Paul Painlevé's father, Léon Painlevé, was a lithographic draughtsman. Paul was :-
... brought up in the simple democratic atmosphere of French skilled artisan family life.
Paul attended the École Primaire where he showed himself to be equally outstanding at both sciences and literature. 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 received his agrégation in mathematics in 1886. While completing the work for his doctoral dissertation he went to Göttingen where he was influenced by Schwarz and Klein; he received a doctorate in mathematics from Paris in 1887 for this thesis. 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 professor of mathematics 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. This was a rapid return to Paris and shows the high regard in which he was held. From 1896 he taught courses at the Collége de France and, from the following year, 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 Poincaré and Émile Picard had failed to solve, showing, as 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.
In 1901 Painlevé married Marguerite Petit de Villeneuve. Their son Jean was born in the following year and, tragically, Marguerite died during the birth.
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, then in the following year 1909 he created the first university course in aeronautical mechanics.
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 :-
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.
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 Doumergue. He remained President of the Chamber of Deputies until April 1925 when M Herriot was defeated on a financial matter. Painlevé then became Prime Minister for a second time :-
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.
In  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 ...
Article by: J J O'Connor and E F Robertson