Cauchy, Abel, and Jacobi — Rigorization of 19th-Century Analysis

The 19th Century — Rigorizing Analysis

The 18th–19th-century giants of the previous article developed calculus explosively. But its foundations were vague:

• What is “infinitesimal”? — Euler manipulated “infinitely small quantities” intuitively
• What is “continuity”? — only the intuition of “an unbroken curve”
• What is “convergence”? — the vague notion of “approaching”

In the 19th century, a movement arose to redefine these rigorously. Its central figure was Augustin-Louis Cauchy (1789–1857). His contemporaries Niels Abel (1802–1829) and Carl Jacobi (1804–1851) opened up a new world in elliptic functions. All three were short-lived geniuses.

Augustin-Louis Cauchy — Rebuilder of Analysis

Augustin-Louis Cauchy (1789–1857) was a French mathematician, born in Paris five weeks after the outbreak of the French Revolution. He rebuilt analysis as a rigorous discipline, establishing the modern definitions of limit, continuity, convergence, differentiation, and integration.

His lifetime output was about 800 papers, second only to Euler. He also made vast contributions to complex analysis, group theory, permutations, elasticity, and optics.

Life

• 21 August 1789 — Born in Paris; his father Louis-François was a lawyer
• Childhood spent in the countryside to escape revolutionary turmoil; educated by his father, met Lagrange and Laplace
• 1805 — Enters École Polytechnique
• 1810–13 — Engineer at Cherbourg harbor works
• 1816 — Professor of analysis at École Polytechnique, member of the Paris Academy (age 27)
• 1821 — Publishes Cours d’Analyse — a monument of the rigorization of analysis
• 1830 — After the July Revolution deposes the Bourbons, Cauchy refuses to swear allegiance to Louis-Philippe and goes into exile in Turin
• 1848 — After the February Revolution abolishes the oath, returns as professor at the Sorbonne
• 23 May 1857 — Dies near Paris, aged 67

Stubborn Royalist and Catholic

Cauchy was a fierce royalist and Catholic, willing to lose his position for his convictions. Leaving France for exile in 1830 out of political principle is an unusual choice for a scholar.

The Prototype ε-δ — Rigorizing Analysis

Post-Euler analysis had handled “infinitesimal” and “infinite” loosely. Cauchy centered the limit and redefined everything around it.

Definition of Limit (Cours d’Analyse, 1821)

“When the successive values attributed to a variable approach indefinitely a fixed value, so that finally they differ from it by as little as one wishes, this last value is called the limit of the variable.”

This is the prototype of the modern ε-δ method (the fully polished ε-δ came later, with Weierstrass in the late 19th century).

Continuous Functions

“If a sufficiently small increment Δx in x makes f(x + Δx) − f(x) also arbitrarily small, then f is continuous at x.”

Convergent Series

“If the partial sums S_n have a limit as n → ∞, the series converges.”

From this comes the Cauchy convergence test.

Cauchy Sequences and Completeness

“A sequence {a_n} is a Cauchy sequence if for every ε > 0 there is N such that m, n > N implies |a_m − a_n| < ε.”

The real numbers are complete — every Cauchy sequence converges. This is a basic concept of modern analysis and functional analysis. The revolutionary idea: decide convergence without appealing to a limit’s existence.

Founding Complex Analysis

Cauchy systematically opened up complex function theory:

Cauchy’s Integral Theorem (1825)

“The line integral of a holomorphic f(z) along a closed curve vanishes: ∮ f(z) dz = 0.”

Cauchy’s Integral Formula

“For f holomorphic in a region, f(a) = (1/2πi) ∮ f(z)/(z − a) dz.”

The central theorems of complex analysis. Riemann and Weierstrass would extend the theory further.

The Residue Theorem

Compute a complex integral as a sum of residues at singularities — with enormous power for physics and engineering. The starting point of the modern technique of computing real integrals by detouring through the complex plane.

Weekly Papers and the Four-Page Rule

Cauchy submitted a paper every week to the Comptes Rendus, until the proceedings were joked to be “filled with Cauchy.” In 1835 the Academy imposed a rule: papers in the proceedings must be no more than four pages long — an unprecedented event: the Academy’s rules were changed for one mathematician. A stark contrast to Gauss’s “Pauca, sed matura”. “Write much and fast” vs. “few, but ripe” — two 19th-century attitudes.

The Loss of Galois’s Manuscript

In 1829 and 1831 the young Évariste Galois submitted papers on his group-theoretic ideas to the Paris Academy. The first referee was Cauchy:

• May 1829 — Cauchy accepted the manuscript, but no report was ever filed, and the manuscript itself was lost
• January 1830 — Cauchy said he would submit “Galois’s new paper” at the next meeting, then missed February’s session, and then went into exile in the July Revolution

Whether Cauchy understood Galois’s genius, was simply too busy, or was politically prejudiced is still debated. Either way, the serious appreciation of Galois’s theory had to wait until Liouville’s 1846 publication after his death. One of the greatest tragedies in mathematical history — and Cauchy’s shadow is on it.

Niels Abel — Nordic Genius Dead at 26

Niels Henrik Abel (1802–1829) was a Norwegian mathematician who died of tuberculosis at 26. In a short life he proved the algebraic unsolvability of the general quintic and laid the foundations of elliptic function theory.

The Quintic (1824)

Cubics had been solved by Cardano and quartics by Ferrari in the 16th century. But the general quintic had remained open for nearly 300 years.

In 1824, at 22, Abel proved that no general formula in radicals exists for polynomials of degree five or higher. He self-published the result, sent copies to Cauchy, Legendre, and Poisson in Paris — and was ignored. “A young man’s work,” and coolly dismissed.

Elliptic Functions

Abel took the revolutionary step of considering the inverse function of an elliptic integral — discovering elliptic functions. He sent the 1826 paper to Paris; the Cauchy Academy again lost the manuscript (!). It was rediscovered and published by Liouville in 1841.

A Short Life

• 5 August 1802 — Born on the island of Finnøy, Norway
• 1815 — Oslo cathedral school; his teacher Holmboe recognizes his talent
• 1824 — At 22, self-publishes the impossibility of the quintic
• 1825–27 — Travels to Berlin and Paris. Crelle, founder of the German Journal für die reine und angewandte Mathematik, recognizes his genius
• 1827 — Returns home; unemployed, in poverty
• 1828 — Publishes major papers on elliptic function theory
• 6 April 1829 — Dies of tuberculosis at 26
• Two days later — a letter from Berlin arrives offering a professorship

“Went unrecognized until days before dying at 26” — one of the great tragedies of mathematics. After Abel’s death his work gained worldwide recognition; today Abelian groups, Abelian varieties, and the Abel Prize (Norway, from 2003, one of the top awards in mathematics) all bear his name.

Carl Jacobi — Abel’s Rival

Carl Gustav Jacob Jacobi (1804–1851) was a German mathematician who developed elliptic functions independently at almost the same time as Abel. Died of smallpox at 46.

After studies in Berlin, he took a post at Königsberg at only 21. His brother Moritz Hermann von Jacobi was a physicist and inventor of electroplating.

Elliptic Functions — Simultaneous Independent Discovery

Abel and Jacobi both proposed reading elliptic integrals as functions of the upper limit and taking the inverse. u = ∫₀ˣ dt/√(…) becomes x = sn(u).

The resulting functions sn, cn, dn turn out to have double periodicity. Doubly-periodic functions on the complex plane were one of the great revolutions of 19th-century analysis, and led into complex analysis, algebraic geometry, and the theory of modular forms.

Jacobi published Fundamenta Nova Theoriae Functionum Ellipticarum in 1829, giving a systematic construction via theta functions. Jacobi noted “Mr. Abel was ahead of me” — a rare, mutual recognition among rivals.

The Jacobian

The determinant of the matrix of partial derivatives of a multivariable map is called the Jacobian. It gives the local rate of volume distortion — indispensable for change of variables in probability, canonical transformations in physics, and everywhere in modern probability and statistics.

Hamilton–Jacobi

Jacobi’s contribution to Hamilton’s canonical formulation of mechanics led to the Hamilton–Jacobi equation, a first-order PDE. It is the most refined form of classical mechanics and the bridge to quantum mechanics (the semiclassical limit of the Schrödinger equation).

A Famous Line

In reply to Fourier:

“The sole aim of science is to honor the human mind.”

A “declaration of independence for pure mathematics” against “science for use” — a slogan of 19th-century German mathematics.

The Königsberg School — A German Revolution in Education

At Königsberg, Jacobi formed the Königsberg school with Bessel (astronomer) and Neumann (physicist) and pioneered research seminars. This became the prototype of the German graduate education model — the shift from “reading a textbook” to “researching together.” A key origin of German mathematical leadership from the 19th into the 20th century.

Weierstrass — Completing the Rigorization

Karl Weierstrass (1815–1897) in Germany completed the ε-δ method into its polished form. A late bloomer who spent his first forties as a provincial secondary-school teacher.

Rigorous ε-δ

“For every ε > 0, there exists δ > 0 such that |x − a| < δ implies |f(x) − f(a)| < ε.” The definitive modern definition.

Pathological Functions

The Weierstrass function (1872): continuous everywhere but nowhere differentiable. It shattered the intuition “continuous ⇒ differentiable.” A precursor of modern fractal theory.

Together with Cauchy’s, Weierstrass’s rigorization completed the modern framework of analysis.

From This Article to the Next

We have traced the rigorization of analysis and the discovery of elliptic functions in the first half of the 19th century. The next article turns to even more dramatic geniuses of the same period — Évariste Galois (1811–1832, dead in a duel at 20) and Srinivasa Ramanujan (1887–1920, dead of tuberculosis at 32). Galois, the opener of group theory, and Ramanujan, the self-taught Indian — the ultimate form of “the tragic genius.”