Augustin Louis Cauchy (1789 - 1857)

Cauchy was born in Paris to a devout Catholic family during the Revolution, a time when Catholicism was not popular in France. Fearing for the safety of his young family, Cauchy’s father moved them to Arcueil. Things were not easy. The Revolution meant that food was scarce, and the whole family lived on small allowances of bread and rice.

Returning to Paris when things were safer, Cauchy was introduced to the world of mathematics. His father was friends with eminent mathematicians including Lagrange and Laplace, who would visit the Cauchy house regularly.

Lagrange took an interest in the talented young man, encouraging him to continue his study of mathematics. However, Cauchy was to specialise in engineering.

After school he got a job as an engineer in Napoleon’s army at Cherbourg. He worked all hours - apparently getting up at 4’o’clock in the morning each day - and still managed to find time for his mathematical research.

All this hard work (early rising, and effectively two careers) led to illness, and Cauchy returned to Paris, hoping to pursue his dream of an academic career.

Turned down for a few university posts, Cauchy committed himself entirely to his research, most importantly on the work that was to become the foundation for his theory of complex functions.

Work in mathematics hitherto, had largely concentrated on real numbers - the numbers that we take for granted. But there are other numbers, involving the square root of -1, that were increasingly being studied.

Functions, the workhorses of mathematics, increasingly involved these new numbers, which are called complex numbers. But little was known about their properties or how they would affect functions and their behaviour.

Cauchy provided many of the answers and the results were astonishing. Things that had been inexplicable before were suddenly understood; powerful new techniques were discovered and exploited, many of which had applications in the ordinary real numbers, providing simple answers to seemingly intractable problems.

Cauchy eventually became a Professor at the École Polytechnique, but it was not until he published his solution of one of Fermat’s theorems (which both Gauss and Euler had unsuccessfully struggled with!) that he became widely accepted in the mathematical community.

He was elected to the French Academy of Science aged only 27, a remarkable achievement for someone so young.

However, his acceptance was controversial, as his predecessor, Gaspard Monge (a supporter of the Emperor), had been removed from the post when the Bourbons returned to the throne after Napoleon was overthrown. Cauchy thought nothing of taking on the position, although it made him many enemies.

After Charles X was deposed in 1830, all members of the Academy were required to swear an oath of allegiance to the new regime. Cauchy refused on the grounds that he had already taken an oath to Charles, and was forced to give up his teaching positions as a result.

He took up a teaching post in Turin, Italy, on theoretical physics, where he stayed for two years. He followed Charles X into exile, tutoring the king’s grandson and heir. Charles X gave Cauchy the impressive but meaningless title of Baron in recognition of his allegiance and support. Eventually Cauchy returned to France, but he was still not allowed to teach, not having sworn the oath of allegiance.

Despite numerous opportunities, his support for the Jesuits and his political opinions prevented appointments that he was more than capable of. Fortunately, the fall of the regime in 1852 provided Cauchy with the opportunity to regain his old positions.

He became increasingly bitter in his old age, and fell out with many of his contemporaries. In 1850 Cauchy lost out to Liouville in the elections for the chair of the Academy.

Relations between the two took a steep nosedive. Cauchy was not an easy man to work with.

His contemporaries found him extraordinarily difficult. He insulted Poncelet and Abel had problems understanding him (though he thought he was brilliant)!

However, his beliefs urged him to charity work, and he was actively involved in good works - during the 1846 famine in Ireland, he appealed to the Pope for help on the Irish peoples’ behalf.

In all Cauchy produced almost 800 papers - an astonishing achievement exceeded only by Euler and Cayley. Despite his notoriety, his name and work live on in a number of mathematical ideas: Cauchy integral theorem, Cauchy-Kovalevskaya existence theorem, Cauchy-Riemann equations and Cauchy sequences as well as in his groundwork for rigor in analysis and mathematics as a whole.

Ironically, Cauchy died in 1857 after catching a fever on a trip to the country to restore his health.

Cauchy’s mathematics

Calculus

He started work on polyhedra and polygons and in 1811 published a paper proving that the angles of a polyhedron are determined by its faces.

At 26, he proved a conjecture of Fermat which had stumped Euler and Gauss: “Every positive integer is the sum of 3 triangular numbers, 4 square numbers, 5 pentagonal numbers, … “. For example, the number 12 is the sum of the triangular numbers:

Modern mathematics is indebted to Cauchy for two of its major interests, each of which marks a sharp break with the mathematics of the eighteenth century.

The first was the introduction of rigor into mathematical analysis. This deals with the underpinning of calculus and how things converge - that is, dealing with sequences, their ultimate values and the sums associated with them.

For example, consider the two sequences:

\left\{1, {1\over4}, {1\over9}, {1\over16}, \cdots \right\} consisting of the reciprocals of the squares

and \left\{1, {1\over2}, {1\over3}, {1\over5}, {1\over7}, {1\over11}, \cdots \right\} consisting of the reciprocals of the prime numbers.

Each of them converges to 0 and this idea was made precise by Cauchy. But their sums behave in very different ways:

{1+{1\over4}+{1\over9}+{1\over16}+\cdots}={{\lambda^2}\over6}

{1+{1\over2}+{1\over3}+{1\over5}+{1\over7}+{1\over11}+\cdots} does not converge - it just gets bigger and bigger.

In this sense, the primes are more numerous than the squares, and they are dense in the number universe.

Combinatrics

The second thing of fundamental importance that Cauchy brought to modern mathematics was of a very different kind.

This concerned the art of counting and finding the subtle relationships between connected objects.

Today we call it combinatorics. An example of this is a formula concerned with polyhedra, relating the number of its faces, edges and vertices.

If the polyhedra has f faces, v vertices and e edges then,

v+f-e=2