Charles Hermite | Biography

Charles Hermite was one of the founders of analytic number theory. This discipline uses the techniques of analysis (the calculus) to handle questions about positive whole numbers. Hermite is also remembered for having shown that one of the central constants of mathematics, e, the base of natural logarithms, belongs in the class of transcendental numbers.

The son of Ferdinand Hermite and Madeleine Lallemand, Hermite was born on Christmas Eve, 1822. His ancestry was both French and German and the town Dieuze, Hermite's birthplace, was at one time claimed by both France and Germany. Nevertheless, Hermite considered himself French all his life and became one of the mainstays of the French academic establishment.

Hermite attended the Collège Henri IV and proceeded from there to the Collège Louis-le-Grand, where he was taught mathematics by the same instructor who had supervised the work of the ill-fated French genius Èvariste Galois. When Hermite decided to continue his studies at the Ècole Polytechnique, he was admitted 68th in his class, thanks to his having neglected geometry. Throughout his life Hermite had a dislike for examinations and preferred to pick up material spontaneously rather than under the pressure of a deadline. Hermite enjoyed corresponding with the best mathematicians of Europe, including Karl Jacobi, and some of the material in Hermite's letters is remarkably sophisticated. In particular, Hermite generalized a result of Niels Abel that applied elliptic functions to the class of hyperelliptic functions as well.

Hermite's family life reflected his increasingly central position in the French mathematical establishment. His wife was the sister of the mathematician Joseph Bertrand, and one of his daughters married the eminent analyst Emile Picard (who was to edit Hermite's works after his father-in-law's death). From the time he became admissions examiner at the Ècole Polytechnique in 1848, Hermite devoted much of his effort to working with students at every level. In addition to Picard, his other distinguished students included Henri Poincare, Camille Jordan, and Paul Painleve. This record attests to his eagerness in welcoming students as colleagues. Emile Borel is said to have remarked that no one made people love mathematics so deeply as Hermite did.

On a professional level, Hermite accomplished some formidable tasks with the analytic apparatus which he had mastered. The solution of the general quadratic (second-degree) equation had been known since ancient times. Solutions to the cubic and quartic equations had been developed during the Italian Renaissance. When Galois showed that ordinary algebraic methods could not solve the general quintic (fifth-degree) equation, the subject appeared to have reached a dead end. Using once again the techniques of elliptic functions, Hermite showed that fifth-degree equations could be solved after all.

The single result for which Hermite is best known was the transcendence of e. A number is said to be algebraic if it is the solution of a polynomial equation with integer coefficients. For example, the square root of 2 is algebraic, since it is a solution of the equation x+b2-2=0. A real number that is not algebraic is called transcendental. The French mathematician Joseph Liouville had shown that there were transcendental numbers but no familiar examples were known. Hermite was able to show that e could not be written as the solution of a polynomial equation and therefore had to be transcendental. His technique was used shortly thereafter to show that was also transcendental, although Hermite does not seem to have recognized just how useful his technique was.

After his appointment as professor analysis at both the Ècole Polytechnique and the Sorbonne, Hermite took to writing textbooks that were widely used and appreciated. Although he resigned his chair at the Ècole Polytechnique after only seven years in 1876, he remained at the Sorbonne for another 21 years. Hermite attached a great value to insight and did not include much rigor in his teaching of elementary material. If his papers suffer from a fault, it is the occasional tendency to allow the details to get in the way of the overall picture. A large number of his ideas were developed by others and the complex generalization of quadratic forms named for him proved to be central in the formulation of quantum mechanics.

At the time of his 70th birthday, the adulation Hermite received from across Europe attests to his reputation as an elder mathematical statesman. His interests were never narrow, and he was awarded an impressive collection of decorations both at home and abroad. Hermite died on January 14, 1901, leaving a solid basis of mathematics and an unmatched collection of students to carry on his work.