In 1866 Appell's family were forced to leave Ritterhus and his father died the following year. Appell rejected the Catholic views of his parents but he retained their strong patriotic French views. He insisted, in 1869, on leaving the Catholic school which he was attending and studying instead at a lycée.
On 14 July 1870 Bismarck provoked France and they declared war five days later. The German offensive was efficient, the French mobilisation was not. Appell's youngest half-brother Charles, who Appell was very close to, joined the French forces. Within four weeks the Germans had surrounded a French field army at Metz. The main French army attempted to lift the siege but was itself surrounded and trapped by the Germans at Sedan. On 2 September 83,000 French troops surrendered. The Germans captured Strasbourg after a 50-day siege in which Appell suffered greatly.
French resistance was carried on by a new French government. On 19 September the Germans began to besiege Paris. New French armies engaged but could not defeat the German forces. Metz surrendered on 27 October and Paris surrendered on 28 January 1871. Germany annexed Alsace and half of Lorraine with Metz. Strasbourg was annexed by the Germans and Appell moved to Nancy to become a French citizen and prepare himself to study at university in Paris.
Appell became a life-long friend of Poincaré while at Nancy. He entered the École Normale Supérieure in Paris in 1873 and graduated in first place in 1876 with a doctorate in mathematics. K O May writes in :-
From this time on, Appell maintained an amazing level of activity in teaching, research, editing and public service.In 1881 Appell married Amelie, niece of Bertrand and of Hermite and a cousin of Émile Picard. One of his three daughters was to marry Borel. In 1885 he was appointed to the Chair of Mechanics at the Sorbonne.
Appell now worked in Paris but returned for each vacation to German held Alsace. There he was given information by his half-brother Charles, which he would report to the French War Office on his return to Paris. Basically Appell and his brother Charles were acting as spies and informants for France against Germany. Charles was to pay a price for this when, in 1889, he was imprisoned for anti-German activities.
In 1892 Appell was elected to the Académie des Sciences. He served as Dean of the Faculty of Science of the University of Paris from 1903 to 1920 and, at the end of his deanship, he was appointed Rector of the University of Paris. He was rector from 1920 to 1925. However he also served in many other roles and his activity in these is described in :-
In various government posts, including membership in the Conseil Supérieure d'Instruction Publique, he was an exponent of educational reform and initiator of numerous large-scale projects ...One of the major political events which gripped France during much of the time that Appell held top posts was the Dreyfus Affair. Dreyfus, like Appell, came from Alsace. Born into a Jewish family, Dreyfus embarked on a military career. In 1894, when he was in the War Ministry, he was accused of selling military secrets to the Germans and he was sentenced to life imprisonment. Although his trial had been highly irregular the anti-Semitic views of many people made the verdict popular. Forged documents and cover-ups soon showed that the legal process had been suspect. In 1898 the novelist Émile Zola wrote an open letter accusing the army of covering up its mistaken conviction of Dreyfus.
The case split France into two opposing camps leading to issues far beyond the guilt or innocence of Dreyfus. There were demands to bring Zola to justice, anti-Semitic riots broke out, and there was a petition demanding that Dreyfus be retried. Zola was sentenced to a year in prison and fined 3,000 francs. By 1899 there had been a confession to the forgeries, followed by a suicide, and Dreyfus was retried, again found guilty, but pardoned.
Appell was very much involved with the case. He himself was from a similar background and had suffered through the Franco-Prussian war of 1870-71, so much centred on Alsace. When Dreyfus was granted a retrial in 1904, Appell served as an expert on the commission that, by July 1906, had cleared Dreyfus and reversed all previous convictions. Hadamard also played a major role in clearing Dreyfus's name.
During World War I Appell founded the Secours National, a semi-official organisation involving all political and religious groups, which gave help to civilian victims of the war. After the war Appell had the ambition of his life fulfilled when his homeland of Alsace was returned to France. Also following the war the League of Nations was set up by the Allies at the Paris Peace Conference in 1919, and Appell served as secretary-general for the French Association during the 1920s when the League had its headquarters at Geneva.
Appell's first paper in 1876 was based on projective geometry continuing work of Chasles. He then wrote on algebraic functions, differential equations and complex analysis. In 1878 he noted the physical significance of the imaginary period of elliptic functions in the solution of the pendulum which had been though to be purely a mathematical curiosity. He showed that the double periodicity follows from physical considerations.
In 1880 Appell defined a series of functions satisfying the condition that the derivative of the nth function is n times the (n - 1)th function. These are now called the Appell polynomials. In 1885 he was awarded half of the Bordin Prize for solving Monge's problem:-
To move a given region into another of equal volume so as to minimise the integral of the element of volume times the distance between its old and new positions.When another prize was offered in 1889 to solve the problem:-
To find an effective method of calculating the Fourier coefficients in the expansion of quadruply periodic functions of two complex variablesAppell submitted a solution which won second place. The winner of this prize was Poincaré.
The article , written by Appell himself, lists 140 works in analysis, 30 works in geometry, 87 works in mechanics as well as many textbooks, addresses, lectures on the history of mathematics and lectures on mathematical education. This is not even the complete list of Appell's publications since he published further works after  was written.
There is an obvious question that we must ask ourselves about Appell. How is it that a mathematician who was so successful in every area of his subject and made significant contributions to many areas outside mathematics is not better known today? Perhaps a clue come from  where Appell himself writes:-
I always had little taste for developing general theories and preferred to study limited and precise questions that might open new paths.This is a pretty accurate assessment: Appell was one of the finest problem solvers there has been in mathematics. However he solved these problems using existing techniques and therefore his work had little lasting impact other than that the problem itself had been solved.
May, writing in , emphasises this point:-
... his scientific work consists of a series of brilliant solutions of particular problems, some of the greatest difficulty. He was a technician who used the classical methods of his time to answer open questions, work out details, and make natural extensions in the mainstream of the late nineteenth century; but his work did not open new doors as he hoped. On the contrary, he does not seem to have looked down any of the new paths that were leading to a period of unbridled abstraction and generalisation. During the last half of his career he was a pillar of a backward looking establishment that was to give way to Nicolas Bourbaki, a namesake of a general who was one of his boyhood heroes.
Article by: J J O'Connor and E F Robertson