Mathematics 1821

Cours d'analyse de l'École Royale Polytechnique

Augustin-Louis Cauchy

Calculus, rebuilt on a single honest idea: the limit.

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In depth · the introduction

For 150 years, calculus worked brilliantly while standing on an idea no one could quite explain — 'infinitely small' numbers. Cauchy replaced the magic with a single honest definition: the limit.

The big idea

Calculus is about quantities that are forever shrinking or forever getting closer — a curve's slope at a point, the area under it, the sum of endlessly many tiny pieces. The mathematicians who invented it, Newton and Leibniz, reasoned with 'infinitesimals': numbers smaller than any real number but not zero. The answers were right, but no one could say what these numbers actually were.

Cauchy's fix was to stop talking about a fixed tiny number and start talking about a process of approach. A variable 'has a limit' if it gets and stays as close to some value as you could ever demand — name any tolerance, however small, and the variable eventually falls within it. From this one idea he rebuilt continuity (no sudden jumps), convergence (an infinite sum that settles down), and the whole calculus — now with reasons you could check, not just results you had to trust.

How it came about

Augustin-Louis Cauchy was a young, fiercely religious, politically conservative engineer turned professor at the École Polytechnique in Paris — the elite school that trained France's engineers and officers. He was asked to teach calculus to first-year students, and he found the subject a mess: powerful, but riddled with vague appeals to the infinitely small and with manipulations that sometimes gave nonsense.

So in 1821 he wrote his lectures into a book, the Cours d'analyse, and rebuilt the foundations from scratch. Students reportedly grumbled that it was too rigorous and too slow. But Cauchy had decided that a proof you could not check was no proof at all — and that conviction reshaped what 'doing mathematics' meant.

Why it mattered

Before Cauchy, calculus was a magnificent machine no one could fully justify. After him, it had foundations: you could prove its theorems instead of merely believing them. That standard — define your terms precisely, then prove what follows — spread to all of mathematics. The 'epsilon-delta' definitions every science and engineering student meets are his, polished. Even his mistakes were productive: a flawed theorem of his pushed others to discover the crucial difference between two kinds of convergence.

A way to picture it

Think of a target with rings drawn ever tighter around a bullseye. Someone challenges you: 'Can you land every shot inside this ring?' You answer: 'Stand me this close to the line, and yes.' They draw a tighter ring; you step closer; you can still do it. A quantity 'has a limit' exactly when, for every ring they can draw — no matter how tight — you can find a distance to stand that keeps every shot inside. The limit is the bullseye you never have to actually hit; you only have to be able to get arbitrarily close on demand.

Where it sits

Cauchy stands at the hinge of the calculus story. Behind him are Newton and Leibniz, who built the machine in the 1600s but left its gears mysterious. Ahead of him are Weierstrass, who turned his words into airtight symbols, and Dedekind and Cantor, who built the real number line his limits secretly relied on. The thread of rigour runs onward to Gödel's question of what proof itself can reach — another document in this Library. Today every limit you compute, every 'as x approaches' you write, is Cauchy's.

The original document

Original source text

Préliminaires — variable & limit

Augustin-Louis Cauchy · Cours d'analyse · 1821 · Préliminaires

On nomme quantité variable celle que l'on considère comme devant recevoir successivement plusieurs valeurs différentes les unes des autres.

(A variable quantity is one that we consider as receiving, in succession, several values different from one another.)

Lorsque les valeurs successivement attribuées à une même variable s'approchent indéfiniment d'une valeur fixe, de manière à finir par en différer aussi peu que l'on voudra, cette dernière est appelée la limite de toutes les autres.

(When the values successively attributed to a variable approach a fixed value indefinitely, so as to end by differing from it by as little as one wishes, this last value is called the limit of all the others.)

The infinitely small

Cours d'analyse · 1821 · Préliminaires

Lorsque les valeurs numériques successives d'une même variable décroissent indéfiniment, de manière à s'abaisser au-dessous de tout nombre donné, cette variable devient ce qu'on nomme un infiniment petit ou une quantité infiniment petite. Une variable de cette espèce a zéro pour limite.

(When the successive numerical values of a variable decrease indefinitely, so as to fall below every given number, that variable becomes what one calls an infinitely small quantity, or an infinitesimal. A variable of this kind has zero for its limit.)

Continuity (Chapter II)

Cours d'analyse · 1821 · Ch. II, § 2 · Des fonctions continues

la fonction f(x) sera, entre les deux limites assignées à la variable x, fonction continue de cette variable, si, pour chaque valeur de x intermédiaire entre ces limites, la valeur numérique de la différence f(x + α) − f(x) décroît indéfiniment avec celle de α.

… un accroissement infiniment petit de la variable produit toujours un accroissement infiniment petit de la fonction elle-même.

(… an infinitely small increase in the variable always produces an infinitely small increase in the function itself. — Cauchy's own restatement of continuity.)

Augustin-Louis Cauchy · Professeur d'Analyse à l'École Polytechnique · Paris, 1821