one appreciated him more than Mme de Staël who loved in him both the man of talent and the man of heart. On the subject of the long friendship which united them, Sainte-Beuve wrote a charming article where he shows us, as Camille Jordan himself says, "in private, loved, enjoyed, almost adored by superior or charming women, and justifying the liveliness of this predilection by qualities and treasures of simplicity, sincerity, candour, honour, devotion and frankness."
From the biographies of Camille Jordan, it seems to result that at the end of the 18th century the Jordans lived in the Lyon region and the Dauphiné; they were notable merchants, well-off bourgeois of whom Camille Jordan seems to be the perfect moral prototype. History does not allow us to go back further; but, alongside this fussy history, which only wants exact facts, there often exists a legendary story probably containing some elements of truth. The Jordan family has its own, based on a certain fact: the Jordans were, in the past, Protestants. The legend, which your illustrious colleague loved to tell and whose family handed down the tradition very carefully and moreover without believing it, has it that the Jordans draw their origin from a Waldensian pastor who lived in Fenestrelle around 1600. The Jordans are said to have suffered from all the persecutions that assailed the Waldensians and Protestants and, during the revocation of the Edict of Nantes, some of them are said to have emigrated to Germany. The German branch is also said to have given birth to distinguished men and, among these, it would be particularly appropriate to mention Charles-Etienne Jordan, born in Berlin in 1700.
After having been a pastor for some time. Charles-Etienne, in despair over the death of his wife, abandoned his holy orders and travelled in France, Holland and England. Returning to Germany, he published a travelogue full of original insights that attracted the attention of Frederick the Great in particular. Frederick, then crown prince, became friends with Charles-Etienne; when he became king, he made him his private councillor and curator of the Berlin academies. Charles-Etienne, who had already shown himself to be a talented writer and an artist with sure taste, demonstrated in his new functions the most valuable qualities of administrator and of high moral value. When Jordan died in 1745, Frederick the Great insisted on delivering his eulogy himself at the Berlin Academy; he had a tomb erected for him with this inscription: "Here lies Jordan, friend of the muses and of the king."
The relationship of Charles-Etienne and Camille is highly problematic. It is known that Charles-Etienne was descended from a family originally from Dauphiné and recently settled in Germany; this, and a certain moral resemblance between Charles-Etienne and Camille, may have been enough to lead to conjecture about the relationship of these two men.
The name Jordan is so widespread in Central Europe that, if I were to indulge in such conjectures, it would be easy for me to associate the geometer Jordan with many men of science. Returning to history, I will content myself with noting his real relationship with the botanist Alexis Jordan, nephew of Camille Jordan the orator, uncle in the fashion of Brittany of Camille Jordan the mathematician.
Son of wealthy Lyon merchants, Alexis Jordan was able to devote himself entirely to the natural sciences, without having to seek paid positions. His whole life, from 1814 to 1897, was spent in Lyon, which he only left for long botanical excursions. It was in Lyon that he died suddenly at the age of 83. Alexis Jordan, a believer, a conservative like his uncle, did not possess the latter's amiable qualities; he left the reputation of an unsociable man, a misanthrope. Very good indeed, like most misanthropes, he generously gave his money and time to charitable, religious, and even political works. He spent a lot of time to hasten the advent of Naundorff (who claimed to be the son of King Louis XVI), prophetically announced and who would save France.
During his botanical excursions, he had noticed undescribable characteristics: could they be some variations of the species? This question, to which he did not hesitate to answer no, oriented his research which until then went from entomology to botany. He answered no because, as he wrote in a scientific memoir, "it is said, in Genesis, that God created herbs bearing seed, each according to its kind, as well as trees bearing fruit, each according to its kind. From this it follows that there are distinct species, created from the beginning and reproducing themselves indefinitely by seed."
The characteristics of species are permanent; by a sort of reciprocal, Alexis Jordan admits that permanent characteristics are characteristics of species. "To reject the criterion of hereditary permanence," he writes, "is to remove the possibility of establishing solid distinctions, it is to reduce everything to simple hypotheses, to arbitrariness, to individual fantasy; it is, in a word, to give scepticism as a foundation for science, which amounts to destroying it."
To study the suspect characteristics, Alexis Jordan transported plants to a garden located in Villeurbanne, one of the suburbs of Lyon. When, obeying the book that guides us, we direct our steps towards the botanical garden of the city we are visiting, we expect to see above all large islands of stony earth, devoid of plants, but copiously planted with iron pins, painted green no doubt to simulate plants. If chance had led us to Jordan's garden, we would have been astonished; it was covered with plants, but only with weeds; or rather, it was almost entirely covered with a single weed, the Draba verna, as vulgar, as unsightly as the dandelion, and much less edible. Alexis Jordan had innumerable specimens of it, all identical, or so the visitors believed. Only, where one had been able to see only one Linnaean species, Jordan was able to perceive small species, Jordanian species as they are now called, differing from each other by minute characteristics, but just as constant as the Linnaean characteristics. These are the species of the Creator; to admit, in fact, that they come from the dismemberment of the Linnaean species would lead to admitting that these derive from larger types and so on; This would open a breach in the rampart of the immutability of species and we would see, announces Jordan, the followers of transformism "rushing there with their beautiful doctrines deployed and all the bad axioms of logic and morality which are their obligatory cortege."
The botanical result that we owe to Alexis Jordan: in a species we can distinguish various small species, was at first mocked and contested; the scepticism was general and similar to that which greeted, a few years ago, the very first works on isotopes. Moreover, the separation of a Linnaean species into small species or of a simple body into isotopes are two almost comparable events; Jordan already compared the mutation of plants to the transmutation of metals. Since Jordan's works were taken up and checked, they are considered as models of patient and skilful research; their results have been used frequently. Whether we consider small species as true species or only as fairly stable forms of equilibrium taken by plants during their evolution, they are nonetheless essential to know for those who wish to experiment on a batch of homogeneous plants. It is also extremely important to know that there is no character, however small, whose fixity we cannot hope to discover or realise. As for the philosophical conclusion of Alexis Jordan, it is not accepted by all. In ten years, Jordan found ten small species of Draba; in twenty years, he found fifty-three; in thirty years, two hundred; where will the dismemberment of the Linnaean species stop, as dangerous as the dismemberment of a genus? Will we not be led to state the principle of immutability by saying: there is something that remains constant? An irrefutable phrase, because empty of meaning, formulated by Poincaré in his critique of the principle of the conservation of energy. In any case, the origin of species remains obscure, we do not know whether species are immutable, nor whether they transform; but what we see is that the theses of the two schools are, themselves, in perpetual evolution. And it is not the least merit of Alexis Jordan to have varied these theses, although this is not what he had proposed.
***
No other member of the Jordan family seems to have devoted himself to scientific research; but Esprit-Alexandre Jordan, born in the Drôme, in Die, in 1800, showed at least enough aptitude in the study of the exact sciences to be received in a good rank at the Ecole Polytechnique and to leave it in the corps of engineers of Bridges and Roads. Alexandre Jordan married Joséphine Puvis de Chavannes, daughter of a chief engineer of the mines Marie-Julien-César-Joseph Puvis de Chavannes of Lyon and sister of the illustrious painter Pierre Puvis de Chavannes. From this marriage were born two children: a daughter, who married in Bresse; a son, the future geometer Camille Jordan.
It was in a very comfortable and educated home that Camille Jordan's calm and happy youth passed, prelude to a calm and happy life. The winter was spent in the city; in the summer, they barely left Lyon, they went to occupy a property bought by Alexandre Jordan to be closer to the Puvis de Chavannes in the small village of Bresse where Camille Jordan now rests. Later, in 1871, Alexandre Jordan was sent to the National Assembly by his native department. Until 1876, he sat there among the monarchist majority. His mandate required him to leave his dear Lyonnais for a few months each year; Camille then lived in Paris and it was at his home that the old family group was more or less reconstituted. After 1876, it was mainly during the holidays that Camille Jordan was still able to enjoy the company of his parents. Alexandre Jordan's wife died in 1882, at the age of 66; Alexandre Jordan died in 1888, at the age of 87. Camille Jordan began his studies in a suburb of Lyon, at the college of Oullins, then run by secular priests. Despite his benevolent nature, he did not have very good memories of his first teachers, whom he considered to be very mediocre professors. The college passed into the hands of the Dominicans at the very end of Jordan's studies; he thus met Father Lacordaire for a few months. At Oullins, Jordan also met his future colleague at the Collège de France, his future colleague at the Academy, the physician, pathologist, anatomist and histologist Louis-Antoine Ranvier, who survived him by so little time.
Jordan's mathematical vocation revealed itself very early; we know, from himself, that he would "steal" mathematics books from his teachers' drawers, to read them in secret; - youth is definitely right to persist in reading forbidden books. To this memory of Jordan is added the testimony of one of his former classmates from Oullins who told M Goursat about his friend's academic successes. He shone in every subject and was, without a doubt, the first in everything; but for mathematics his mastery was such that the professor would never have ventured to speak of the solution to a problem without having first studied Jordan's copy.
The chief engineer Puvis de Chavannes would have liked his son Pierre to go through the École Polytechnique; an unfortunate illness interrupted Pierre's mathematical studies, which turned out badly, as we know, and he became a painter to the great scandal of the family. Grandfather Puvis pushed Camille into the path abandoned by Pierre; he probably did not have to insist much. Camille entered the Lycée de Lyon in the special mathematics class; in 1855, at the age of 17 and a half, he was accepted first at the École Polytechnique. The jury was composed of Didion, Hermite, Lefebure de Fourcy, Serret and Wertheim; Serret, in particular, had the reputation of being very difficult; however, he gave the young Camille a mark of 19.8 out of 20. This shows us the exceptional value of the candidate Jordan, and also the illusions that Serret had about the precision of his examinations.
Jordan left the École Polytechnique with the number two. Like Hermite, he was useless in the graphic arts and this made him lose first place. After leaving the École des Mines in 1861, Jordan went for a while to the provinces, not far from Lyon, to Privas then to Châlon-sur-Saône; in 1867, he was called to Paris.
If, at this time, Jordan abandoned the vicinity of Lyon, if he expatriated himself and settled permanently in Paris, it was because his career was oriented; it would be a career as a scientist rather than an engineer. From the École des Mines, Jordan was put to scientific work; as so often happens for mathematicians, most of the research to which he subsequently devoted himself was already begun in his very first memoirs. There was such a vast field of reflection there, the promise of such a rich harvest of results, that Jordan found himself naturally inclined to devote his life to it. Moreover, considering the old man, modest to the point of self-effacement, reserved to the point of shyness that Jordan had become, I readily imagine him deciding, when still young, that he would never be a man of action, never a leader of men. His task as an engineer was reduced to the supervision of the quarries of Paris, to the control of the equipment of the Orleans railway. In 1870 he was, as they say now, mobilised in his job; he participated in the defence of Paris in the dual capacity of engineer and captain of engineers. In 1885 he retired as an engineer: he was then chief engineer.
In 1873, he had been called to the École Polytechnique as examiner of students, for analysis, replacing Ossian Bonnet, who was moving to the director of studies. He was immediately highly regarded; he left the memory of an amenity, an equanimity, a firm benevolence quite unique; in front of him, each student was aware of being supported by his examiner and of giving his true measure. I do not know if this is proof that Jordan was as appreciated by the administration as by the students, but his salary, originally set at 7500 francs, was increased two years later, by three francs sixteen centimes.
A better proof of the esteem in which Jordan was held by all at the École Polytechnique, is that he was appointed professor of analysis there, on 25 November 1876, replacing Hermite who was from then on to devote himself entirely to his course at the Sorbonne. As examiner, he was replaced by Moutard.
Jordan thus became Joseph Bertrand's colleague in the house where he had been his pupil. Having to give the same teaching as Joseph Bertrand was, for Jordan, a novice teacher, a severe test. He underwent it with complete success; if his course did not have the brilliance of that of the sparkling Joseph Bertrand, it was much richer, solid and coordinated. The test that one might have feared dangerous for Jordan, was much more so for Bertrand; students began, rather unjustly, to complain about him. Moreover, this judgment was perhaps somewhat affected by the attraction of novelty; Jordan's star paled slightly in turn when he had as a colleague his former student Georges Humbert, a professor less profound than Jordan, but less difficult to follow and possessing essential qualities for speaking before a large audience.
A professor must be, in his own way, a trainer of men; Jordan was too reserved to succeed fully in creating the desired atmosphere around him; he was nevertheless appreciated and listened to. His audience, won over in advance by the reputation for kindness and conscientiousness that Jordan had acquired as an examiner, was sensitive to his charm, made of naturalness, sobriety and ease. His students forgave him the effort that his lessons demanded; only in the songs of the "Ombres" session did one recall the abundance and heterogeneity of the professor's notations.
If one judged Jordan according to his caricatures at the École Polytechnique, one would declare him a fervent disciple of Bacchus, because he is always depicted with a glass in his hand. This is because he was very fond of sugared water; he drank two or three glasses of it per session, without ever removing the teaspoon, hence an effect that the caricaturists did not forget. In the special language of the École Polytechnique, Jordan has become a common noun: it refers to the professor's glass of water.
To get a clear idea of the teaching given by Jordan, it is not enough to leaf through his Traité d'analyse. "Course of the École Polytechnique," he told me one day, "we put that on the cover to please the publisher"; already, in fact, the first edition of the Traité goes far beyond the course of the École, yet it makes it quite clear. The following editions were mainly influenced by the lessons taught by Jordan at the Collège de France, first as Serret's substitute in 1875, then, from 1883, as Liouville's successor. Jordan's qualities suited the audience of the Collège even better than that of the École Polytechnique; certain artifices, intended to reduce the effort and which Jordan hardly used, were no longer important. He was dealing here with young people who were already very well-educated in mathematics, who wanted to know and really understand, and who knew full well that this required work. Jordan's reasoning, difficult but solid, facing up to difficulties, was precisely what they needed.
An audience of young workers with the qualities I have mentioned is necessarily small; Jordan's was sometimes very small and partly through his own fault. The author of this notice does not want to claim that it is easy to have a large number of students at the Collège de France, but Jordan had played on the difficulty by holding his classes at noon.
In 1912, Jordan retired as a professor. At the École Polytechnique, M Hadamard succeeded him; at the Collège de France, it was Georges Humbert; Humbert had already been replacing him there for several years.
In 1885, Jordan took on another task; he took over from Résal as director of the Journal de Mathématiques that Liouville had founded. It was a role that suited him perfectly. His vast erudition allowed him to judge all the papers; he had the necessary notoriety to attract the good copy which, as M Emile Picard says, drives out the bad and if, unfortunately, some bad manuscript got lost in the mail of the editor of the newspaper, he had the necessary authority to be able to say: no. The war compromised the existence of the newspaper for a moment; it was a great sorrow for Jordan and the occasion for great concerns. Many contributions were then offered to him as so many expressions of sympathy and gratitude; Jordan had been very sensitive to them. To deserve them even more, he believed he had to give up a task that had been so dear to him; on 1 January 1922, he handed over the destiny of the newspaper, Jordan's newspaper, as we all said, into the hands of M Henri Villat who, at the critical hour, had given him the most judicious and effective help.
* * *
It was in July 1902 that I had the opportunity to speak with Camille Jordan for the first time; I brought him a copy of my thesis. "Persevere in scientific research," he told me more or less, "you will experience great joys there. But you will have to learn to enjoy them alone. You will be a subject of astonishment for your own people. You will not be much better understood by the learned world; mathematicians have a special place there and they do not even always read each other." These words, long forgotten, I hear them clearly at the moment when it would be appropriate that, by putting their somewhat disillusioned philosophy at fault, I succeed in giving here an exact idea of the extent, the beauty, the importance of Camille Jordan's work. When the magnitude of the task and the skill of the worker are so disproportionate, he is not ashamed to admit his inadequacy; also, it is not to apologise, it is to better warn of the difficulty of Jordan's work that I point out letters exchanged between Jordan and Hermite. Jordan, perhaps on the occasion of a memoir that he would have submitted to the judgment of the Academy, had complained to Hermite that he did not read his work; Hermite replied that Jordan's work is too difficult, too abstract, he will resign as a member of the Academy if anyone tries to force him to read it!
Mathematicians are so accustomed to hearing "too abstract" as a reproach or an excuse, that it is the excuse and not the reproach that comes to their mind first; it is nonetheless very curious to see Hermite declare that Jordan engages in abstract research. In fact, Jordan's reasoning and subjects of study sometimes differ greatly from those of Hermite, but I would willingly say for my part that they are more concrete. Without arguing further about the most appropriate qualifier, I will try to highlight how Jordan differs from other geometers of his time.
In Jordan's work there are two parts, closely linked. One is in some way classical, it deals with the theory of forms, algebraic equations and related questions. It could have been considered difficult, but its importance was understood immediately. It is considerable, rich in results; it shows us Jordan's rare technical skill, the acuity of his sense of observation, the fertility of his imagination; we see Jordan further advancing old theories, almost completing others, such as that of algebraic equations where men of genius had not been able to succeed in solving certain fundamental problems.
Jordan's reputation is based almost entirely on this first part of his work. The second is composed of apparently unrelated memoirs, dealing with questions that one would readily describe as bizarre; it is in the choice of these questions that Jordan showed himself to be particularly innovative. Posterity, faithful to the compromise of justice and injustice to which it is accustomed, will perhaps somewhat forget the masterpieces of the master mathematician Jordan to glorify above all notes or memoirs written in passing, one might say, and to which Jordan himself cared the least.
To examine Jordan's work, I must speak of mathematics; I humbly apologise for this, as Darboux did when he pronounced before you the eulogy of Hermite; but this is the only way to honour a great mathematician. And if I must speak at length, it is because Jordan's long life was very full. I will, however, leave aside numerous and important works; I will not speak of Jordan's research on the kinematics of our space and of higher spaces, nor of the studies relating to the stability of bodies resting on supports or floating, nor of the problems of geometric probability that he treated, nor of his definition of angles in higher spaces and of the theorems which generalise those of Euler and Meusnier, nor of his multiple studies on binary forms, on quadratic forms, on bilinear forms, considered either from the analytical point of view, or from the arithmetic point of view, and which extend the work of Gauss, Hermite, Gordan, Kronecker, nor even of the determination of groups of motions, used immediately by theorists of crystallography like Leonhardt Sohncke and in which, as M Picard said here, is found the first study of a group of transformations. This is where the very important notion of the fundamental domain of a group is introduced, this is where we see the appearance of those denominations now universally adopted in group theory and which Jordan has drawn from crystallography. And, leaving aside many other memories, I will only speak, as regards the part of Jordan's work that I have called classical, of his two principal studies.
At the end of the eighteenth century, the question of the resolution by radicals of the general equation of the fifth degree was on the agenda. After having proved that this resolution is impossible, Abel posed the following two problems:
To form the equations of a given degree which are solvable by radicals;
Given an equation, recognise whether or not it is solvable by radicals and carry out this resolution when it is possible.
Abel succeeded in solving these problems for equations of a prime degree. "When the degree of the equation is a prime number," he wrote to his former professor, who had become his friend, Holmboe, "the difficulty is not so great, but when the number is composite, the devil gets involved."
Jordan's main result is the resolution of these two problems; but there are many others that Jordan deals with at the same time. For example, those relating to the resolution of equations, no longer with the help of radicals, but with the help of roots of equations of lower degrees.
Among all of Jordan's statements I choose the following because it generalises Abel's theorem, the origin of the whole theory: the general equation of degree cannot be resolved by means of equations of lower degree.
In his research, Jordan uses the brilliant method of Galois, the essential point of which is the introduction of a certain group of substitutions, already perceived by Lagrange, which can be attached to each algebraic equation and in which the properties of the equations are faithfully reflected. But to know how to observe in this mirror, one must have learned to distinguish the various qualities of the groups of substitutions and to reason with them. This is what Jordan did with skilful tenacity and rare happiness; in his Treatise on substitutions and algebraic equations, where he brought together and coordinated his research, the properties of the equations are derived immediately from those of the groups of substitutions.
The main qualities of the groups that Jordan uses are characterised by the qualifiers transitive or intransitive, primitive or imprimitive, simple or composite. Jordan's theorem on the composition of groups is the best known of all his results; it leads to this fundamental consequence: there is no need to choose between the different methods of algebraic resolution of an equation, they are all equivalent and lead to the same calculations, up to the order. If Jordan's theorems on transitivity and primitivity are less well known, it is because, until now, didactic treatises have only dealt with the general properties of equations and not with Abel's problems. They therefore do not intend to prepare the reader for effective calculations, which is on the contrary Jordan's aim. Also, a quantity of information can only be found, even today, in the Treatise on Substitutions. The concern for complete preparation for calculation, as if algebraic resolution were of practical use, has also led Jordan a little far. One is sometimes tempted to smile when one sees him enumerate congruences up to the degree 12000, enumerate types of groups solvable up to the degree 1000000, or apologise for the fact that a calculation becomes a little tedious when the number on which one operates has more than a thousand billion digits!
This concern for generality, which made Jordan return to many questions studied in his Treatise after its publication, has also been fruitful. Solving the first of the extremely difficult questions, Jordan would have been justified in declaring himself satisfied as soon as he had found a solution; quite the contrary, he strived to have solutions presenting varied advantages and he thus finds himself ready for all applications. To applications, he was able to devote a significant part of his Treatise; a very attractive part, because it brings us into contact with all the questions studied at the time by algebraists and geometers and, by advancing each of them, it shows to what rare degree of completion Jordan had pushed the general theory.
The study of the groups of algebraic equations encountered in the most diverse questions allowed him to conclude the existence of certain relations hitherto unsuspected, a little in the way in which the comparison of equations of dimensions foreshadows relations between quantities; this is how Jordan connects the equation to the 16 straight lines of Kummer surfaces, that to the 27 straight lines of cubic surfaces, and that of the trisection of functions with four periods.
The results obtained attracted the attention of those who had previously worked on the different subjects thus touched upon by Jordan. To find these results in another way, or to extend them, Klein, Geiser, Brioschi, Clebsch, Cremona, and Sylvester wrote memoirs which hastened the diffusion of Jordan's theorems and demonstrated their importance; also, before he was even forty years old, Camille Jordan was universally considered as one of the very first geometers of his time.
A few words on the work relating to the algebraic integration of linear differential equations. Frobenius had shown the link between this question and the theory of linear substitutions. The formation of the various types of equations with algebraic integrals results from the construction of the finite groups contained in the linear groups. M Klein had thus formed the five possible types of equations of the second order; Jordan showed that, for any order, the number of types of equations is finite and he undertook the enumeration of these types for equations of the third and fourth order. He thus provided us with fundamental results and yet, this enumeration was for him the occasion of an unpleasant misadventure. The table of finite groups given by Jordan, for the case of three variables, was incomplete and, unfortunately, Jordan had omitted the one that of all the groups perhaps which, from the geometric point of view, presented the most interest: the group of 168 collineations found shortly after by M Klein. Jordan returned to the question, discovered where his error lay, and resumed his enumeration. It was not yet perfect, however; twice since then, it has been rectified. And as we now possess up to three demonstrations of the fact that the list is complete, it is not impossible, in fact, that there is no longer any need to revise it. But why persist in using a process that requires such constant and meticulous attention that the best cannot hope to be without failures? It is because calculation, that precious tool of mathematics, the one that Descartes proudly claimed should be universal and all-powerful, is lacking here. Paradoxical as it may seem, the study of the algebraic resolution of algebraic equations does not fall within the calculus of algebra; the study of the algebraic resolution of linear differential equations does not fall within the calculus of analysis. Both questions require synthetic reasoning, akin to that of arithmetic or pure geometry.
When he cannot resort to calculation, the mathematician must explore the field in which he works, observe the role of the different mathematical beings he encounters there, watch them live, one might say, in order to discern their qualities and recognise the contributions of each of these qualities. This even obliges him to dissections, to experiments whose results are often translated into the establishment of a classification or an enumeration; in short, the mathematician transforms himself into a naturalist. If we did not judge the sciences according to the expository treatises, if we thought above all of their elaboration, we would recognise that in geometry and arithmetic, the scientist has always used this sort of experimental method where logic, while remaining the supreme argument of demonstrations, appears as the instrument par excellence of observation and mathematical experimentation. Deprived of calculation, the analyst must also have recourse to so-called synthetic reasoning. One would be surprised to learn that Jordan was a revolutionary or that he sought to distinguish himself in order to capture attention; if, in the main part of his works, Jordan used almost exclusively synthetic reasoning, it is because he followed the path and example of Galois. The training he thus acquired, and also some natural tendency, encouraged him to tackle unexpected problems. The mathematician who hardly uses calculation substitutes for the mathematics of quantities a mathematics of qualities; this raises new questions or questions that had hitherto been excluded from mathematics. It had indeed become customary to reject many problems outside of mathematics; to close one's eyes and sometimes to take on a scandalised air, was this not the best tactic to reinforce the dogma that calculation regulated everything? Jordan did not let himself be stopped by these prejudices: the solution of certain questions would be useful to the progress of analysis, he was able to undertake this study, that was enough for him; his love of analysis obliged him to broaden its framework. It is in this, above all, that Jordan differs from his contemporaries; through his research he has proven to us that geometry could render services to analysis comparable to those it received from it. In this turnaround that we are witnessing, Jordan's name will remain eternally linked with that of the brilliant Riemann.
It must also be added that, in Riemann, the role of spatial intuition appears clearly only on the occasion of Riemann's surfaces, while it is constantly evident, in Jordan, in his detailed studies of certain primary notions of mathematics.
What is an area, he asks himself, what is a volume, an integral, the length of an arc of a curve, what even is a curve or a domain? He studies these questions as a mathematician and not as a metaphysician, he does so in his analysis course and with a view to analysis; thus, alongside the magnificent exposition he gives of the theory of complex variables, he begins the construction of a theory of real variables so intimately linked and so useful to its neighbour that the barriers, raised between them by habits or prejudices, fall of their own accord. After Jordan, we dare to study general real functions, somewhat forgotten during the 19th century, we admit again that the aim of analysis is the study of reality, of that which does not allow itself to be extended into the complex domain.
At present, we are publishing, and in all languages, large works devoted only to real functions, the fruit of the work of Jordan and his numerous disciples. Camille Jordan must be considered as the first artisan of this renaissance of the study of reality which has just given our science a new vigour. He had, certainly, predecessors, numerous and of great value; only, while before him there were ingenious remarks, isolated results, profound conceptions but often as obscure as profound, after him, there was a clear and coordinated science. Jordan knew how to draw from previous works notions that he made simple, precise, immediately useful; on the other hand, he himself introduced entirely new notions, like that of a function with bounded variation, so important that M de la Vallée Poussin devoted one of the general lectures of the last International Congress of Mathematicians to it.
Time is pressing, I cannot dwell on these works, as I would like to do, to say again and more fully all that I owe to the thought of my illustrious master, to relate how it often happened to me to note that such an idea of which I was very proud was clearly and simply expressed in Jordan. It was from there that I had taken it, but to savour all its content, it had taken me so much effort that I had believed, for a moment, to have done personal work. The more one reflects on questions relating to real variables, the more one realises how deeply Jordan has penetrated them; he has done it without apparent effort and so simply that, despite recent writings, full justice has not yet been done to him.
Moreover, works on real variables have until now left aside the most original and personal part of his work, that which deals with the geometry of situation, analysis situs. In this new mathematics of which I have spoken, the qualities with which one deals derive above all from the notion of order; Jordan was able to write that his research "has almost constantly aimed at deepening the theory of order from the dual point of view of pure geometry and analysis". This idea of order, at the same time as it introduces into analysis a notion of geometric essence, broadens the field of classical geometry; for if the geometry of order has been, since Leibniz, called Analysis situs or geometry of situation, it is still to be constituted. Jordan has made brilliant contributions to it, the importance of which will be affirmed even more with the progress of science.
His research on the notions of curve, of domain, are short and simple; they were so right that they constitute today the first paragraphs of the richest, most nourished and best coordinated chapter of the new science. With M Schoenfliess, we have learned to see in Jordan's theorems the reason for the existence of implicit functions; since then, we have gradually recognised that existence theorems all have qualitative premises belonging to the Analysis situs. To emphasise the importance of this new idea, I will recall that Poincaré's last memoir is related to it; this memoir so suggestive from a mathematical point of view, so admirable from a moral point of view where he told us: this is what it would be important to do; I did not know how to do it.
As early as 1867, Camille Jordan had dealt with the Analysis situs. His first results are still little known: to clearly show their wealth and their scope, which are considerable, I had to devote to them the entire course that I gave this year at the Collège de France; here I must limit myself to a few assertions. The search, for polyhedra, for symmetries in the sense of Analysis situs dominates and resolves the question of the determination of finite groups of transformations as long as there are only two real variables to consider. It is to extend this powerful method to the general case that Poincaré undertook his long research on Analysis situs.
Jordan demonstrates that two surfaces having the same genus are applicable to each other, if we assume them to be extensible and deformable. From this it follows that we cannot obtain, for an algebraic curve, an invariant other than its genus by the sole consideration of its Riemann surface. An extremely important proposition that we had not even thought of considering. Poincaré, examining the analogous question for hypersurfaces, showed that the application, demonstrated possible by Jordan for surfaces, is in general impossible for hypersurfaces, which underlines the interest of Jordan's result.
The persistence with which I must cite Poincaré here after Jordan, for all these researches on Analysis situs, will perhaps suffice to make us suspect the primordial importance of the questions addressed by Jordan.
In the course of his research, Jordan studied this unfortunate theorem of Euler on polyhedra, whose erroneous demonstrations are legion; to limit myself to authors belonging to the high mathematical nobility, I will only cite those of Euler, Cauchy, Poinsot, and Cayley. In Camille Jordan we find the correct, simple, concise, elegant demonstration, now adopted in almost all didactic works. This shows that Jordan was capable on occasion of combining with the rigour of reasoning this elegance of form for which, in general, he cared little.
He cared little for the choice of notation, the arrangement of calculations; he never aroused in us admiration, or rather stupor, by resolving a question by considerations with no apparent connection with it; on the contrary, he always showed very clearly the progress of his thought and succeeded in giving us the illusion that we had carried out his difficult work as well as he had; if not better, because we would have taken more care than he had with the details of form. In him, elegance, high and powerful, was generality, clear sequence of ideas, courageous audacity in the face of difficulties, disdain for artifice.
These master qualities give his Treatise on Analysis a particular appeal. Jordan worked on it with predilection, using in the course of successive editions, as perhaps only he could do, the most recent works and dealing with extremely varied subjects. Thus, in the second edition, we find both an exposition of theories on sets, due to Cantor, and a real treatise on elliptic functions, the first to have been constructed in France from the ideas of Weierstrass.
In 1912, he deals with the completely new theory of integral equations and, of all the works, he uses above all the most recent, those of M Goursat.
If Jordan's Treatise is rich in innovations, we also find there the treasures of the past and even his memories. Jordan is willingly a traditionalist innovator; he keeps the outdated division into differential calculus and integral calculus; but, as his reflections have made him recognise in the integral the simplest, the most intuitive, the most primitive of all the notions of analysis, he begins the exposition of differential calculus with the definition of the integral. It is interesting to note that at the time when Jordan was thus restoring the notion of the integral to its first place, the one it has in the history of science, certain nationalists, in order to oppose Weierstrass to Cauchy, to exalt Hilbert, Hurwitz, Gordan and depreciate Hermite, were going on repeating that the integral is a learned notion; according to them, the idea of a series alone would be simple and primitive.
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By his Treatise on Analysis, Camille Jordan had students all over the world; his fame was considerably increased. The list would be long of the academies and learned bodies that insisted on counting him among their members. However, he was not one of those who facilitate the attribution of these distinctions by going to meet them; his friends had to insist for several years before he accepted the presidency of this Academy where he had succeeded Chasles in 1881.
Jordan had the academic honours he deserved; why should other honours have been so sparingly measured to him? Camille Jordan was only an officer of the Legion of Honour. Our leaders, who know how to speak of science with such eloquence and in such an emotional tone, should have done more for him. Not content with being a civil servant with the triple title of engineer, professor at the Ecole Polytechnique, and professor at the Collège de France, he had become one of the undisputed masters of science, he had thus worked for the intellectual renown of France; finally, he was one of the most representative figures of our scholarly world as former president and senior dean of this Academy.
You are very demanding, I might be told, Public Education is even very rarely so generous. Let there be no mistake; Jordan was named knight on 8 February 1877 on the proposal of the Ministry of Public Works, he was promoted to officer on 12 July 1890 on the proposal of the Ministry of War, never did Public Education, which had left to other ministries the honour of these decorations, find the opportunity to include Jordan on the lists, during the thirty-two years that he remained an officer. Public Education did not forget Jordan, however, although he was neither an administrator nor an office manager, but only a scholar and professor; it named him an officer of the academy in 1885 and an officer of Public Education on 28 January 1906: Camille Jordan had just entered his 69th year.
I had the duty to point out this oversight of the ministry which should encourage scholars; but I must add that Jordan certainly did not suffer from this oversight. He may not have even noticed it. Could a few moments of satisfaction, procured by a new decoration, count alongside the calm and lasting happiness that he had been able to find in his family life?
In 1862, he married Miss Isabelle Munet whom he had known in Lyon circles. From this marriage were born eight children; he knew twenty-five of his grandchildren, three of his great-grandchildren.
It was at 48 rue de Varenne, in the quiet neighbourhood near the Invalides, that Jordan raised his large family. When his son Edouard, himself the father of a very large family, was appointed professor of medieval history at the Sorbonne, Jordan gave him his home and moved in next door, at number 46, with his son Camille, the plenipotentiary minister. The home at 48 rue de Varenne was the site of an old hotel, built around 1700 by Charles Skelton, a field marshal. This is where Mgr de Ségur was born at the beginning of the 19th century. The hotel at 46 is that of Narbonne Pelet.
If, at number 46, Jordan's study had become quite stylish, at number 48, it was very simple; relegated to the second floor, with the outbuildings, it was part monk's cell and part student's attic. The only slightly luxurious object there was a large, deep armchair where Jordan would seat his visitor, suffocated by the smell of tobacco, for Jordan was an incorrigible smoker. It was in this room, which would hardly be enough for very young professors, that Jordan made most of his great discoveries while surrounding himself with clouds of smoke and chalk, for he preferred to work on two blackboards. It is said that the valet was only allowed to dust and put away books and papers twice a year; certainly, like many of us, Jordan had a horror of these systematic disturbances carried out in the name of order.
When he went down to see his family, Jordan did not always manage to leave his mathematical concerns upstairs. He was then very distracted and the children were very happy to catch their father red-handed in his absence. They had devised a curious psychological experiment; when they saw him very preoccupied, they would slowly read an article of some kind in front of their father; a few days later, the conversation would turn to the subject treated in the article; frequently, to the great joy of all, Jordan would use information drawn from this reading of which he had never been aware and of which he had nevertheless recorded all the essentials.
Camille Jordan gladly received at his home the scholars passing through Paris; his children remember seeing Borchardt, Schwartz, Brioschi, Klein before the manifesto of the 93, Sophus Lie, Mittag-Leffler, Guccia, Volterra and many others. One evening, Jordan had gathered a few people to dine in honour of Sylvester; when Poincaré arrived, Sylvester monopolised him without giving him time to greet the mistress of the house; "I have a beautiful theorem to demonstrate to you," he declared, and he demonstrated his theorem to him. From that moment on, Poincaré did not utter a word, he ate like an automaton, which did not fail to cast a chill. After dinner, Poincaré suddenly became aware of the outside world again, he discovered Sylvester, rushed towards him, exclaimed: "but your theorem is false," and he proved it to him peremptorily. It was not enough for Jordan to be distracted, he also had to bring other distracted people to his home!
Apart from his moments of preoccupation, Jordan was cheerful without ever being very talkative. He was very interested in the lives of his family and, in particular, in the studies of the children, for whom he had become a tutor in both literature and science.
According to his classmates, at the École Polytechnique, Camille Jordan devoted all his time to the study of literature and Greek; there is undoubtedly some exaggeration here, but Jordan had at least managed to read the ancient authors fluently in the text. Edouard Jordan, the professor at the Faculty of Letters, can attest that his father was an excellent teacher of Greek.
Jordan did not read the classics only to help his children; this reading remained his main distraction until the end. He hardly went to the theatre or to concerts; the only game he played was chess. As a good theorist of order, he was very interested in this game, he collected everything that was published on the subject; for a long time he was a regular in the circle of chess players, at the Café de la Régence.
He frequented museums and knew all those in Europe perfectly, because he travelled a lot. His long holiday hikes were his great joy; he visited all of Europe several times, he went to Algeria, Egypt, Palestine, the United States. But above all he travelled, and in all directions, the Alps for which he had a passion.
Accompanied most often by one or the other of his sons, he made all the climbs, at least all those that did not involve acrobatics. He climbed for the pleasure of climbing and left for the planned excursion even when he was certain that, from the summit, nothing would be seen.
Until the end of his life, he remained very alert. At the Strasbourg congress, in 1920, an excursion to Sainte-Odile had been organised. Jordan, disdaining cars, made the climb on foot. Each bend in the road revealed new horizons; while admiring them, the tourists staggered. At one point, Jordan found himself alone; he thought he was behind, with a firm and steady step he began to climb. He arrived at Sainte-Odile more than a quarter of an hour before his companions who, at each bend in the road, saw him always further ahead and wondered why he was going so fast. At 82 years old, Jordan was still a good walker and somewhat distracted.
Jordan was actively involved all his life in charitable works in his neighbourhood; very generous, he gave to many other works, however he was more sparing of his time than of his money and only agreed to be part of a committee if he believed he could be really useful.
He was interested in private education; he was part of the Civil Society of the Stanislas College and the Patronage Committee of this college; at the time of the creation of the Catholic Institute, he was one of its first professors with his friends and future colleagues, Lemoine and de Lapparent. After about a year, having been ordered to opt, he devoted himself to teaching at the École Polytechnique.
Lemoine and de Lapparent, with whom Jordan became friends from the École des Mines, Jenner and Vicaire, whom he knew from the École Polytechnique, later Humbert, his former student, Duhem, his former auditor at the Collège de France and the classmate of his son Edouard, are among the best friends of Camille Jordan to whom they were linked in particular by affinities of belief. Jordan also knew how to have non-Catholic friends; he had as much affection as admiration for Halphen. The loss of Halphen, to whom he devoted a fine study in the Journal de Mathématiques, caused him great sorrow.
It is the sad lot of those who live a long time to have to mourn many deaths. Jordan saw many of those he loved disappear: almost all his friends; one of his daughters in 1912; during the war, from 1914 to 1916, three of his sons and the eldest of his grandsons, a glorious and painful tribute to a family where one always knew how to do one's duty; in 1918, finally, he lost his faithful companion. Under these repeated blows, Camille Jordan seemed to weaken for a moment; he seemed to have regained his strength when, suddenly, death struck him.
All those who knew him will remember him as a perfect man of goodness, dignity and uprightness. All those who know how to read him will admire the depth of his views, the generality of his considerations, the power and originality of his reasoning. The progress of science will further enhance the name of the man whom all the allied mathematicians, gathered in Strasbourg in 1920, had named by acclamation their honorary president.
Last Updated March 2025