# Paul Painlevé

### Quick Info

Paris, France

Paris, France

**Paul Painlevé**worked on differential equations. He served twice as prime-minister of France.

### Biography

**Paul Painlevé**'s father, Léon Louis Painlevé (1832-1906), was a lithographic draughtsman, born in Paris, the son of Euphrosine Célestine Leroy and Jean Baptiste Painlevé. He became the owner of a printing ink factory and was quite well-off financially. Paul's mother Antoinette Élisabeth Détang (1835-1893), the daughter of Élisabeth Étiennette Hayot and Denis Détang, was born in Meaux, Seine-et-Marne. Léon and Antoinette were married in Paris on 4 October 1855 in the Roman Catholic Church. They had three children, Marie Mathilde Painlevé, born on 6 August 1856, Blanche Julie Painlevé, born on 18 September 1861, and Paul Painlevé, the subject of this biography. Let us say a little about Paul's two sisters since they enter his biography later.

Marie Mathilde Painlevé worked in lace and corset supplies. She married Ferdinand Lamy and had four children, Jeanne, Fernande, Pierre and Suzanne. Ferdinand Lamy died before 1901, as did one of the children. Blanche Julie Painlevé married Maurice Louis Dainville (1856-1943), a painter and architect, and had a daughter Jeanne Dainville (1889-1939) who married Pierre Appell (1887-1957), the son of Paul Appell.

The three Painlevé children were baptised into the Roman Catholic Church but Léon Painlevé stopped being a practising Catholic, distancing himself from the Church. Paul was [24]:-

... brought up in the simple democratic atmosphere of French skilled artisan family life.Paul attended the lycée Saint-Louis, where he won many prizes, then in 1877 he began his studies at the lycée Louis-le-Grand [23]:-

Leaving primary school to which he would always remain so deeply attached, he went through high school where everything seemed equally easy ...At Louis-le-Grand he showed himself to be equally outstanding at both sciences and literature. Again he won many prizes across the whole range of the syllabus. By the time his secondary education was completed he was still undecided on the direction that he wanted to take in life, feeling that he would like to take up politics or engineering but in the end chose to embark on the research career.

He entered the École Normale Supérieure in 1883, and was immediately drawn to mathematics through the influence of outstanding professors like Paul Appell, Gaston Darboux, Charles Hermite, Émile Picard, Henri Poincaré, and Jules Tannery. He studied differential and integral calculus as well as descriptive geometry and mechanics. In 1885 he obtained the licence in mathematical sciences and in physical sciences. He received his agrégation in mathematics in 1886. While completing the work for his doctoral dissertation, advised by Émile Picard, he went to Göttingen where he was influenced by Hermann Schwarz and Felix Klein whose courses he attended; he received a doctorate in mathematics from Paris in 1887 for his thesis

*Sur les lignes singulières des fonctions analytiques*Ⓣ. Before giving a little information about his thesis, let us recount an important event in Painlevé's political development which occurred while he was in Göttingen.

Guillaume Schnaebelé was a French police inspector. He was born near Strasbourg in Alsace, and, after the Franco-Prussian war of 1870-71, opted to live in France after Strasbourg became part of Germany. Schnaebelé was arrested by the Germans in April 1887 when on his way to meet with a German police inspector. It was unclear whether he was arrested on French territory or on German territory and the incident nearly led to war between France and Germany. Painlevé was a pacifist and he was appalled both at the strength of nationalism shown by both sides and also the strength of militarism. He felt that the incident had been very badly handled by the French government, and he was also aware the France was in a weak position militarily, which had influenced their actions.

We promised to return to Painlevé's outstanding thesis [23]:-

Painlevé's thesis focused on the important and remarkable property that the individuality of a function resides, in a way, in its singularities, and that it suffices to know these, which can be in more or less limited number, to be informed about the entire function. He dealt more particularly with functions which have whole lines of singular points; on this subject he demonstrated several new and important theorems and began to orient his results towards the study of new transcendents defined by differential equations.Here is an extract from Painlevé's introduction:-

The first part of this work is devoted to the study of a function in the vicinity of a singular line or cut. The notion of cut occurs in the discussion of the Cauchy integral; it can even be introduced, as shown by M Hermite using definite integrals where the integration variable is real. The Taylor series, converging in a certain circle, also offers a simple example of a cut, and MM Weierstrass, Jules Tannery, Appell have formed numerous series which represent two distinct functions in two different areas of the plane. We list, in the first Chapter, the various singularities that symbols (and particularly series) can present which define in a certain space, an analytical function of z. ...The standard career path for a leading French academic at this time was to obtain a first post in the provinces, then later to attempt to return to Paris. Painlevé followed this route, being appointed as a lecturer in rational and applied mechanics at Lille in 1887, and then returning to Paris in 1892 where he taught both at the Faculty of Science and at the École Polytechnique taking up his appointment on 23 July. This was a rapid return to Paris and shows the high regard in which he was held. In 1895 he was invited to deliver lectures at the University of Stockholm giving the first on 2 October. The lectures were published in 1897 under the title

In the second Chapter, we extend to the most general uniform functions the forms of decomposition into sums and products, given in the theory of functions with singular points. In 1881, before M Mittag-Leffler's theorem, M Émile Picard indicated, in the case of a circular cut, a form of development into a product, applicable to any cut, and, shortly after, decomposed a function F(z), having for cuts line segments, into a sum of n functions having only one cut, then developed these functions in series. After the discovery of the Mittag-Leffler theorem, M Goursat extended this theorem to uniform functions with any singularities. Finally, M Mittag-Leffler has himself devoted a Memoir to the study of these proposals. We give, with a slightly different demonstration of these theorems, several modes of development in series of functions presenting in the plane a single cut, to which we find ourselves returning. A first development, analogous to the Taylor series, is based on conformal representation; the others generalise the expansions, indicated by M Appell in the case of a holomorphic function inside a contour of arcs of circles. It follows from this theorem that any holomorphic function in a convex region can be developed in this region in a series of polynomials.

The notions of residue, order, remainder (defined either as integrals or as coefficients) are therefore easily generalised with the propositions to which they are attached. In particular, the residue theorem and Liouville's theorems remain for doubly periodic functions with arbitrary singularities.

*Leçons sur l'intégration des équations différentielles professées à Stockholm*Ⓣ. These lectures [23]:-

... shine with a particularly vivid brilliance ... He put the best of himself into these lessons and reading them will remain a source of inspiration for our youth for a long time.An important feature of these lectures is the Painlevé Conjecture about the $n$-body problem. We quote from [15]:-

Advised by Gustav Mittag-Leffler, King Oscar II of Sweden and Norway, a protector and supporter of science and especially of mathematics, established in 1887 an important prize for solving the 3-body problem. The formulation was very precise: one must obtain, for any choice of the initial data, a solution expressing the coordinates as a power series, convergent for all real values of the time variable. ... Unexpectedly, nobody could provide the desired solution. ... In 1895, at 32 years of age, Paul Painlevé was already one of the most famous mathematicians of his time, and King Oscar II invited him to give a series of lectures at the University of Stockholm in September-November of that year. The event was considered of paramount importance, and even the King attended the introductory lecture. The notes were published in 1897 in handwritten form ... The last pages contain an application of the results to the 3-body problem and an opinion of the author concerning the n-body case, formulated as a statement which was known afterwards as the Conjecture of Painlevé.From 1896 Painlevé taught courses at the Collége de France as a Professeur suppléant and at the École Polytechnique as a Répétiteur d'analyse. From the following year, he was a Maître de Conférences at the École Normale Supérieure.

Painlevé's first area of interest in mathematics was rational transformations of algebraic curves and surfaces. In this topic he introduced the notion of a biuniform transformation. He worked on differential equations, particularly studying their singular points, and on mechanics. His interest in mechanics was a natural one since this subject provided a natural setting for applications of the results which he had proved for differential equations. He solved, using Painlevé functions, differential equations which Henri Poincaré and Émile Picard had failed to solve, showing, as Jacques Hadamard wrote, that:-

... continuing the work of Henri Poincaré was not beyond human capacity.For his outstanding mathematical work Painlevé received many awards. In 1890 he was awarded the Grand Prix des Sciences Mathématiques, then in 1894 he received the prestigious Prix Bordin followed two years later by the Prix Poncelet. In 1900 he was elected to the geometry section of the Académie des Sciences. For his candidacy to the Academy, Painlevé wrote a notice which was published in 1967 as

*Analyse des travaux scientifiques jusqu'en 1900*Ⓣ.

For extracts from reviews of this 1967 publication, and of other books by Painlevé, see THIS LINK.

A step towards Painlevé's political career came in 1899 when he gave evidence at the trial of Alfred Dreyfus in Rennes. Dreyfus had been convicted of treason in December 1894 and sentenced to life imprisonment. Evidence came forward that he was innocent but documents were forged in a deliberate miscarriage of justice prompted by anti-Semitism. A new court martial was held in Rennes beginning in August 1899 at which Dreyfus was again convicted. The affair became something of great national interest with most people against Dreyfus but he also had some high powered supporters. Painlevé was a witness at the second Dreyfus court martial in Rennes. He said that his statements had been manipulated to make them prosecution evidence against Dreyfus. He also denounced the hand writing expert Alphonse Bertillon. Painlevé would continue to fight for justice for Dreyfus until he was exonerated in 1906. Jean Perrin writes [28]:-

... the Dreyfus affair led a few rare intellectuals to oppose almost the whole of the Country, first ill-informed, then gradually won over. In this drama of which France can remain proud, because each of the two groups that clashed fought selflessly for its belief, because also no other Nation would have allowed the rectification of the error once committed, Painlevé recognised where Justice was, and where from then on was the true interest of the Fatherland, of our Fatherland. ... This passionate love of justice, which the Dreyfus affair revealed in Painlevé, was perhaps the most striking feature of his moral character.On 8 November 1901 Painlevé married Julie Marie Marguerite Petit de Villeneuve (1868-1902), known as 'Gaette', the daughter of the architect André Jules Edmond Petit de Villeneuve and Marie Marguerite Léodie Clairin. Paul and Marguerite Painlevé's son Jean Marie Léon Painlevé (1902-1989) was born on the 20 November of the following year and, tragically, Marguerite died six weeks after the birth, on 31 December 1902, of puerperal fever. One of Jean's friends wrote:-

It would be hard to find a young man of more radical views and more opposed to the church. ... Jean Painlevé, resourceful and irresponsible as he was, found a way of taking part in everything that bore even the faintest trace of social protest and disorder.Jean became a filmmaker and is sometimes known today as the father of the documentary film. After Painlevé's wife died, Jean was looked after for a short time by Painlevé's sister Blanche but quite soon Painlevé and his widowed sister Marie decided to live together with their four children. They lived first at 18 rue Séguier, then at 81 rue de Lille in a seven-room apartment (with three maids' rooms). Marie Lamy, who wished to be independent, continued to carry on her trade in lace and corset supplies while servants took care of the running of the household.

At the International Congress of Mathematicians held in Paris in 1900, Painlevé was chairman of the Analysis Section. In 1904 Painlevé was a plenary speaker at International Congress of Mathematicians in Heidelberg. He delivered his lecture

*Le problème moderne de l'intégration des équations différentielles*Ⓣ on 11 August 1904.

You can read the first section of the lecture at THIS LINK.

Painlevé took a special interest in aviation, applying his theoretical skills to study the theory of flight. He approached the Chamber of Deputies in 1907 arguing that it was necessary to set up a branch of the military involved with aviation; he was successful and the military aviation service was set up. He was Wilbur Wright's first passenger making a record 1 hour 10 minute flight at Auvours in 1908, became the first person to fly on two different planes when he was a passenger with Henry Farman, then in 1909 he created the first university course in aeronautical mechanics. After his flight he wrote [23]:-

The signal is given; here we are launched into space. Sensation of delights and dizziness. We fly, we fly ... but it is no longer on the Auvours camp that we hover in the growing night, it is on the indefinite face of the Earth, dominated, conquered by the big bird. The conquest of the air is now accomplished. Tomorrow, on grander aircraft, safe and powerful engines, free from weight restrictions, will take away otherwise heavy burdens. The greatest challenge that nature had brought to man is finally met.In 1910 Painlevé and Émile Borel published the book

*L'Aviation*. They begin their Introduction as follows:-

It is some 120 years since, first on hot air balloons, then on balloons, man ventured across the ocean by air. Exceeding the eagle and the condor, the balloon made sublime holes in the azure. Poets sang of its glories and its martyrs; they celebrated the audacity of those conquerors of height who dared to complete: "by the wicker basket, the attack started by the rock of the Titans." But the balloon is a toy of the wind, it goes where the air leads it and not where the pilot leads it. By providing it with an engine, by lengthening it into the shape of a fish, Kerbs and Renard, in 1884, gave it direction, or at least, allowed it to move in a certain measure.Although less skilled in politics than mathematics he began a political career in 1906 which led to two periods as French Prime Minister. It may seem unfair to say he was less skilled in politics than mathematics when he achieved the highest possible office in politics, but this statement is more meant to comment on his truly outstanding mathematical contributions. Although Painlevé began his political career in 1906, this was not the year he left mathematics. It was the year in which he was elected to the Government as a Paris Deputy for the fifth arrondissement, the Latin Quarter [24]:-

He was soon distinguished both by the excellent matter of his speeches and by the interest he displayed in military, naval, and aeronautical affairs, and served on several Parliamentary committees concerned with the national forces.By 1910 he had given up all his mathematical posts and had become a full-time politician. His expertise in military affairs meant that after World War I started in 1914 he chaired many committees with a military remit, such as those set up to reorganise munitions, the navy, and aeronautics. He joined the Cabinet in 1915 as Minister of Public Instruction and Inventions. By early 1917 he was appointed as head of the Ministry of War and accepted, against his better judgement, the advice of his Commander-in-Chief to launch an all-out attack on the German lines. The attack rapidly failed and Painlevé had to replace his Commander-in-Chief.

On Sunday 20 May 1917 a monument was inaugurated to honour the memory of Marcelin Berthelot and Painlevé, as Minister of War, made a speech. In it he gave high praise to Science [23]:-

It is Science which will assure to human societies fair and rational laws and organisation. It will solve social problems by multiplying the industrial forces of man and his grip on nature, constantly creating new wealth that will not have been taken from anyone, however that it will bring the final softening of manners by its lessons of fraternity and by the development of intelligences. Already his essentially collective effort has brought to the bottom of our hearts and our minds the life-giving lesson of high solidarity.Writing about Painlevé in 1917, Henry Baker wrote [11]:-

On the one hand, he is at present one of the very few men whose pronouncements mark the destiny of our Western civilisation; on the other, he lived, not so long ago, in one of the most abstract realms of the modern theory of analytic functions, and very few were, or will ever be, those who fully followed the subtlety of his thought.After a disagreement with the French Socialists, Prime Minister Ribot was forced out and on 7 September 1917 and Painlevé became Prime Minister. He played a leading role in the Allied Conference at Rapallo in Italy, but was defeated after returning to Paris and he resigned as Prime Minister on 13 November 1917. He played little part in political affairs from this time until the election of November 1919 when he came to the fore as a strong critic of the elected Government. At the next election of May 1924 Painlevé was part of the winning alliance and was elected President of the Chamber of Deputies. The alliance had been forged by Painlevé and M Herriot and the latter became Prime Minister.

Painlevé was put forward for election as President of the Republic but lost out to M Gaston Doumergue. He remained President of the Chamber of Deputies until April 1925 when M Édouard Marie Herriot was defeated on a financial matter. Painlevé then became Prime Minister for a second time [24]:-

His new Government was weak from the first. Serious disorders in Syria further discredited it. His schemes for financial reform, which fell disappointingly short of what had been expected from a man of his ability, failed to meet with the approval of the Chamber, and on 21 November 1925 he had to resign.Painlevé still retained high office, however, for he returned to his position as Minister of War. In May 1932 his name was put forward in the election for President of the Republic but he withdrew before voting took place. After this he held the position of Minister of Air and in this role he made proposals for an international agreement to end the production of bombers in all countries, and for an international air force to be set up to be used against any aggressor. His plans could be taken no further after the Government fell in January 1933. This ended his political career.

For a much more detailed look at his political career, see some of the documents produced by the French National Assembly to celebrate the 150th anniversary of Paul Painlevé's birth in December 2013 at THIS LINK.

In [24] his personality is described in these terms:-

Painlevé had a naturally simple and unaffected manner, and was possessed of a singular charm that few persons, even among his opponents, were able to resist. His energy was untiring ...Jean Perrin writes in [28] about what:-

... made him loved, the charm of his conversation, his laughter, his extraordinary youth, the strength which was in his gaiety, his verve, his vitality, the flow of his generous words, the Flame which emanated from him, and all that was his soul?Thomas Greenwood writes in [20] about Painlevé's death:-

When I went to see him shortly before his untimely death, Prof Paul Painlevé was editing, with the aid of an assistant, the second part of his famous lectures on the "Mecanique des Fluides" Ⓣ recently delivered at the Sorbonne. In the dusk of his life, the 'President', as he was familiarly called by his friends, was thus returning to his favourite studies, for it was as a mathematician that M Painlevé began his extraordinary career. He was slowly recovering from a long and dangerous breakdown and was hoping to give an inaugural lecture in the great hall of the Conservatoire des Arts et Metiers, which was recently named after him in honour of his scientific genius. The hope was not to be fulfilled: instead, it was his coffin which was placed in that very hall before it was borne to the Pantheon. "I am still holding on to life," he was heard to say recently; "and if I have to let go, I shall try to do it as elegantly as I can!" These prophetic words became true when on October 29, Prof Painlevé died in his own home from heart failure. In him, France loses one of her most distinguished sons, and the world one of the greatest mathematicians and statesmen of the day.$$

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Written by J J O'Connor and E F Robertson

Last Update July 2020

Last Update July 2020