Leonard Euler: How He Solved the Basel Problem and Shaped the Language of Mathematics

Euler and the Basel Problem: A Century-Old Puzzle Cracked

We close our 10-day experiment by expanding on yesterday’s note about Leonard Euler. If Kolmogorov broke into serious mathematics by constructing a pathological integrable function with a divergent Fourier series, Euler became the “superstar” of mathematics by solving the century-old Basel problem.

In 1735, the young Euler computed the sum 1 + 1/2² + 1/3² + … and arrived at the exact answer π²/6. Before him, mathematicians knew only approximations and could not believe that a geometric constant like π was hiding inside a sum of squared reciprocals. Euler boldly applied the idea of infinite products and, by comparing coefficients, reached a result that stunned his contemporaries. This was the beginning of Euler’s reputation as someone who could hear the “music” in infinite series.

Here are a few more equally remarkable facts about this extraordinary man.

Blindness That Sharpened His Mind

Euler’s eyesight deteriorated over many years, and by his mid-fifties he could barely see at all. Rather than slowing down, he restructured his entire workflow: he dictated treatises from memory and verified calculations “by ear.” By 1775, his pace had become legendary — roughly one paper per week, with material going straight to print without drafts. His phenomenal memory and strict domestic discipline made it possible: assistants wrote everything down, family members read aloud, and he corrected formulas on the fly. Even in this condition, Euler remained unbothered. “One more distraction removed,” was how he described the loss of his sight — mathematics came before everything else.

How Euler Built the Alphabet of Modern Mathematics

Today we write f(x), use Σ for sums, e for the exponential base, and i for the imaginary unit — and rarely stop to wonder where any of it came from. In Euler’s time, mathematical notation sprawled across paragraphs, and its inconsistency made communication between fields painfully slow. He made notation compact and portable: a single line replaced a page of explanation. This changed not just convenience but the speed of science itself — formulas stopped being a local dialect and became a shared language.

The Euler Characteristic and the Birth of Topology

For any convex polyhedron, the number of vertices minus edges plus faces always equals 2, regardless of size or angles. This is a striking hint that shapes possess some hidden “essential constancy.” Euler had intuited the idea of an invariant that survives any deformation short of tearing. The world of shapes suddenly became less about lengths and areas and more about skeletons and holes. This is where the foundations of topology were laid.

Euler’s Equations and the Physics of Fluids

When we watch a jet of steam or the wake of a ship on water, we are effectively looking at solutions to Euler’s model of an inviscid fluid. The same ideas carry over into ballistics and early atmospheric modeling. Entire schools of approximation methods grew up around these equations. Euler essentially formulated the core framework for what would become hydrodynamics and continuum mechanics — and demonstrated that in mathematics, writing the right equation is sometimes the most important act of all.

Mental Arithmetic as a Superpower

Colleagues liked to test him for speed: they would argue about a distant term in a series or ask him to name numbers with rare properties. Euler would pause for a moment and produce an answer that was later confirmed on paper. François Arago wrote that he “calculated as naturally as he breathed” — and this was not a figure of speech but a record of direct observation. The secret lay not in tricks but in a trained memory and a deep understanding of structure: he seemed to see long calculations fully unfolded, all at once.

Science Until His Final Breath

On September 18, 1783, in Saint Petersburg, Euler was discussing the orbit of the newly discovered planet Uranus with Anders Lexell. After dinner he played with his grandson, laughed and joked — and then was gone. In his obituary, Condorcet wrote the phrase that became both epitaph and portrait: “He ceased to calculate and to live.” It reads like a statement of fact that joins the man to his life’s work: for Euler, mathematics was not a profession but a natural state of being.

The Man Who Gave Mathematics Its Rules

Euler did not merely solve hard problems — he assembled mathematics into a unified system. He gave it a language, habits, and tools. If there is one figure from the past who set the rules of the game for centuries to come, it is Euler. And the more closely you study his life, the more clearly you understand mathematics today.