Leonhard Euler
Based on Wikipedia: Leonhard Euler
If you've ever written a mathematical equation, you've probably used notation that Leonhard Euler invented. The letter π for the ratio of a circle's circumference to its diameter? Euler popularized it. The notation f(x) for a function? That was Euler. The letter i for the imaginary unit—the square root of negative one? Euler again. The sigma symbol Σ for summations? Euler. The constant e, that mysterious number approximately equal to 2.718 that appears everywhere in mathematics from compound interest to radioactive decay? We call it "Euler's number" for good reason.
This Swiss mathematician, born in 1707, didn't just contribute to mathematics. He essentially designed the language we use to write it.
A Mind Without Limits
Euler has been called a "universal genius" equipped with "almost unlimited powers of imagination, intellectual gifts, and extraordinary memory." These aren't the exaggerations of enthusiastic biographers—they're assessments shared by the greatest mathematicians who came after him.
Pierre-Simon Laplace, one of the most important mathematicians of the following generation, gave this advice to his students: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss, often considered the greatest mathematician in history, wrote that "the study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it."
Consider the scope of his output: 866 publications. Not papers—publications, including massive multi-volume treatises. His collected works, still being compiled today under the title Opera Omnia Leonhard Euler, run to over 80 volumes. He is regarded as the most prolific contributor in the history of mathematics and science, and unquestionably the greatest mathematician of the eighteenth century.
But raw output doesn't capture his importance. Euler founded entirely new fields of mathematics. Graph theory—the study of networks and connections that now underlies everything from social media algorithms to GPS navigation? Euler invented it. Topology—the branch of mathematics concerned with properties preserved under continuous deformation? Euler launched that too. He made foundational contributions to analytic number theory, complex analysis, and infinitesimal calculus.
He wasn't just a mathematician, either. He was a physicist, astronomer, logician, geographer, and engineer. He reformulated Newton's laws of motion, developed the equations governing fluid flow that engineers still use today, wrote three volumes on optics that influenced the design of microscopes and telescopes, and studied the structural behavior of beams and columns.
The Pastor's Son Who Became a Mathematician
Leonhard Euler was born in Basel, Switzerland, on April 15, 1707. His father Paul was a pastor in the Reformed Church, and the family had scholarly roots—his mother's ancestors included well-known classical scholars. Soon after Leonhard's birth, the Eulers moved to Riehen, where his father had accepted a position as the local pastor.
From a young age, Leonhard received mathematics lessons from his father. This wasn't unusual religious education padding—Paul Euler had studied mathematics at the University of Basel under Jacob Bernoulli, one of the leading mathematicians of the era. The Bernoulli family was a mathematical dynasty, and the connection would shape young Leonhard's entire future.
Around age eight, Euler was sent to live with his grandmother in Basel and enrolled in the local Latin school. He also received private tutoring from Johannes Burckhardt, a young theologian with a passion for mathematics. At thirteen, Euler enrolled at the University of Basel. This sounds precocious today, but attending university at such a young age wasn't unusual in early eighteenth-century Europe.
At Basel, Euler's elementary mathematics course was taught by Johann Bernoulli—the younger brother of Jacob Bernoulli, who had taught Euler's father. The mathematical circle was closing.
Johann Bernoulli recognized something extraordinary in young Euler. As Euler later described it in his autobiography, Bernoulli made it "a special pleasure for himself to help me along in the mathematical sciences." Bernoulli was too busy for private lessons, but he offered something more valuable: a method for self-education.
He gave me a far more salutary advice, which consisted in myself getting a hold of some of the more difficult mathematical books and working through them with great diligence, and should I encounter some objections or difficulties, he offered me free access to him every Saturday afternoon, and he was gracious enough to comment on the collected difficulties, which was done with such a desired advantage that, when he resolved one of my objections, ten others at once disappeared, which certainly is the best method of making happy progress in the mathematical sciences.
This is a remarkable pedagogical insight: the teacher who can resolve your deepest confusions in such a way that ten related confusions evaporate simultaneously has given you something far more valuable than rote instruction.
With Bernoulli's backing, Euler persuaded his father to let him become a mathematician rather than following him into the ministry. In 1723, Euler received a Master of Philosophy with a dissertation comparing the philosophies of René Descartes and Isaac Newton. He then enrolled in the theological faculty—perhaps a compromise with his father—but his heart was elsewhere.
The Seven Bridges of Königsberg
Before we follow Euler to Russia, let's understand one of his most famous contributions: the problem that invented graph theory.
The city of Königsberg in Prussia—now Kaliningrad, Russia—was built on both banks of the Pregel River, with two islands in the middle. Seven bridges connected these landmasses. The citizens of Königsberg had a puzzle: Could you walk through the city crossing each bridge exactly once, no more and no less?
People tried and failed. They tried different starting points, different routes. No one could do it, but no one could prove it was impossible either.
In 1736, Euler proved that it couldn't be done—and more importantly, he proved why. He stripped away all the irrelevant details: the shapes of the landmasses, the lengths of the bridges, the winding streets. What mattered were only the connections. He represented each landmass as a point (what we now call a "vertex" or "node") and each bridge as a line connecting two points (an "edge"). This abstract representation—a graph—captured everything essential about the problem.
Euler then reasoned as follows: If you're going to walk through a landmass without starting or ending there, you need to enter it the same number of times you leave it. That means every landmass you pass through must have an even number of bridges. The only exceptions can be your starting point and ending point, which might have an odd number.
In Königsberg, every landmass had an odd number of bridges. That meant you couldn't even have two valid endpoints, let alone create a complete walking tour. The problem was impossible, not through bad luck or lack of cleverness, but through mathematical necessity.
This might seem like a trivial puzzle, but Euler had created something profound. He'd invented a new way of analyzing situations based purely on connections, ignoring distances and shapes entirely. Today, this approach underlies computer science, network analysis, social media algorithms, epidemiology models, and countless other applications. When Facebook suggests friends you might know, or Google Maps finds the shortest route, or epidemiologists model how diseases spread—they're all using descendants of Euler's insight about the bridges of Königsberg.
Russia: The First Act
In 1726, Euler completed a dissertation on the propagation of sound, hoping to secure a professorship at Basel. He was rejected. That same year, he entered the Paris Academy's annual prize competition for the first time—the problem was to find the optimal placement of masts on a ship. Pierre Bouguer, who would become known as "the father of naval architecture," won first place. Euler came second.
Second place in a naval architecture competition might seem like a strange stepping stone for the greatest mathematician of the century, but it foreshadowed Euler's practical bent. He would eventually enter this competition fifteen times, winning twelve.
Meanwhile, Johann Bernoulli's two sons, Daniel and Nicolaus, had taken positions at the Imperial Russian Academy of Sciences in Saint Petersburg. They promised to recommend Euler for a position when one opened up. In 1726, Nicolaus died of appendicitis after less than a year in Russia. Daniel moved into his brother's position in the mathematics and physics division and recommended Euler for the physiology post he'd vacated.
Physiology? Euler was a mathematician. But opportunities in eighteenth-century academia were scarce, and Euler accepted eagerly. He delayed his departure briefly to apply for a physics professorship at Basel—rejected again—and arrived in Saint Petersburg in May 1727.
The Russian Academy had been established by Peter the Great to improve education in Russia and close the scientific gap with Western Europe. To attract talent, it offered foreign scholars generous salaries and living conditions. Euler was promoted from his medical department position to the mathematics department, lodged with his friend Daniel Bernoulli, and collaborated closely with him. He learned Russian, settled into Saint Petersburg life, and took on additional work as a medic in the Russian Navy.
But politics intervened, as politics always does. Catherine I, Peter's widow, had continued her husband's progressive policies. She died before Euler arrived. Power passed to the twelve-year-old Peter II, and the conservative Russian nobility reasserted itself. Suspicious of the academy's foreign scientists, they slashed funding and restricted enrollment.
Conditions improved somewhat when Peter II died in 1730 and Anna of Russia, who had German sympathies, took the throne. Euler rose quickly through the ranks, becoming a professor of physics in 1731. He left the Russian Navy, refusing a promotion to lieutenant—mathematics was calling. In 1733, when Daniel Bernoulli left Saint Petersburg in frustration with the censorship and hostility, Euler succeeded him as head of the mathematics department. In 1734, he married Katharina Gsell, daughter of a painter at the Academy.
Berlin: The Productive Years
Frederick the Great of Prussia was building a new Berlin Academy and wanted Europe's top scholars. He tried to recruit Euler in 1740, but Euler preferred Saint Petersburg. Then Empress Anna died, Russia's political instability resumed, and Frederick sweetened the offer. Euler cited his need for "a milder climate for his eyesight"—a diplomatic excuse—and in June 1741, he left for Berlin.
He would stay for twenty-five years. They were astonishingly productive years.
Euler wrote 380 works during his Berlin period, 275 of which were published. This included his famous Introductio in Analysin Infinitorum in 1748—a foundational text on mathematical functions—and his Institutiones Calculi Differentialis in 1755, a comprehensive treatment of differential calculus. He was elected to the Royal Swedish Academy of Sciences and the French Academy of Sciences. He supervised the Berlin Academy's library, observatory, and botanical garden. He oversaw the publication of calendars and maps that generated academy income.
He even designed water fountains.
This last task would become a source of embarrassment. Frederick wanted a spectacular water feature at Sanssouci, his summer palace. Euler calculated the specifications for the pumps and channels. The fountain was built—and barely worked.
Frederick was not amused. "I wanted to have a water jet in my garden," he wrote. "Euler calculated the force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci. My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry!"
Modern engineers who have examined Euler's calculations believe they were probably correct. The problem was almost certainly in the execution—leaky pipes, poorly built components, miscommunication with the builders. But Frederick had found a grievance, and he nursed it.
The truth was that Frederick and Euler were mismatched. Frederick surrounded himself with sophisticated intellectuals and witty conversationalists. Voltaire held a place of honor at his court. Euler was none of these things. He was, by all accounts, a simple man, devoutly religious, who never questioned conventional beliefs or existing social order. He wasn't a skilled debater and had an unfortunate habit of arguing about subjects he knew little about, making him an easy target for Voltaire's famous wit.
Frederick found him unsophisticated, "ill-informed on matters beyond numbers and figures." Despite Euler's immense contributions to the academy's prestige, despite being nominated for the presidency by the eminent mathematician Jean le Rond d'Alembert, Frederick named himself as president instead.
Yet Euler's work transcended these petty conflicts. Among his most popular publications were the Letters to a German Princess—over two hundred letters written to Frederick's niece, Friederike Charlotte of Brandenburg-Schwedt, explaining physics and mathematics in accessible terms. These letters were translated into multiple languages, published across Europe and in the United States, and became more widely read than any of his mathematical works. Euler could explain science to a lay audience, a rare skill for a research mathematician then as now.
The Basel Problem and Euler's Identity
To understand why mathematicians revere Euler, consider the Basel problem. Named after the Bernoullis' hometown, it had stumped mathematicians for nearly a century. The question was simple to state: What do you get if you add up the reciprocals of all the perfect squares?
That is: 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36, and so on forever. Each term is 1 divided by some whole number squared. The sum clearly gets larger as you add more terms, but it doesn't grow without bound—it approaches some specific value. What is that value?
In 1735, Euler proved that this infinite sum equals exactly π²/6—pi squared divided by six.
The appearance of π here is genuinely surprising. Pi is the ratio of a circle's circumference to its diameter. What does that have to do with adding up fractions involving perfect squares? There are no circles anywhere in sight. Yet somehow this constant, defined by circles, emerges from a completely unrelated calculation. Euler had discovered one of the deep hidden connections that run through mathematics like underground rivers.
Even more famous is Euler's identity, often called the most beautiful equation in mathematics:
e^(iπ) + 1 = 0
This single equation connects five of the most important numbers in mathematics: e (Euler's number, the base of natural logarithms), i (the imaginary unit, the square root of negative one), π (the ratio of a circle's circumference to its diameter), 1 (the multiplicative identity), and 0 (the additive identity). These numbers come from completely different areas of mathematics—analysis, complex numbers, geometry, arithmetic—yet Euler showed they're related by this elegant formula.
What does it mean to raise a number to an imaginary power? That's not obvious, and explaining it requires Euler's formula: e^(ix) = cos(x) + i·sin(x). When you plug in π for x, the cosine of π is negative one and the sine of π is zero, giving you e^(iπ) = -1, or equivalently, e^(iπ) + 1 = 0.
The identity reveals that exponential functions and trigonometric functions—growth and rotation—are secretly the same thing when you extend your view to include imaginary numbers. This isn't just aesthetically pleasing. It's practically essential. Modern electrical engineering, quantum mechanics, and signal processing all depend on this connection Euler made explicit.
Return to Russia
In 1760, with the Seven Years' War raging across Europe, Russian troops sacked Euler's farm in Charlottenburg. General Ivan Petrovich Saltykov paid compensation for the damage, and Empress Elizabeth of Russia later added 4,000 rubles—an enormous sum at the time. Even in war, Euler's reputation commanded respect.
The political situation in Russia had stabilized under Catherine the Great, and in 1766 she invited Euler to return. His conditions were extraordinary: a 3,000 ruble annual salary, a pension for his wife, and high-ranking appointments for his sons. Catherine agreed.
Euler returned to Saint Petersburg, now assisted by a younger mathematician named Anders Johan Lexell. In 1771, a fire destroyed his home. His eyesight, which had troubled him for years, finally failed completely. By 1771, he was totally blind.
For most mathematicians, blindness would end their careers. Mathematical work seems to require seeing equations, drawing diagrams, manipulating symbols on paper. But Euler's mental abilities were so extraordinary that blindness barely slowed him. His memory was legendary—he could recite the entire Aeneid and state the first and last lines of every page in his edition. He dictated his mathematics to assistants, performing complex calculations entirely in his head.
His productivity in his final years actually increased. Of his 866 publications, roughly half were produced after he went blind.
Euler's Characteristic and the Shape of Space
Here's another of Euler's discoveries, one that seems almost too simple to be important. Take any polyhedron—a three-dimensional shape with flat faces, like a cube or a pyramid. Count its vertices (corners), edges, and faces. Now calculate: vertices minus edges plus faces.
For a cube: 8 vertices, 12 edges, 6 faces. 8 - 12 + 6 = 2.
For a tetrahedron (triangular pyramid): 4 vertices, 6 edges, 4 faces. 4 - 6 + 4 = 2.
Try any polyhedron without holes. The answer is always 2.
This number is now called the Euler characteristic, and it's one of the foundational concepts of topology—the branch of mathematics that studies properties preserved under continuous deformation. A coffee cup and a donut have the same Euler characteristic (they both have one hole), which is different from a sphere's (no holes). This seemingly simple observation about counting corners and edges opened up entirely new ways of thinking about shape and space.
The End
Euler's first wife, Katharina, died in 1773 after thirty-nine years of marriage. Three years later, he married her half-sister, Salome Abigail Gsell. He continued working, continued producing mathematics, continued dictating papers to his assistants.
On September 18, 1783, Euler spent the day discussing the recently discovered planet Uranus, calculating the orbit of a hot air balloon, and working through a mathematical problem. In the evening, while playing with one of his grandchildren, he suffered a brain hemorrhage.
"I am dying," he said, and lost consciousness. He died a few hours later at age seventy-six.
The French mathematician Condorcet, delivering Euler's eulogy, said: "He ceased to calculate and to live."
The Exponential Function and Beyond
Why does Euler matter for understanding the exponential function—the subject of the article that led you to this Wikipedia deep dive?
Because Euler essentially invented our modern understanding of it. The number e had been discovered earlier, arising naturally from compound interest calculations. If you invest money at 100% annual interest, compounded more and more frequently, the amount after one year approaches a limit: e, approximately 2.71828. But it was Euler who recognized e's central importance, gave it its name, and developed its properties systematically.
He showed that e^x is its own derivative—the only function with this property. He revealed its connection to trigonometry through complex exponentials. He demonstrated how it appears in solutions to differential equations throughout physics. The exponential function, as we understand it today, bears Euler's fingerprints everywhere.
More broadly, Euler represents something remarkable in intellectual history: a mind of such scope and productivity that he touched nearly every area of mathematics and left each one richer than he found it. When you use the notation f(x), you're using Euler's invention. When you write Σ for a sum or Δ for a difference, you're speaking Euler's language. When you encounter π, e, or i, you're working with concepts he codified and connected.
He didn't just advance mathematics. He gave mathematicians the vocabulary to talk about it.
Read Euler, Laplace said. He is the master of us all.