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As Europe flourished during the Enlightenment, advancing in literacy, math, and science, so, too, did Leonhard Euler’s brilliance. He published over 800 works in his lifetime, and more than half of them were completed after he had gone blind.
Leonhard Euler was born on April 15, 1707, in Basel, Switzerland, to pastor Paul and his wife Margaretha. He spent his early years in the nearby village of Riehen. Growing up in the quiet countryside is said to have shaped Euler's calm and steady temperament, enabling him to maintain extraordinary focus and resilience, even after losing his sight.
At the age of seven, he was already studying an advanced algebra textbook far beyond the typical understanding of most children - or even adults - at that time. At 13, Euler enrolled at the University of Basel, which, like many European universities, admitted students much younger than is typical today. While the university itself was not particularly prestigious, Euler had the good fortune of being tutored by a brilliant mathematician. Johann Bernoulli was the leading mathematician of the era. By then, Isaac Newton was in his final years, and Gottfried Wilhelm Leibniz had already passed away.
Although Bernoulli declined to provide private lessons due to his busy schedule, he invited Euler to visit every Saturday afternoon to discuss any challenges he encountered, and Euler was grateful: “...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.” After receiving his bachelor's and master’s, he began studying theology, as his father wanted him to become a pastor but Leonhard’s heart lay in mathematics. Bernoulli is said to have visited Paul Euler, his former college classmate, to convince him that his son was a prodigy destined to be his successor. Paul agreed to let his son dedicate himself fully to mathematics, yet Leonhard never abandoned his Christian faith. Euler viewed the harmony in mathematics as evidence of a divine creator. His famous equation, Euler’s Identity, unites five fundamental constants, blending a mix of real and imaginary numbers in a simple yet profound way that results in zero.
Establishing a career in math in early 18th-century Switzerland was difficult as opportunities were scarce in the small country. He applied to become a physics professor at the University of Basel, but at 19, he was deemed too young to make the final cut. In hindsight, being rejected was a blessing. To reach his full potential, Euler needed the collaborative environment of a larger institution. As Ronald S. Calinger, author of a comprehensive biography on Euler, explains: “The social, institutional, and intellectual setting throws light on the intensity of collaboration and debate, and it is a principal factor in determining what questions and problems will be central.”
There was one place where such a setting existed - Russia, then part of a vast and powerful empire. When a position opened up in St. Petersburg at the Academy of Sciences, Euler’s good friend Daniel Bernoulli, the son of Johann Bernoulli and a member of the academy, recommended him. On April 5, 1727, he boarded a boat in Basel, traveling along the Rhine River, before arriving in St. Petersburg in May. Although Euler was offered an adjunct position in the Department of Physiology, he redirected his energy toward math, collaborating with brilliant mathematicians like Jakob Hermann and his friend Daniel Bernoulli, and corresponding often with Christian Goldbach who was in Moscow.
He tackled many fascinating challenges in St. Petersburg. One of the most famous was the Seven Bridges of Königsberg. The Prussian city of Königsberg, now modern-day Kaliningrad, Russia, was divided by a river with seven bridges connecting four regions. Residents often wondered whether it was possible to cross each bridge exactly once and return to their starting point. Euler approached the problem by simplifying the layout. He represented each area as a dot, or node, and each bridge as a line, or edge, connecting the nodes. The key to solving the problem lies in understanding the number of bridges or edges connected to each area, referred to as the degree of a node. For example, Node A has five connection points, while Node B has three. Node C and Node D each have three connection points as well. For someone to cross each bridge only once and return to their starting point, they must leave each area - or node - via a different bridge than the one they arrived on. So every node must have an even degree - that is, an even number of bridges connected to it - except for the beginning and end of the journey. When Euler analyzed his graph of Königsberg, he found that all four nodes had an odd degree, meaning each area was connected to an odd number of bridges. He therefore concluded that it was impossible to find a path that crosses each bridge exactly once and returns to the starting point. This was groundbreaking because it showed that the physical layout of the city was irrelevant - the solution depended solely on the connections between nodes, a fundamental concept in graph theory.
A big opportunity to advance his career came in 1733 when Daniel Bernoulli decided to leave St. Petersburg, hating the harsh Russian winters and haunted by the death of his brother Nikolaus just eight months after arriving. Euler succeeded Bernoulli as the senior chair of mathematics at the Academy. The promotion came with a substantial pay raise, which put him in a position to court Katharina Gsell, the daughter of a Swiss-born artist working in St. Petersburg. They married in January 1734, and their first child, Johann, was born that November. They went on to have 13 children altogether, though only five survived infancy. Euler is said to have made some of his greatest mathematical discoveries with “a child on his knee and a cat on his shoulder...”
One of these was a problem that had stumped mathematicians for nearly a century. The Basel Problem deals with summing the infinite series of reciprocals of integers squared. In other words, if you keep adding up smaller and smaller fractions forever, what number do you get? He worked on the problem on and off for years and solved it in 1735. His solution revealed a surprising link between an abstract problem in number theory and the geometric constant π (pi), traditionally associated with circles and geometry. Uncovering connections between seemingly unrelated areas of math was a testament to his brilliance.
While solidifying his reputation as a genius, Euler suffered from severe fevers in his thirties that caused his right eye to go blind. His grandson-in-law, mathematician Nikolas Fuss blamed his blindness on eyestrain from an intense three days of work in 1735. However, historian Ronald Calinger suggests it began with a sudden health crisis in 1738. As Calinger explains, “...great stress leads to a high fever, which together with an infection, produces an eye abscess that contributes to the evolution of blindness…” Regardless of its origin, Euler was not inclined to slow down. As Fuss noted, “He would give up his food before work...”
In the midst of his health problems, the Prussian King Frederick the Great recruited him to transform the Berlin Academy into a leading scientific institution. Euler only became interested after the death of the Russian empress Anna in 1740 triggered a power struggle and increased hostility toward foreigners like himself. After 14 years in Saint Petersburg, he packed his bags and arrived in Berlin on July 25, 1741.
There, he made groundbreaking contributions to celestial mechanics, particularly in tackling the challenging three-body problem - which involves predicting the motion of an object influenced by the gravity of two others. He applied this to the moon, which is gravitationally pulled by both the Earth and the Sun. The Moon’s elliptical orbit around Earth not only shifts in orientation, but the line of nodes, where the Moon’s orbit crosses Earth’s orbital plane, rotates a full cycle every 18.61 years. Euler’s refined calculations greatly improved the accuracy of these predictions, crucial for astronomy and navigation.
The following year, in 1748, Euler published one of the most groundbreaking works in mathematics: Introduction to the Analysis of the Infinite. In it, he defined a function as a rule that assigns one output to each input, such as y = x squared, where the rule squares x and gives the result y. This broad concept of a function applying to all values of x laid the foundation for modern algebra, calculus, and graphing.
One of the most important functions Euler explored is the exponential function, where e is a special constant approximately equal to 2.718. Exponential functions model real-world phenomena, such as population growth and compound interest. Euler expanded on the idea of exponential growth to uncover something profound. His formula shows that when you use imaginary numbers, represented by i, in an exponential function, instead of just growing, the results follow a circular path in the complex plane. Here’s the fascinating part: as the point moves around this circle, the real and imaginary parts of the function form smooth, wave-like patterns. These patterns are none other than the sine and cosine waves - familiar in sound, light, and the signals that power modern technology. It was a stunning revelation, showing that exponential functions and trigonometry, which might seem unrelated, are actually deeply connected.
Euler excelled in so many different areas of mathematics. In geometry, Euler discovered that for any polyhedron, meaning a 3-dimensional figure with flat faces, straight edges, and sharp corners, the number of vertices (V) minus the number of edges (E), plus the number of faces (F), always equals 2.
Euler had complete freedom to pursue his work as the King, who was preoccupied with the Silesian Wars against Imperial Austria, didn’t interfere with him and the academy. Euler was content during this period: “The king calls me his professor and I think I am the happiest man in the world.” But that happiness would not last. After the war’s end in 1745, Frederick redirected his attention toward domestic matters, including the academy. The King was rather dismissive of Euler, favoring French intellectuals like Voltaire and preferring literature and philosophy over abstract mathematics.
One incident in particular revealed his disdain for math. In 1749, the King asked Euler to design a hydraulic system to power the fountains at his Sanssouci palace in Potsdam so they could soar higher than those at the Palace of Versailles. But the wooden pipes were unable to withstand the high-pressure water, and reinforcing them with metal rims didn’t help. The pipe diameter was also too small to provide enough water flow. Euler lacked practical experience and had not accounted for issues like material limitations or friction. Frustrated by the failure, Frederick reportedly quipped to Voltaire: “Vanity of vanities! Vanity of mathematics.”
Euler lacked practical experience and hadn’t accounted for friction and material limitations. But he didn’t let the setback deter him. He would later master the behavior of fluids in his seminal work on fluid dynamics in 1757. His equations of motion reveal how forces like pressure and gravity work together to control the flow of fluids. Imagine water flowing through a slightly tilted pipe. Let’s focus on a small section of the water within the pipe. Two key forces act on this section: the pressure at the lower end pushes the fluid upward, while the opposing pressure at the higher end resists the flow. As the fluid moves upward, the pressure decreases along its length, creating a pressure difference that drives the flow. Additionally, gravity adds another resisting force, pulling the fluid downward. Despite these opposing forces, the water flows upward because the pressure difference at the lower end is strong enough to overcome both the pressure resistance and gravity.
Even with all that Euler accomplished, the king still snubbed him. When the academy’s president, French mathematician and philosopher Pierre Louis Maupertuis, passed away, the King refused to consider Euler his successor, preferring the French intellectual Jean le Rond d’Alembert. When d’Alembert declined, unwilling to leave his comfortable life in Paris, the King appointed himself deputy president. When Euler requested the King to secure military positions for his sons, he refused. As author Calinger put it: “He now decided that he and his family should leave; to remain would be disgraceful.”
And it just so happened there was one place eager to welcome him. Catherine the Great wanted him to return to St. Petersburg to revitalize the Academy that had fallen into decline. She also agreed to secure positions for Euler’s three sons. When Euler asked to be let go from his contract, Frederick was livid. He ignored Euler’s first two requests and tried to placate him after receiving a third, writing in response, “I would like to let you know that you will please me if you desist from this demand.” Eventually, to avoid a public scandal and so as not to antagonize the Russian Empress, Frederick reluctantly agreed to release Euler, writing: “...I permit you to quit in order to go to Russia.” He didn’t even so much as offer a thank you for his service.
Euler arrived in St. Petersburg on July 28, 1766. By then, a cataract in his good left eye had impaired his vision to the point where he couldn't distinguish between a blank page and one with writing. Euler would jokingly say, “Now I will have fewer distractions.” Despite being blind, he produced nearly half of his life's work during this period, authoring over 400 papers, thanks to his incredible memory and ability to perform complex calculations entirely in his head.
With the help of an unemployed tailor, Euler dictated his popular text Elements of Algebra, aimed at making mathematics accessible to a wider audience. By the end of their collaboration, the tailor could solve complex algebraic problems himself - a testament to Euler's teaching ability. His blindness made him deeply reliant on those around him not only for dictating his work but also in times of crisis. In May 1771, when a devastating fire swept through St. Petersburg, destroying over 500 homes including his own, his servant, Peter Grimm, scrambled to find a ladder and climb to the second floor to rescue the blind mathematician.
Later that year, in September 1771, Euler underwent cataract surgery in the hope of restoring vision to his left eye. While the surgery initially improved his sight, a lack of proper post-surgical care combined with a large incision led to complications, likely an infection, leaving him completely blind once again. Euler was determined not to rely solely on his children for help. When his 66-year-old wife, Katharina, passed away in November 1773, he remarried her half-sister, Salome Abigail Gsell.
Toward the end of his life, Euler suffered from frequent bouts of vertigo as his health declined. On September 18, 1783, he spent his final day much like any other. He taught mathematics to his grandchildren and discussed the upward motion of hot-air balloons with his guests. He took a nap after lunch and in the afternoon, lit his pipe. Moments later, the pipe dropped to the floor. While reaching for it, he suddenly clasped his hands to his forehead and uttered his final words, “Ich sterbe”. “I am dying.” He had suffered a stroke and passed away that night at the age of 76. 11pm
In death, Euler believed, “we will find ourselves in a more perfect state of dreaming.” Euler’s work laid the foundation for every area of math. If you’re looking to deepen your understanding of algebra, calculus, geometry, or vectors, Brilliant is the perfect place to start. What makes Brilliant stand out is its hands-on approach - it’s not about passive learning but actively solving problems as you go so you truly understand concepts. With Brilliant, you can learn at your own pace. Whether you’re a beginner or advanced, there’s something for everyone. You can also learn how to program by familiarizing yourself with Python and start building programs on day one with a built-in drag-and-drop editor. One of my favorites is How Large Language Models Work, which offers a fascinating look behind the technology that powers ChatGPT. The best part is: you can try Brilliant for FREE for 30 days by signing up using my link in the description: brilliant.org/newsthink. Or scan the QR code on your screen. That’s brilliant.org/newsthink for a FREE 30-day trial, plus you’ll save 20% on an annual Premium subscription. Thanks for watching. For Newsthink, I’m Cindy Pom.