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 himself a cup of tea, and worked on the mathematics of the recently discovered planet Uranus. He had spent the afternoon discussing science with colleagues. He had, by any account, 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. He had lost the sight of his right eye to a near-fatal fever in his late twenties. He had lost the sight of his left eye to a cataract when he was 59. And in those seventeen years of total blindness, working by dictation and by the extraordinary precision of his memory, he had produced more mathematics than most sighted mathematicians produce 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 of Sciences, which published his work, continued to release new Euler papers for more than thirty years after his death. He is the mathematician who gave us the symbols e, i, f(x), and Σ. He invented graph theory with a single recreational puzzle about bridges. He wrote the equation that physicists and mathematicians have called, across three centuries, the most beautiful in all of mathematics.

And yet most people who know the name Einstein or Newton have never heard of Euler. This is one of history’s more curious oversights. Because the mathematics Euler created is not 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. You almost certainly have no idea.

Table of Contents

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 Euler family moved to Riehen, a village near Basel, when Leonhard was one year old, and it was there that he grew up — in a household where intellectual seriousness was normal and mathematical thinking was 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 the most eminent mathematicians in Europe. He gave his son a solid mathematical grounding before sending him to the University of Basel at the age of thirteen. There, Euler took his Saturday afternoon tutorials from Johann Bernoulli himself — the arrangement that Bernoulli offered when he recognised that his young student was asking questions that could not be answered in regular lectures.

Bernoulli’s assessment of Euler at this period was precise and prescient. He reportedly told Paul Euler that his son would become the greatest mathematician in Europe. This was not a polite compliment. Bernoulli was competitive, occasionally difficult, and not given to flattering students unnecessarily. He meant it as a factual prediction.

Euler completed his Master’s degree in philosophy in 1723 at the age of sixteen, with a dissertation comparing the philosophical systems of Descartes and Newton. He then began studying theology at his father’s request, intending to follow him into the ministry. But mathematics kept pulling him back. Johann Bernoulli intervened, persuading Paul Euler that his son’s destiny lay elsewhere. Euler abandoned theology and returned to mathematics — a decision whose consequences for science are difficult to overstate.

By the age of nineteen he 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. The man who beat him, Pierre Bouguer, was an experienced naval mathematician and hydrographer. For a first submission, from a nineteen-year-old who had never been at sea, placing second was remarkable.

St. Petersburg: Where the Real Work Began

In 1727, Euler accepted an invitation to join the newly founded St. Petersburg Academy of Sciences in Russia, created by Catherine I on the model of the Paris and Berlin Academies. Two of Johann Bernoulli’s sons, Nicolaus and Daniel, had gone there first. Nicolaus died of appendicitis within months of arriving. Daniel, grief-stricken, wrote to Euler asking him to come and fill the vacancy.

Euler arrived in St. Petersburg at the age of twenty. He would spend the most creative decades of his life there, working with an intensity and breadth that astonished even his contemporaries. Russia under Peter the Great’s successors was not always stable — political upheavals, foreign wars, and the periodic death of monarchs kept the Academy in uncertain territory — but Euler kept his head down and his pen moving. He is said to have joked that political turmoil was actually helpful: the silence of people afraid to speak in public allowed him to concentrate on mathematics.

Within a few years of arriving, he had solved the Basel Problem — a question that had defeated the best mathematicians in Europe 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 result was not just a solution to a specific problem but a demonstration of a new connection between the prime numbers, the natural numbers, and the fundamental constant π — a connection that opened an entirely new branch of mathematics, eventually leading to the Riemann hypothesis, still unsolved, which the Clay Mathematics Institute has designated one of the seven Millennium Prize Problems, each with a $1 million prize for solution.

He lost the sight of his right eye in 1735, following a near-fatal fever that left him bedridden for weeks. He attributed the loss of the eye to the strain of an intensive calculation he had completed in three days that others estimated would take months. He was not especially troubled by it. “Now I will have fewer distractions,” he reportedly said. He kept working.

The Seven Bridges of Königsberg: How a Recreational Puzzle Founded a New Branch of Mathematics

In 1736, Euler turned his attention to a puzzle that had been circulating in the city of Königsberg (now Kaliningrad, Russia). The city was divided by the Pregel River into four land areas, connected by seven bridges. The puzzle asked: was it possible to walk through the city crossing each bridge exactly once?

It sounds like a weekend diversion. It founded graph theory and topology.

Euler’s approach was to abstract the problem entirely. The specific geography of Königsberg — its streets, its river, its buildings — was irrelevant. What mattered was the relationship between the land areas and the bridges connecting them. He represented the land areas as points (what we now call vertices or nodes) and the bridges as lines connecting them (what we now call edges). The question became: is there a path through this abstract network that traverses every edge exactly once?

Euler proved that such a path exists if and only if the network has exactly zero or two vertices with an odd number of edges connected to them. Königsberg’s network had four vertices with odd numbers of edges — so no such path was possible. The puzzle had no solution.

This was published in 1736 in the journal of the St. Petersburg Academy, in a paper that the mathematician Robin Wilson of the Open University has described as “the paper that founded graph theory.” Every network analysis that has been conducted since — from the structure of the internet to social network analysis to the routing algorithms that determine how data packets find their way across the globe — traces its intellectual lineage to Euler’s Königsberg paper.

When Google’s PageRank algorithm — the mathematical engine that determined the relevance of web pages in the early internet — was developed by Larry Page and Sergey Brin at Stanford University in 1998, it was built on the mathematics of directed graphs: networks of pages linked to other pages, analysed using precisely the kind of graph-theoretic thinking that Euler initiated in Königsberg in 1736. The founders of Google were, in a meaningful sense, implementing Euler.

Berlin: Productivity at Scale

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

His relationship with Frederick was productive but sometimes strained. The king valued scientific prestige more than science itself, and Euler’s quiet, methodical personality — he was deeply religious, devoted to his family, and entirely uninterested in the philosophical controversy that Frederick preferred in his courtiers — did not make him a favourite of the royal salon. Frederick reportedly called him a “mathematical Cyclops,” a reference to Euler’s missing right eye that was intended as a slight but inadvertently captured something true: Euler’s mathematical vision was so singular that the normal rules of what a person could accomplish simply did not apply to him.

During the Berlin years, Euler developed his formulas for the motions of the moon — work of enormous practical importance for navigation, since accurate lunar tables allowed sailors to determine longitude at sea. He submitted a set of lunar tables to the British Board of Longitude in 1755 and received a prize of £300. The same tables were used by navigators for decades and contributed to the practical solution of the longitude problem that had occupied European science for more than a century.

He also wrote the Introductio in Analysin Infinitorum (1748), a foundational textbook in mathematical analysis that introduced the modern concept of a function and established the notation f(x) — the standard way of writing a function applied to a variable that every student of mathematics learns today. The historian of mathematics Carl Boyer at Brooklyn College, City University of New York, described the Introductio as “the greatest modern textbook in mathematics,” a judgement that has been widely echoed.

Euler’s Identity: The Equation That Connects Everything

Of all Euler’s contributions, the one most frequently cited as his greatest — by mathematicians, by physicists, by people who have spent lifetimes thinking about what mathematics means — is the equation now known as Euler’s identity:

e^iπ + 1 = 0

Five numbers. One equation. Zero redundancy.

e is Euler’s number, approximately 2.71828 — the base of the natural logarithm, appearing wherever growth and decay occur in nature: in compound interest, radioactive decay, population dynamics, the charging of capacitors, and the mathematics of probability.

i is the imaginary unit, the square root of −1 — the number that “should not exist” in ordinary arithmetic but that, once accepted, opens an entirely new dimension of mathematics and is indispensable to electrical engineering, quantum mechanics, and signal processing.

π is pi, approximately 3.14159 — the ratio of a circle’s circumference to its diameter, appearing wherever circles, waves, oscillations, and rotations are involved: in the analysis of sound, light, and heat, in the mathematics of the planets’ orbits, in the formula for the normal distribution that underlies all of statistics.

1 and 0 are the multiplicative and additive identities — the most elementary numbers in arithmetic, the building blocks of counting itself.

These five numbers come from completely different areas of mathematics. They were not discovered together. They were not expected to be related. Euler showed that they are connected by a single, simple, exact equation. In a 1988 poll by The Mathematical Intelligencer, Euler’s identity was voted the most beautiful theorem in mathematics. In a 2004 poll by Physics World, it tied with Maxwell’s equations of electromagnetism as the greatest equation ever.

A study published in the journal Frontiers in Human Neuroscience in 2014, conducted by researchers at University College London led by Dr. Semir Zeki — a professor of neuroaesthetics — found that viewing Euler’s identity activated the same regions of the medial orbito-frontal cortex that respond to beautiful music and art in trained mathematicians. The beauty is not metaphorical. It is neurological.

Richard Feynman — Nobel Prize-winning physicist, widely regarded as one of the greatest scientific communicators of the twentieth century — called Euler’s formula (from which the identity is derived) “our jewel” and “the most remarkable formula in mathematics” in his famous Feynman Lectures on Physics, delivered at Caltech between 1961 and 1963. For the full story of Feynman and his approach to physics, see our article on Richard Feynman: the Nobel Prize physicist who called curiosity his greatest scientific instrument.

Going Blind and Working Faster

In 1766, Euler returned to St. Petersburg — Frederick the Great had proved too frustrating to work under, and Catherine the Great of Russia offered significantly better conditions. Shortly after his return, he developed a cataract in his remaining left eye. 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 possessed a remarkable memory — he could recite the Aeneid from beginning to end and recall the page number of any passage in any book he had read. Blindness, it seemed, concentrated this capacity further. He could hold long calculations entirely in his head and dictate the results to a scribe with the same accuracy others achieved with pen and paper in front of them.

The St. Petersburg Academy, to its credit, provided him with a team of assistants. His son Johann Albrecht became one of his most important collaborators. Euler would lie in bed in the morning, working through problems in his head, then dictate his results. By afternoon, the calculations were checked and the papers were drafted. The flow was continuous.

During his years of blindness, he produced more than half of his total output. The papers included fundamental contributions to celestial mechanics, differential equations, and the theory of functions. His Letters to a German Princess — a series of letters written to the Princess of Anhalt-Dessau explaining scientific concepts in accessible terms — became one of the bestselling science books of the eighteenth century, translated into eight languages and reprinted dozens of times. 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 virtually 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, French mathematician and astronomer, one of the leading scientists of the generation after Euler, speaking to young mathematicians in the early nineteenth century

“Euler’s identity is a real classic and you can do no better than that… It is simple to look at and yet incredibly profound, it comprises the five most important mathematical constants.”

— Professor David Percy, Institute of Mathematics and its Applications, in an interview with the BBC

“Euler’s formula reaches down into the very depths of existence.”

— Keith Devlin, mathematician at Stanford University and author of The Language of Mathematics

“Our jewel… the most remarkable formula in mathematics.”

— Richard Feynman, Nobel Prize-winning physicist, from The Feynman Lectures on Physics, California Institute of Technology, 1963

“The greatest modern textbook in mathematics.”

— Carl Boyer, historian of mathematics, Brooklyn College, City University of New York, on Euler’s Introductio in Analysin Infinitorum

Euler’s Mathematics in the Modern World: A Guided Tour

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

Your smartphone’s signal processing. Every audio and video signal processed by your phone — every phone call, every streamed song, every video — is manipulated using the Fourier transform, a mathematical tool that decomposes complex signals into their component frequencies. The Fourier transform is formulated using Euler’s formula connecting exponentials and trigonometric functions. Without Euler’s formula, there is no digital audio, no digital video, no wireless communication as we know it.

The internet’s structure. The mathematical analysis of networks — how information flows, how to find the shortest path between two points in a network, how to identify the most connected or most influential nodes — draws directly on graph theory, which Euler founded in Königsberg in 1736. The routing protocols that direct internet traffic, the algorithms that power social networks, the analysis of biological networks in systems biology: all of it is graph theory. All of it is Euler.

Electrical engineering. The analysis of alternating current circuits — the kind of electrical engineering that makes power distribution, radio transmission, and electronics possible — uses complex numbers in exactly the way Euler developed them. Electrical engineers use Euler’s formula as a matter of routine. The phasor representation of sinusoidal signals, the impedance of capacitors and inductors, the analysis of filters and resonance: all of these use eiωt — Euler’s exponential — as the fundamental mathematical representation.

Quantum mechanics. The Schrödinger equation — the fundamental equation of quantum mechanics, governing the evolution of quantum states — is a complex differential equation whose solutions involve Euler’s imaginary unit i throughout. The wave functions that describe electrons, photons, and all other quantum particles are complex-valued functions. Without Euler’s development of complex analysis, quantum mechanics could not be written down in its modern form. For a look at how quantum mechanics underlies the technologies of the twenty-first century, see our article on the quantum world beneath everyday reality.

Structural engineering. Euler’s formula for the critical load at which a slender column will buckle under compression — derived in 1744 — is still used by structural engineers designing bridges, buildings, and aerospace components. Every structural calculation that involves a thin column or beam under compressive load uses Euler’s buckling formula. It is taught in every undergraduate engineering programme in the world.

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

Why This Matters: Euler and the Language of the Universe

Euler’s significance is not merely historical. It is not a matter of giving appropriate credit to a dead mathematician. It is a question of understanding how the world works at a fundamental level.

Mathematics is sometimes described as the language of the universe — the claim that the patterns and regularities of nature are, at their deepest level, mathematical in character. If that claim is true — and the extraordinary success of mathematics in describing physical reality suggests it is at least partly true — then the question of who built that language, and how, is not a minor historical question. It is a question about the structure of knowledge itself.

Euler built more of that language than anyone else in history. The notation that every mathematician and physicist uses — function notation, summation notation, the symbols for e and i and π — was standardised or introduced by Euler. The mathematical tools that underlie electricity, quantum mechanics, network science, and structural engineering were developed by Euler. The connections between different areas of mathematics — the connection between complex exponentials and trigonometric functions that Euler’s formula expresses, the connection between prime numbers and infinite series that led to the Riemann hypothesis, the connection between geometry and combinatorics that graph theory opened — were established by Euler.

When mathematicians today work on the Riemann hypothesis — the deepest unsolved problem in mathematics, with a $1 million prize for its solution — they are working in territory that Euler opened. When physicists use complex numbers to write down quantum mechanical wave functions, they are using the mathematical framework Euler developed. When network scientists analyse the structure of the internet, they are using the conceptual vocabulary Euler established with a recreational puzzle about seven bridges.

The comparison with Ramanujan is instructive. Both were mathematicians of extraordinary intuition who saw patterns invisible to their contemporaries. 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 a productivity that has never been matched. For the full story of the other great mathematical visionary of the self-taught tradition, see our article on Srinivasa Ramanujan: the man who knew infinity.

Euler’s death, that evening in September 1783, was described by the mathematician and 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 Euler understood, a precise description of how he had always lived: calculation and life, for Euler, were the same thing.

Frequently Asked Questions

Who was Leonhard Euler and why is he important?

Leonhard Euler (1707–1783) was a Swiss mathematician and physicist, widely regarded as the greatest mathematician of the eighteenth century and the most prolific in history. His 886 papers and books span virtually every area of mathematics and physics. He invented graph theory, founded the analysis of complex functions, standardised the mathematical notation used worldwide today, and developed tools that underlie modern electrical engineering, quantum mechanics, network science, and structural engineering. His equation e^iπ + 1 = 0 is regularly cited as the most beautiful in mathematics.

What is Euler’s identity and why is it called beautiful?

Euler’s identity is the equation e^iπ + 1 = 0. It connects five of the most fundamental constants in mathematics — e (the base of natural logarithms), i (the imaginary unit), π (pi), 1, and 0 — in a single, exact equation. Each constant comes from a completely different area of mathematics, and their connection was not anticipated. A 1988 poll by The Mathematical Intelligencer voted it the most beautiful theorem in mathematics. A 2014 study at University College London found it activates the same brain regions as beautiful art and music in trained mathematicians.

What did Euler invent that we use today?

Euler’s most widely used contributions include: the mathematical notation f(x) for functions, the symbol e for the base of natural logarithms, the symbol i for the imaginary unit, the symbol Σ for summation, graph theory (the foundation of network analysis and internet routing), the formula for complex exponentials used in signal processing and electrical engineering, Euler’s buckling formula used in structural engineering, and the foundations of number theory that underlie modern cryptography and internet security.

How did Euler continue working after going blind?

Euler lost the sight of his right eye in his late twenties and the sight of his left eye to a cataract at age 59. He continued working by dictating to a team of secretaries and students, relying on an extraordinary memory that allowed him to hold long calculations entirely in his head. During his seventeen years of total blindness, he produced more than half of his total mathematical output. His son Johann Albrecht was among his most important collaborators in this period.

What is the connection between Euler and modern computing?

The connection is multiple and direct. Graph theory, which Euler founded, underlies internet routing, search algorithms, and social network analysis. Euler’s formula for complex exponentials is the mathematical basis of the Fourier transform, which is used in all digital signal processing — audio, video, wireless communications. The analysis of networks, the mathematics of optimisation, and much of the discrete mathematics used in computer science all trace their origins to Euler’s work.

How does Euler compare to other great mathematicians like Newton or Gauss?

Newton and Gauss both made revolutionary contributions in specific areas — Newton in calculus, mechanics, and optics; Gauss in number theory, statistics, and differential geometry. Euler’s distinction is breadth combined with depth: he made foundational contributions to more areas of mathematics and physics than either Newton or Gauss, and he did so while producing more published work than any mathematician before or since. Carl Friedrich Gauss himself described Euler’s work as the best school for learning mathematics — a judgement that remains the consensus among historians of mathematics.

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About the Author

Baryon is the founder and editor of Web News For Us. Driven by a deep fascination with the biggest unanswered questions in science — from quantum physics and cosmology to the nature of consciousness and the genetic code written into every living cell — he has spent years studying modern physics, biology, and the history of scientific thought. He covers Science & AI, Space, Genetics & Research, and the timeless wisdom of history’s greatest thinkers and mystics.

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