Leonhard Euler

Chapter 1 · Saturday Afternoons with Bernoulli

A pastor's son, argued out of the pulpit

1707 – 1727 · Basel · Riehen

Leonhard Euler was born at Basel in 1707. His father Paul was a pastor who had studied under Jacob Bernoulli and had a genuine interest in mathematics, but who intended his son for the church.

Euler entered the University of Basel at thirteen, which was less remarkable then than it sounds, and took a master's degree at sixteen comparing Descartes's philosophy with Newton's. He then enrolled in theology, as planned.

What changed things was Johann Bernoulli. Jacob's brother, by then the leading mathematician in Europe, agreed to see Euler on Saturday afternoons — not to teach him, exactly, but to answer questions about whatever he had failed to understand during the week. It was an arrangement that required Euler to read on his own and come with difficulties, which turned out to suit him exactly.

Bernoulli went to Paul Euler and told him the boy would be a great mathematician and not a pastor. That conversation is the hinge of the whole life, and it is worth noticing that it took a famous man's intervention to overturn a father's plan; talent alone would not have done it.

In 1726 Euler competed for a prize from the Paris Academy on the best arrangement of masts on a ship — he had never seen a seagoing vessel — and came second. The following year, at twenty, he left for the new Academy at St Petersburg, where Johann Bernoulli's sons Daniel and Nicolaus had gone before him. He arrived on the day Catherine I died, into a city where the Academy's future was suddenly uncertain, and briefly considered joining the navy.

Why this matters

The most productive mathematician in history nearly became a pastor, and was saved from it by somebody else's word to his father.

Saturday afternoons with the best mathematician in Europe. What would you ask him?

Ask Euler

• “How did Johann Bernoulli teach you?”
• “What did your father say when he was told you would not be a pastor?”
• “You won a prize about ships' masts having never seen a ship. How?”
• “What was St Petersburg like when you arrived at twenty?”
• “Did you ever regret leaving theology?”

Chapter 2 · The Symbols

f(x), and most of the rest of it

1734 – 1748 · St Petersburg · Berlin

Open a modern mathematics textbook and a surprising amount of what is on the page comes from one man.

In 1734 Euler began writing f followed by a quantity in brackets to mean a function of that quantity. Before that, functions were described in words or written out as expressions every time; there was no way to refer to one without saying what it was. Naming a function made it an object you could talk about, compose with another, or invert — and the notation is the reason a modern student can write fg(x) at all.

He popularised the Greek letter for the ratio of a circle to its diameter, which had been used occasionally before him and became universal because he used it in the Introductio. The letter for the base of natural growth is his. The symbol for the square root of minus one is his. The summation sign is his. The notation for a finite difference is his. The convention of writing the sides of a triangle as lowercase letters against the uppercase letters of their opposite angles — which is how the sine rule is written in every school textbook — is his.

His attitude to this was deliberate, not accidental. He thought about notation, wrote in a style designed to be read by people who were not already experts, and produced textbooks so much better than the alternatives that Europe simply adopted them. The Introductio in analysin infinitorum (1748), the Institutiones calculi differentialis (1755) and the integral calculus volumes taught the subject for a century.

He also wrote, over three years, more than two hundred letters explaining light, sound, gravity, magnetism and logic to a German princess, in language a non-mathematician could follow. It became one of the best-selling science books of the eighteenth century.

Why this matters

Notation is not decoration: naming a function is what makes it possible to compose one with another, and that idea has an inventor and a date.

Half the symbols on the page, from one man. What would you ask him?

Ask Euler

• “Why write f with a quantity in brackets?”
• “How do you decide whether a symbol is a good one?”
• “Why did you spend time writing for a princess?”
• “What did mathematicians do before functions had names?”
• “Which of your symbols are you most pleased survived?”

Chapter 3 · What Is a Function?

A definition he had to abandon

1748 – 1760s · Berlin

The Introductio of 1748 did something more important than introduce notation: it changed what analysis was about. Before it, the objects of study were CURVES, treated geometrically. After it, they were FUNCTIONS, treated as objects in their own right, with the curve merely a picture of one.

Euler's definition in the Introductio is precise and, as it turned out, too narrow: a function of a variable quantity is an analytic expression composed in any way from that variable and from numbers. In other words, a function is a FORMULA.

The definition broke on a physical problem. D'Alembert had written down an equation for a vibrating string and solved it. The question was what initial shapes the string could be given. A violin string is plucked: it is pulled aside at one point and released, so its initial shape is two straight segments meeting at a corner. That shape is not given by a single formula. Is it a function?

D'Alembert said no — the mathematics required a formula, and shapes drawn freehand were outside it. Euler said yes: the physical problem plainly has a solution for a plucked string, so the mathematics must be widened to admit shapes drawn by hand, in pieces if necessary. Daniel Bernoulli offered a third answer, that any such shape could be built from an infinite sum of simple waves, which both the others rejected and which turned out to be the beginning of something enormous.

Euler widened his definition. That matters for a modern student because the definition eventually reached — a function is a rule pairing each input with one output, however given — is the one the machine picture in a textbook describes, and it exists because a real physical case broke the tidier version.

Why this matters

The definition of a function that school teaches is the survivor of an argument about a plucked violin string, and its inventor lost the first round.

A definition broken by a violin string. What would you ask him?

Ask Euler

• “What was your first definition of a function?”
• “Why did the plucked string cause a problem?”
• “Was d'Alembert wrong, or was he being consistent?”
• “How hard is it to widen a definition you have published?”
• “What did Daniel Bernoulli propose, and why did you reject it?”

Chapter 4 · The Bridges of Königsberg

A puzzle solved by throwing away the map

1735 – 1736 · St Petersburg · Königsberg

The town of Königsberg sat on both banks of a river with two islands in it, connected by seven bridges. The question people asked was whether a walk could be devised crossing every bridge exactly once.

Euler's paper on it, presented in 1735, opens by saying the problem seems to belong to geometry, but to a kind of geometry in which distances do not matter and no calculation helps — a geometry of position, which he says had barely been mentioned by anybody.

His solution is to throw away almost everything. The size of the islands is irrelevant. The length of the bridges is irrelevant. Where anything is relative to anything else is irrelevant. All that matters is which land masses are joined to which, and how many bridges arrive at each.

Once it is reduced that far the answer is immediate. Every time a walker enters a land mass by one bridge they must leave by another, so except at the start and the finish the bridges at each land mass must pair up — the number arriving must be even. At most two land masses can have an odd number, because a walk has two ends. In Königsberg all four have an odd number. No such walk exists, and the paper proves it rather than merely reporting that nobody has found one.

The move — discard everything except what is connected to what — is now the foundation of graph theory, and the same instinct runs through topology, network analysis, circuit design and the mathematics of the internet. Euler was modest about it and treated it as a curiosity.

Why this matters

Solving a problem sometimes means deciding what to ignore, and this is the clearest example of that decision in the history of mathematics.

Seven bridges, and an answer found by discarding the map. What would you ask him?

Ask Euler

• “How did you solve the bridges problem?”
• “Why does it not matter how long the bridges are?”
• “How do you prove that something is impossible?”
• “Did you think this was serious mathematics?”
• “What other problems would this way of thinking settle?”

Chapter 5 · Seventeen Years Blind

And half the output

1738 – 1783 · St Petersburg · Berlin

Euler lost the sight of his right eye in 1738, in his early thirties, after a severe illness. He is said to have remarked that he would have fewer distractions.

In 1771 a cataract took the left. A surgeon restored some vision briefly; infection destroyed it. From then until his death twelve years later he was effectively blind.

His productivity went up. Roughly half of his enormous output — nine hundred works in total — comes from after he lost his sight. He dictated to assistants, to his eldest son Johann Albrecht and to a young colleague, working out entire papers in his head and then giving them out. He is described as writing formulae in chalk on a slate for others to copy, and as being able to hold a computation of many terms without notes; he knew the Aeneid by heart and could give the first and last lines of any page of the edition he had learned from.

His personal life in these years was hard: his first wife died in 1773 after forty years, and thirteen children had been born to them of whom only five survived infancy. He worked, by all accounts, in a room full of children and grandchildren, apparently without difficulty.

On 18 September 1783 he spent the morning discussing the newly discovered planet Uranus and the mathematics of balloons, had dinner, and died in the afternoon of a brain haemorrhage while playing with a grandchild. Condorcet's eulogy records that he ceased to calculate and to live.

The St Petersburg Academy went on publishing his backlog of papers for another forty-eight years.

Why this matters

The idea that mathematics is done by looking at things is refuted by the man who produced half his work without being able to see.

Twelve years blind and more productive than ever. What would you ask him?

Ask Euler

• “How did you work after you lost your sight?”
• “How much can you actually hold in your head?”
• “Did losing your sight change the kind of mathematics you did?”
• “How did you work with children in the room?”
• “Is there anything you left unfinished?”

What Euler changed

Euler gave mathematics more of its notation than anybody else — f(x) from 1734, the summation sign, the symbol for the square root of minus one, the lettering of a triangle's sides against its angles — and naming a function is what made composing and inverting them expressible at all. The Introductio of 1748 made the function rather than the curve the central object of analysis, which is the viewpoint the whole modern subject rests on. His solution of the seven bridges founded graph theory. Roughly half of his nine hundred works were produced after he went blind.

A debate that continues

His 1748 definition of a function as an analytic expression was too narrow and he had to widen it after the dispute over the vibrating string — a revision sometimes glossed over. By nineteenth-century standards he was often cavalier with infinite series, reaching correct results by methods that do not bear inspection. He was careless about priority in both directions, and lost credit for several results he had first.

Keep exploring — ask Euler

• “How much of what a subject can think is decided by the symbols it uses?”
• “What should happen to a definition when a real case breaks it?”
• “Does mathematics require sight, or only memory and attention?”

Related lives

• Jacob Bernoulli — Against My Father's Will
• Pierre de Fermat — The Margin Was Too Narrow
• Alan Turing — Father of Computer Science

Related themes

Functions and notation · Graph theory · The eighteenth-century calculus