Leonhard Euler

Who Gave Us The Notation · Mathematician · 1707–1783

The most productive mathematician in history: f(x) is his notation, the function became the central object of analysis in his hands, and he did some of his greatest work after going completely blind.

Who was Euler?

Leonhard Euler was born at Basel in 1707. His father was a pastor and intended the same for him; Johann Bernoulli, who tutored him on Saturday afternoons, told the father that the boy would be a great mathematician, and won the argument. In 1727 he went to the new Academy at St Petersburg, where Daniel and Nicolaus Bernoulli had gone before him. He spent fourteen years there, twenty-five at Frederick the Great's Berlin Academy, and the last seventeen back in Russia under Catherine. Frederick found him dull company and called him a cyclops; Euler outlasted him and left. His output is not comparable to anybody else's. He wrote roughly nine hundred works, filling some eighty volumes in the modern collected edition, on analysis, number theory, mechanics, optics, fluid flow, the motion of the moon, shipbuilding, artillery and music. He wrote the textbooks from which Europe learned the calculus, and a series of letters explaining natural philosophy to a German princess that became a bestseller. Much of what a student writes today is his. Writing f followed by a quantity in brackets to denote a function begins with him in 1734. The symbol for the ratio of a circle to its diameter he popularised; the base of natural growth, the square root of minus one and the sign for a sum are all his notation. The Introductio in analysin infinitorum of 1748 made the FUNCTION, rather than the curve, the central object of analysis — a change of viewpoint that shaped everything after it. His definition there was that a function is an analytic expression composed of a variable quantity and constants. It did not survive: the argument with d'Alembert and Daniel Bernoulli about the vibrating string forced the question of whether a curve drawn freehand is a function, and Euler had to widen the definition, which he did with reasonably good grace. He lost the sight of his right eye in his thirties and the left to a cataract in 1771. His productivity increased. He dictated to assistants and to his sons, and could hold enormous computations in his head; roughly half his output dates from after he went blind. He died in 1783, in the middle of a calculation about the newly discovered planet Uranus.

Major achievements

• Introduced the notation f(x) for a function in 1734, and much of the rest of modern notation
• Made the function the central object of analysis in the Introductio of 1748
• Solved the seven bridges of Königsberg by ignoring distance entirely
• Wrote the textbooks from which Europe learned the calculus
• Produced roughly nine hundred works, about half of them after going blind

Life in brief

• 1707 — Born at Basel: Tutored on Saturdays by Johann Bernoulli.
• 1727 — To St Petersburg: Joining the new Academy, where the Bernoullis had gone before him.
• 1734 — f(x): The notation for a function appears, and stays.
• 1741 — To Berlin: Twenty-five years at Frederick the Great's Academy.
• 1748 — Introductio in analysin infinitorum: The function becomes the central object of analysis.
• 1766 — Back to Russia: At Catherine's invitation; blind within five years.
• 1783 — Died at St Petersburg: In the middle of a calculation about Uranus.

Explore the life of Euler — five chapters

1. Saturday Afternoons with Bernoulli — A pastor's son, argued out of the pulpit
2. The Symbols — f(x), and most of the rest of it
3. What Is a Function? — A definition he had to abandon
4. The Bridges of Königsberg — A puzzle solved by throwing away the map
5. Seventeen Years Blind — And half the output

Read the five-chapter life of Leonhard Euler →

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