Who Was Leonhard Euler? Man Behind the Most Beautiful Equation in Mathematics

On the evening of 18 September 1783, a 76-year-old man in St. Petersburg sat down with his grandson, poured a cup of tea, and worked on the mathematics of the newly discovered planet Uranus. He had spent the afternoon discussing science with colleagues. By any account, he had lived a full day. Then he suffered a brain haemorrhage, said the words “I am dying,” and was gone within hours.

His name was Leonhard Euler. He had been almost completely blind for the last seventeen years of his life — the right eye lost to a near-fatal fever in his late twenties, the left to a cataract at fifty-nine. And in those seventeen years of total blindness, working by dictation and by the extraordinary precision of his memory, he produced more mathematics than most sighted mathematicians manage in a lifetime.

Euler is the most prolific mathematician in history. His collected works fill 886 papers and books — so many that the St. Petersburg Academy, which published him, kept releasing new Euler papers for more than thirty years after his death. He gave us the symbols e, i, f(x), and Σ. He founded graph theory with a single recreational puzzle about bridges. He wrote the equation that mathematicians have called, across three centuries, the most beautiful in all of mathematics.

And yet most people who know the names Einstein or Newton have never heard of him. This is one of history’s stranger oversights, because Euler’s mathematics is not a historical curiosity — it is the foundation on which GPS, electrical engineering, quantum mechanics, network science, and modern cryptography are built. You encounter Euler’s mathematics every day of your life, and you almost certainly have no idea.

A Basel Childhood and the Education That Shaped a Century

Leonhard Euler was born on 15 April 1707 in Basel, Switzerland, the eldest child of Paul Euler, a Protestant minister with a strong interest in mathematics, and Marguerite Brucker. The family moved to the village of Riehen when Leonhard was one, and it was there he grew up, in a household where intellectual seriousness was normal and mathematical thinking encouraged.

Paul Euler had studied under Jacob Bernoulli at the University of Basel and was friends with Jacob’s brother Johann Bernoulli, then one of Europe’s most eminent mathematicians. He gave his son a solid grounding before sending him to the University of Basel at thirteen. There, Euler took Saturday-afternoon tutorials from Johann Bernoulli himself — an arrangement Bernoulli offered once he saw that his young student was asking questions ordinary lectures could not answer. Bernoulli’s assessment was blunt and prescient: he reportedly told Paul Euler that his son would become the greatest mathematician in Europe. Coming from a competitive and famously difficult man, it was not flattery but a factual prediction.

Euler completed a Master’s degree in philosophy in 1723, aged sixteen, with a dissertation comparing the systems of Descartes and Newton. He then began theology at his father’s wish, intending to enter the ministry — but mathematics kept pulling him back, and Johann Bernoulli intervened to persuade Paul Euler that his son’s destiny lay elsewhere. By nineteen, Euler had submitted his first paper to the Paris Academy of Sciences, a competition entry on the optimal placement of masts on a ship. He came second, behind an experienced naval mathematician — a remarkable result for a nineteen-year-old who had never been to sea.

St. Petersburg: Where the Real Work Began

In 1727, Euler joined the newly founded St. Petersburg Academy of Sciences. Two of Johann Bernoulli’s sons, Nicolaus and Daniel, had gone there first; when Nicolaus died of appendicitis within months, a grief-stricken Daniel wrote to Euler asking him to fill the vacancy. Euler arrived at twenty and would spend the most creative decades of his life there. Russia in those years was politically turbulent, but Euler kept his head down and his pen moving — he is said to have joked that the turmoil helped, since the silence of people afraid to speak in public let him concentrate.

Within a few years he had cracked the Basel Problem — a question that had defeated Europe’s best for nearly a century: what is the sum of the reciprocals of all the perfect squares, 1 + 1/4 + 1/9 + 1/16 + …? Euler showed it equals π²/6. The answer was not just a solved puzzle but a startling new bridge between the whole numbers, the primes, and the constant π — a connection that opened a whole branch of mathematics and led, eventually, to the Riemann hypothesis, still unsolved and now one of the Clay Mathematics Institute’s seven Millennium Prize Problems, each carrying a $1 million reward.

He lost the sight of his right eye in 1735, after a near-fatal fever, and attributed the damage to the strain of an intensive calculation he had finished in three days that others thought would take months. He was not much troubled by it, reportedly remarking that he would now have fewer distractions. He kept working.

The Seven Bridges of Königsberg

In 1736, Euler turned to a puzzle circulating in the city of Königsberg (now Kaliningrad). The city was split by the Pregel River into four land areas, linked by seven bridges. The question: could you walk through the city crossing each bridge exactly once? It sounds like a weekend diversion. It founded graph theory and topology.

Euler’s insight was to abstract the problem entirely. The streets, the river, the buildings — all irrelevant. What mattered was the relationship between the land areas and the bridges. He represented the land areas as points (vertices) and the bridges as lines connecting them (edges), and asked whether a path exists that traverses every edge exactly once. He proved such a path is possible only if the network has exactly zero or two vertices with an odd number of connections. Königsberg had four — so no such walk was possible.

Every network analysis since — the structure of the internet, social-network analysis, the routing algorithms that steer data packets across the globe — traces its lineage to that 1736 paper. When Larry Page and Sergey Brin built Google’s PageRank algorithm at Stanford in 1998, they built it on directed graphs: pages linked to pages, analysed with precisely the graph-theoretic thinking Euler began in Königsberg. The founders of Google were, in a real sense, implementing Euler.

Berlin: Productivity at Scale

In 1741, Euler moved to Berlin at the invitation of Frederick the Great, and spent twenty-five years producing mathematics at a rate that defies easy description. He wrote textbooks that defined the field for a generation and worked on optics, music theory, lunar motion, shipbuilding, map-making, and the mathematics of lotteries. He even advised the Prussian government on pension design, using probability in ways that anticipated modern actuarial science.

His relationship with Frederick was productive but strained. The king prized scientific prestige more than science, and Euler’s quiet, devout, family-centred temperament did not suit the philosophical banter Frederick liked in his courtiers. Frederick reportedly called him a “mathematical Cyclops” — a jab at his missing right eye that nonetheless caught something true: Euler’s mathematical vision was so singular that the ordinary limits of what one person could accomplish simply did not apply.

In Berlin he developed his formulas for the motion of the Moon — of enormous practical value for navigation, since accurate lunar tables let sailors find their longitude at sea. He submitted a set to the British Board of Longitude in 1755 and received a £300 prize; navigators used them for decades. He also wrote the Introductio in Analysin Infinitorum (1748), a foundational text that introduced the modern concept of a function and established the notation f(x) that every mathematics student now learns. The historian of mathematics Carl Boyer regarded the Introductio as the finest modern textbook mathematics had produced — a judgement widely echoed since.

Euler’s Identity: The Equation That Connects Everything

Of all Euler’s contributions, the one most often called his greatest is the equation now known as Euler’s identity:

e^iπ + 1 = 0

Five numbers. One equation. Zero redundancy. e, Euler’s number (~2.71828), is the base of the natural logarithm, appearing wherever growth and decay occur — compound interest, radioactive decay, population dynamics. i, the imaginary unit, the square root of −1, is the number that “should not exist” yet opens an entire new dimension of mathematics indispensable to electrical engineering and quantum mechanics. π (~3.14159) is the ratio of a circle’s circumference to its diameter, appearing wherever circles, waves, and oscillations arise. And 1 and 0 are the multiplicative and additive identities, the building blocks of counting itself.

These five numbers come from completely different areas of mathematics. They were not discovered together and were not expected to be related. Euler showed they are joined by a single, simple, exact equation. In a 1988 poll by The Mathematical Intelligencer, his identity was voted the most beautiful theorem in mathematics; in a 2004 Physics World poll it tied with Maxwell’s equations as the greatest equation ever. And the beauty is not merely metaphorical: a 2014 study at University College London, led by the neuroaesthetics researcher Semir Zeki and published in Frontiers in Human Neuroscience, found that viewing Euler’s identity activated the same brain regions in trained mathematicians that respond to beautiful music and art.

Richard Feynman called the underlying formula “our jewel” and the most remarkable formula in mathematics in his famous lectures at Caltech. For the full story of Feynman and his approach to physics, see our article on Richard Feynman, the Nobel physicist who called curiosity his greatest instrument.

Going Blind and Working Faster

In 1766, Euler returned to St. Petersburg — Frederick had proved too frustrating, and Catherine the Great offered far better conditions. Soon after, a cataract clouded his remaining left eye, and within a few years he was effectively blind. His output increased.

This is not a misprint. Freed from the physical effort of writing, working entirely by dictation to a rotating team of secretaries and students, Euler produced mathematics at a pace that staggered his contemporaries. He had always had a phenomenal memory — he could recite the Aeneid from beginning to end and recall the page number of any passage in any book he had read — and blindness seemed only to concentrate that power. He could hold long calculations entirely in his head and dictate the results with the same accuracy others achieved with pen and paper before them.

He would lie in bed in the morning, work through problems in his head, then dictate; by afternoon the calculations were checked and the papers drafted. During his years of blindness he produced more than half of his total output, including fundamental work on celestial mechanics, differential equations, and the theory of functions. His Letters to a German Princess — explaining science in accessible terms — became one of the bestselling science books of the eighteenth century, translated into eight languages. It is, in retrospect, one of the first great works of popular science writing, and it was written blind, by dictation, by a man in his sixties who had simultaneously mastered nearly every area of mathematics known to his era.

What Scientists Say

“Read Euler, read Euler, he is the master of us all.”
— Pierre-Simon Laplace, to young mathematicians of the generation after Euler

The admiration has never really faded. Professor David Percy of the Institute of Mathematics and its Applications, speaking to the BBC, called Euler’s identity a genuine classic — simple to look at yet profoundly deep, drawing the five most important mathematical constants into a single statement. The Stanford mathematician Keith Devlin has written of the formula’s almost unsettling depth, comparing its economy and beauty to that of a great poem. And Richard Feynman, as noted, singled out the formula behind it as the most remarkable in all of mathematics. Across two and a half centuries, the people best equipped to judge have kept reaching for the same superlatives.

Euler’s Mathematics in the Modern World

Euler’s contributions are not museum pieces. They are active infrastructure. Here is where you meet his mathematics in daily life, whether you know it or not.

Your phone’s signal processing. Every phone call, streamed song, and video is manipulated using the Fourier transform, which decomposes complex signals into their component frequencies — and the Fourier transform is formulated using Euler’s formula connecting exponentials and trigonometric functions. Without it, there is no digital audio, video, or wireless communication as we know it.

The internet’s structure. How information flows through a network, how to find the shortest path between two points, how to spot the most influential nodes — the routing protocols directing internet traffic and the algorithms powering social networks all draw on graph theory, which Euler founded in Königsberg in 1736.

Electrical engineering. The analysis of alternating-current circuits — power distribution, radio, electronics — uses complex numbers exactly as Euler developed them. The phasor representation of signals, the impedance of capacitors and inductors, the analysis of filters and resonance all use e^iωt, Euler’s exponential, as their fundamental representation.

Quantum mechanics. The Schrödinger equation, governing the evolution of quantum states, is a complex differential equation whose solutions carry Euler’s imaginary unit i throughout; the wave functions describing electrons and photons are complex-valued. Without Euler’s development of complex analysis, quantum mechanics could not be written in its modern form — the same framework the physicist Paul Dirac used to predict antimatter from pure equations. For how the quantum world underlies everyday reality, see our article on the quantum tapestry beneath the everyday world.

Structural engineering. Euler’s formula for the critical load at which a slender column buckles under compression — derived in 1744 — is still used by engineers designing bridges, buildings, and aircraft, and taught in every undergraduate engineering programme in the world.

Topology. Euler’s formula for polyhedra — that vertices minus edges plus faces always equals 2 for any convex solid (V − E + F = 2) — founded the field of topology. The Euler characteristic generalised from it is a fundamental invariant appearing throughout modern mathematics and physics, from the classification of surfaces to the compactification of extra dimensions in string theory.

Why This Matters: Euler and the Language of the Universe

Euler’s significance is not merely a matter of giving a dead mathematician his due. It is a question about how the world works at a fundamental level. Mathematics is often described as the language of the universe — the claim that nature’s deepest patterns are mathematical in character. If that is even partly true, then the question of who built that language, and how, is not a minor historical footnote. It is a question about the structure of knowledge itself.

Euler built more of that language than anyone else in history. The notation every mathematician and physicist uses; the tools underlying electricity, quantum mechanics, network science, and structural engineering; the connections between distant areas of mathematics — between complex exponentials and trigonometry, between prime numbers and infinite series, between geometry and combinatorics — were established by Euler. When mathematicians work today on the Riemann hypothesis, they work in territory he opened. When physicists write quantum wave functions in complex numbers, they use his framework. When network scientists map the internet, they use the vocabulary he created with a puzzle about seven bridges.

The comparison with Ramanujan is instructive. Both possessed extraordinary intuition and saw patterns invisible to their peers. But where Ramanujan worked alone, in poverty, producing results he could not always prove, Euler worked systematically across an entire civilisation’s worth of mathematics, building and connecting and clarifying with a rigour and productivity never matched. For the other great self-taught visionary, see our article on Srinivasa Ramanujan, the man who knew infinity.

Euler’s death, that evening in September 1783, was described by the philosopher Nicolas de Condorcet — permanent secretary of the French Academy of Sciences — with a phrase that has become famous: “He ceased to calculate and to live.” It was meant as a tribute. It was also, as those who knew him understood, a precise description of how he had always lived. For Euler, calculation and life were the same thing.

Frequently Asked Questions

Sources

• MacTutor History of Mathematics — Leonhard Euler (University of St Andrews)
• Encyclopaedia Britannica — Leonhard Euler
• Wikipedia — Euler’s Identity
• Wikipedia — Seven Bridges of Königsberg
• Frontiers in Human Neuroscience — Zeki et al., The Experience of Mathematical Beauty (2014)
• Kronecker Wallis — Euler: The Most Prolific Mathematician in History