1822 – 1901 • Transcendence, Polynomials & the Beauty of Form
The mathematician who proved e transcendental, solved the quintic via elliptic functions, and gave his name to matrices, polynomials, and operators across all of mathematics.
Born on December 24, 1822 in Dieuze, Lorraine, Charles Hermite was the sixth of seven children. He was born with a deformity in his right leg that required a cane throughout his life, exempting him from military service but also causing him social difficulties.
Hermite studied at the College de Nancy and then at the College Henri IV and Lycee Louis-le-Grand in Paris. Even as a student, he began corresponding with Jacobi about elliptic functions — at age 18, he sent Jacobi original results that impressed the master.
He entered the Ecole Polytechnique in 1842 but was nearly expelled due to his disability and his neglect of non-mathematical subjects. He passed the examinations only on the second attempt.
Hermite's letters to Jacobi (1843) contained a proof that the general quintic can be solved using elliptic modular functions — a result published only later. Jacobi was deeply impressed by the unknown student.
Hermite walked with a limp his entire life. Far from limiting him, this may have deepened his focus on intellectual pursuits. His physical disability contrasted with extraordinary mathematical agility.
Hermite served as repetiteur and examinateur at the Ecole Polytechnique. His early work on quadratic forms, abelian functions, and the transformation theory of elliptic functions established him as a leading analyst. He was elected to the Academie des Sciences in 1856.
Appointed to the Sorbonne in 1869, Hermite became the dominant figure in French mathematics for three decades. His lecture courses, meticulously prepared, influenced an entire generation including Poincare, Picard, Appell, and Borel.
Hermite's proof that e is transcendental was a landmark achievement. It showed that e satisfies no polynomial equation with rational coefficients, settling a question that had been open since Euler and establishing a new method that Lindemann extended to prove pi transcendental.
In his later years, Hermite was revered as the patriarch of French mathematics. His 70th birthday in 1892 was celebrated with tributes from mathematicians worldwide. He corresponded with virtually every leading mathematician of his era.
Hermite worked during France's recovery from the mathematical dominance of the early 19th century (Cauchy, Fourier, Laplace). The Franco-Prussian War of 1870 was a national trauma that also affected mathematics: Germany, especially Berlin and Gottingen, had surged ahead.
Hermite was determined to maintain France's mathematical tradition. He translated and promoted the work of German mathematicians (especially Riemann), while developing French expertise in analysis, number theory, and algebra.
The key mathematical questions of his era involved the nature of numbers (algebraic vs. transcendental), the structure of quadratic forms, and the deep connections between elliptic functions and number theory.
Liouville (1844) had shown that transcendental numbers exist but only through artificial constructions. Proving that specific important constants (e, pi) are transcendental required entirely new methods that Hermite pioneered.
Hermite was politically conservative, deeply Catholic, and temperamentally opposed to excessive abstraction. He once wrote: "I turn away with fear and horror from this lamentable plague of continuous functions which do not have derivatives." Yet his own work opened doors to abstraction.
In 1873, Hermite proved that e is transcendental: it is not a root of any polynomial with integer coefficients. This was the first proof that a "naturally occurring" mathematical constant is transcendental.
The proof works by assuming e satisfies a polynomial equation and deriving a contradiction. Hermite constructed clever auxiliary polynomials that, combined with the Taylor expansion of e^x, produce an integer that is simultaneously between 0 and 1 — an impossibility.
Hermite himself refused to tackle pi, saying the task was too difficult. But nine years later, Lindemann adapted Hermite's method to prove pi transcendental, resolving the ancient problem of squaring the circle.
Lindemann's extension of Hermite's method to pi proved that squaring the circle with compass and straightedge is impossible. This resolved a problem that had been open for over 2,000 years, since the ancient Greeks.
If a1,...,an are algebraic numbers linearly independent over Q, then e^{a1},...,e^{an} are algebraically independent. This vast generalization of Hermite's result is central to transcendental number theory.
Hermite's proof technique involved constructing rational approximations to e^x with precisely controlled error. This approach — Pade approximation — became a major tool in approximation theory and numerical analysis.
Despite Hermite's breakthrough, many basic transcendence questions remain open. We still do not know whether e + pi, e * pi, or e^e are transcendental. Schanuel's conjecture, if proven, would resolve all such questions.
The Hermite polynomials H_n(x) are a classical family of orthogonal polynomials defined by:
H_n(x) = (-1)^n e^{x^2} (d^n/dx^n) e^{-x^2}
They satisfy the orthogonality relation:
∫ H_m(x) H_n(x) e^{-x^2} dx = sqrt(pi) 2^n n! δ_{mn}
Hermite polynomials are the eigenfunctions of the quantum harmonic oscillator, making them fundamental in quantum mechanics. The wave functions of the oscillator are Hermite polynomials times a Gaussian.
A matrix A is Hermitian if A = A* (equal to its conjugate transpose). Hermite proved that all eigenvalues of such matrices are real. This property is the foundation of quantum mechanics, where observable quantities correspond to Hermitian operators.
Every Hermitian matrix can be diagonalized by a unitary matrix. This spectral theorem, rooted in Hermite's work, is the basis of principal component analysis (PCA), quantum measurement theory, and vibration analysis.
Hermite interpolation matches not just function values but also derivatives at given points. This gives smoother approximations than Lagrange interpolation and is used in computer graphics (Hermite splines), robotics, and CAD.
Every integer matrix can be put into Hermite normal form (upper triangular with specific properties). This is fundamental in computational number theory, lattice reduction, and integer programming.
Abel and Galois proved the general quintic cannot be solved by radicals. But Hermite showed it can be solved using elliptic modular functions — a higher class of functions beyond radicals.
Just as the cubic can be solved by trigonometric functions (Vieta's substitution), the quintic can be solved by Jacobi's elliptic functions. Hermite reduced the general quintic to Bring-Jerrard form x^5 + x + a = 0, then expressed the roots using values of elliptic modular functions.
This result was a triumph of the 19th-century program connecting algebra, analysis, and number theory through elliptic functions.
Hermite made fundamental contributions to the theory of quadratic forms, proving that every positive-definite quadratic form in n variables has a basis where all basis vectors have length at most a constant times the n-th root of the determinant. This "Hermite bound" is central to lattice theory.
Hermite contributed to the theory of algebraic number fields, working on generalizations of quadratic reciprocity that Eisenstein and others had initiated.
Hermite developed the theory of continued fractions for complex numbers and used them to study approximation of irrational numbers — work that anticipated the modern theory of Diophantine approximation.
"I turn away with fear and horror from this lamentable plague of continuous functions which do not have derivatives."
— Charles Hermite, letter to Stieltjes (1893)Hermite prized beauty and elegance above all in mathematics. He sought "the simple idea" behind a proof and would rework arguments repeatedly until they achieved what he considered the right form. His published proofs are models of clarity.
Unlike later abstractionists, Hermite worked with explicit formulas, specific functions, and concrete calculations. He believed the path to truth lay through computation and that excessive abstraction obscured rather than revealed.
Hermite's strength was connecting disparate areas: using elliptic functions to solve algebraic equations, applying number theory to analysis, and finding algebraic structure in analytic objects. These connections often revealed deep mathematical unity.
Hermite's Sorbonne lectures were legendary. He presented mathematics as a living, developing subject rather than a finished body of knowledge, often incorporating recent discoveries and open questions into his courses.
Hermite's enormous correspondence (over 2,000 letters to Stieltjes alone) reveals a mathematician deeply connected to his community. His students Poincare and Picard became the next generation's leaders.
Hermite's famous remark about functions without derivatives reveals his deep conservatism about the direction of mathematics. He shared Kronecker's unease about Cantor's set theory and Weierstrass' pathological examples, though he never engaged in personal attacks.
His relationship with Kronecker was complex: they shared constructivist sympathies and a love of explicit computation, but Hermite was more willing to accept results that went beyond finite methods. His proof of e's transcendence used infinite processes that a strict constructivist might reject.
Hermite's refusal to tackle pi's transcendence, despite having all the necessary tools, remains puzzling. Was it modesty, fear of failure, or genuine belief that the problem was beyond reach?
Hermite was a devout Catholic who saw mathematics as revealing divine truth. He believed that mathematical objects existed independently of human minds and that mathematicians discovered rather than invented. This Platonic view influenced his aesthetic preferences.
Despite the Franco-Prussian War, Hermite maintained warm relations with German mathematicians. He translated and promoted Riemann's work in France and corresponded extensively with Weierstrass and Kronecker, earning respect on both sides of the divide.
Hermite recognized Poincare's genius early. When asked about his most gifted student, he reportedly said: "The one I can teach the least." Poincare went on to transform mathematics far beyond Hermite's conservative vision.
Hermitian operators represent physical observables. Hermite polynomials give the energy eigenstates of the harmonic oscillator. The spectral theorem for Hermitian matrices is the mathematical core of quantum measurement.
Hermite created the methods that proved e and pi transcendental. Modern transcendence theory (Baker's theorem, the Gelfond-Schneider theorem) descends directly from his techniques.
Hermite's constant and Hermite normal form are fundamental in the geometry of numbers and lattice-based cryptography (NTRU, lattice-based post-quantum crypto).
Hermite interpolation, Hermite-Gauss quadrature, and Hermite functions are standard tools in numerical analysis, spectral methods, and scientific computing.
Hermite functions (Hermite polynomials times Gaussian) are eigenfunctions of the Fourier transform, making them natural for time-frequency analysis and wavelet theory.
Hermite splines, which interpolate both positions and tangent vectors, are used throughout computer animation, CAD, and font rendering for smooth curve generation.
Gaussian-type orbital basis sets used in quantum chemistry computations are based on Hermite functions. The integrals that arise in molecular calculations are evaluated using Hermite polynomial recurrence relations.
Hermite polynomial features appear in kernel methods and in the analysis of neural networks. The Hermite expansion of functions provides a natural basis for studying functions of Gaussian random variables.
Lattice-based cryptography (leading candidate for post-quantum security) relies on the hardness of finding short vectors in lattices. Hermite's constant bounds the shortest vector in terms of lattice volume.
Hermite-Gaussian modes describe the transverse intensity patterns of laser beams in optical cavities. These modes are fundamental to fiber optic communication and laser design.
Hermite spline interpolation is used for smooth trajectory planning in robotics and for keyframe interpolation in computer animation, ensuring both position and velocity continuity.
Hermite polynomial expansions (Gram-Charlier and Edgeworth expansions) adjust the Gaussian distribution to model skewness and kurtosis in financial returns, improving option pricing models.
Hermite / ed. Picard (1905–1917) — Four volumes of collected works. The primary source for all of Hermite's mathematics.
Burger & Tubbs (2004) — An accessible introduction to transcendental number theory, starting from Hermite's proof and building toward modern results.
Baillaud & Bourget, eds. (1905) — The extraordinary correspondence between Hermite and Stieltjes, revealing the day-to-day development of 19th-century analysis.
Andrews, Askey & Roy (1999) — The definitive modern reference on special functions, with thorough coverage of Hermite polynomials and their properties.
"There exists, if I am not mistaken, an entire world which is the totality of mathematical truths, to which we have access only with our minds, just as a world of physical reality exists, the one like the other independent of ourselves, both of divine creation."
— Charles Hermitee = 2.71828... is transcendental — Hermite, 1873