By July 1870 Napoleon III, the French emperor, was trying to improve his popularity. Thinking that there is nothing like a war to get people behind you, and being advised that France could win against Prussia, Napoleon was keen to start a war. Otto von Bismarck, the Prussian chancellor, saw a war as an excellent opportunity to unite the German states. Bismarck sent a provocative message to France and, as he had hoped, they declared war on 19 July.
Halphen served in the French army in the conflict. It soon became obvious that Napoleon III had been badly advised and the French were no match for the Prussian forces. The French forces were defeated at the Battle of Sedan and, on 2 September, 83,000 French troops surrendered. Two weeks later the Germans besieged Paris which surrendered on 28 January 1871. It was a war in which France had been humiliated, and the terms of the treaty which ended the war reflected this. Halphen, however, had served his country with great distinction.
In 1872, after leaving the army, Halphen married the daughter of Henri Aron. They had seven children, three daughters and four sons. Also in 1872 Halphen was appointed as répétiteur at the École Polytechnique and he was soon making major contributions. The first result which brought him to the attention of mathematicians world-wide was his solution in 1873 of a problem of Chasles :-
Given a family of conics depending on a parameter, how many of them will satisfy a given side condition? Chasles had found a formula for this but his proof was faulty. Halphen showed that Chasles was essentially correct, but that restrictions on the kinds of singularity were necessary. Halphen's solution was ingenious ...Halphen took a different view on the problems of enumeration from his contemporaries. He defined the concepts of proper and improper solutions to an enumerative problem involving conics. Then a particular number associated with a problem about conics has enumerative significance when it counts the number of proper solutions.
In fact Halphen was well ahead of his time in the ideas which he brought to these problems. This did not mean, however, that his ideas were accepted by everyone around. Halphen and Schubert engaged in a heated debate on whether an enumerative formula should be allowed to count degenerate solutions along with the nondegenerate solutions. This was, in the end, simply a special case of an old argument: is a mathematical theory important because of its external applications or because of its internal beauty?
Next Halphen classified singular points of algebraic closed curves thus extending the work of Riemann. He was led to extend results due to Max Noether which, in turn, had him examine projective transformations which fix certain differential equations. A characterisation of such invariant differential equations appeared in Halphen's doctoral dissertation On differential invariants which he presented in 1878. Poincaré writes in  that:-
... the theory of differential invariants is to the theory of curvature as projective geometry is to elementary geometry.Halphen made major contributions to linear differential equations and algebraic space curves. He examined problems in the areas of systems of lines, classification of space curves, enumerative geometry of plane conics, singular points of plane curves, projective geometry and differential equations, elliptic functions, and assorted questions in analysis. He gave a formula for the number of conics in a 1-dimensional system which properly satisfy a codimension 1 condition, and also a proof of his formula for the number of conics which properly satisfy five independent conditions. This last result appeared in a paper Halphen published in the Proceedings of the London Mathematical Society in 1878.
He received great honours and prizes for his work on these topics. For example, in 1880 he won the Grand Prix of the Académie des Sciences for his work on linear differential equations. Then, in 1882, he won the Steiner Prize from the Berlin Academy of Sciences for his work on algebraic curves.
In 1884 Halphen was made an examinateur at the École Polytechnique, then two years later he was elected to the Académie des Sciences. Sadly he died in 1889 at age 44 when at the height of his creative powers.
A major figure in his time, much of Halphen's work was in areas which have fallen out of favour. Other work such as that on linear differential equations was overtaken by Lie group methods. Bernkopf writes in :-
The amount and quality of Halphen's work is impressive, especially considering that his mathematically creative life covered only seventeen years. Why, then, is his name so little known? ... he worked in analytic and differential geometry, a subject so unfashionable today as to be almost extinct. Perhaps with its inevitable revival, analytic geometry will restore Halphen to the eminence he earned.
Article by: J J O'Connor and E F Robertson
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