Charles Hermite (1822-1901): Celebrated algebraist

CHARLES HERMITE was born on December 24, 1822 at Dieuze, France, in a family of five sons and two daughters. His father, though educated is an engineer, settled down in business as a cloth merchant. Due to the growing demands of the business, Hermite was sent, at the age of six, to the boarding school at Nancy. He studied for a short time at Lycee Henri IV and then shifted in 1840 to the famous school Louis-le-Grand to sit for the entrance examination for Ecole Polytechnique.

Instead of preparing for the competitive examination, he studied the memoir of Lagrange and `Disquisitones Arithmetical' of Gauss; what is more, he mastered the works a few before or since have mastered it. This prompted Professor Richard to call him `a young Lagrange'. It was by the Grace of God and the devoted efforts of this good Professor - like what he did earlier to save Galors (1811-32) for science, that Hermite was not `tossed out by stupid examiners to rot on the rubbish heap of failure.'

Hermite was admitted to the Ecole Polytechnique, passing out with a poor rank. This humiliating outcome made an impression on the young man which all the triumphs in his later career could never efface. He stayed only one year here, but the year was not wasted.

Instead of slaving over elementary mathematics and descriptive geometry, which he hated, Hermite devoted his time to Abelian functions, then the topic of interest to the great mathematicians of Europe. He also made the acquaintance of Joseph Louisville (1809-82), an outstanding mathematician of that time. The latter was optimistic that the great algorist Jacobi (1804-51) would show appreciation to Hermite; he was not mistaken.Having finally evaded his rapacious examiners, Hermite settled down to a peaceful and uneventual career, which led him to become a great mathematician. Hermite now cast longing eyes on the teaching profession where he could earn his livelihood, simultaneously pursuing research. But the inexorable rules and regulations came in the way.

Intervention of Sturm

At the age of 24, he abandoned his pioneering research in Abelian functions to devote his time for obtaining his first degree in science. Two harder ordeals would have normally followed but for the intervention of two of his eminent friends sturm and Bertrand both fine mathematicians, the latter got him appointed to a position where he could mock the examiners! To reach this position, he had sacrificed nearly five years of his youth (1843- 48), the most inventive period in a man's career.

After spells of service in the college de France, the Ecole Normale and other institutions, be became professor at the Sorbonne in 1870, by which time he reached the age of 47. He held this influential position for 27 years, when he trained a whole generation of distinguished mathematicians in France. Henri Poincare is one among this group. Hermite's influence extended, by his classic works, beyond the shore of France.

One marvels at the shifts of emphasis in Hermite's research work:

- Transformation of Abelian and elliptic functions (1843- 47).

- Arithmetic of the quadratic forms, (1847-51) Hermitian forms.

- Theory of invariants, fifth degree equation (1854-64).

- Approximation of functions (1873).

- Application of elliptic functions (1877-81).Hermite displayed his originality in the novel methods which he introduced into higher arithmetic. As an example, the spirit of one of them is briefly presented.

Fundamental discoveries

Arithmetic in the sense of Gauss deals with the properties of rational integers 1, 2, 3..., irrationals like the square root of 2 are excluded. Gauss investigated the integer solutions of indeterminate equations such as ax2+2bxy+cy2 = m where a, b, c and m are integers. This equation in x and y is to be solved entirely in the domain of rational integers, that is, in the realm of discrete number.

To fit analysis, which is adapted to the investigation of a continuous number, to such a discrete would seem impossible; yet this is what Hermite did.

His work for solving the general equation of the fifty degree, using modular functions, created a sensation in the mathematical world: it inaugurated a new branch of algebra.

The other sensational result (1873) is that which established the transcendence of the number denoted by the letter `e', the base of the natural system of logarithms. The mathematical world was astonished at the marvellous ingenuity of the proof that Hermite gave.

Hermite was born lame and he could move about only with a cane. This deformity never affected his sweet disposition and his vigour in research.

He attacked problems which would have baffled Gauss in 1800. He died, loved the world over, on January 14, 1901.

Outstanding unsolved problems demand new methods, while powerful new methods, in their turn, beget new problems. It is the man, observed his student Poincare, not the method that solves a problem.

All three theses are sustained in the life of Charles Hermite. (E.T. Bell: Men of Mathematics, Penguin Books, 1953).

R. Parthasarathy

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