The Master of Us All — 1707–1783
The most prolific mathematician in history. Euler's work spans analysis, number theory, graph theory, mechanics, optics, and astronomy — producing over 800 papers and dozens of books.
Born April 15, 1707, in Basel, Switzerland, Leonhard Euler was the son of Paul Euler, a Calvinist pastor who had studied mathematics under Jakob Bernoulli. The family moved to the nearby village of Riehen when Leonhard was an infant.
At age 13, Euler enrolled at the University of Basel, receiving a Master of Philosophy in 1723 at just 16. His thesis compared Descartes' and Newton's philosophies. At Basel, Johann Bernoulli recognized Euler's extraordinary talent and gave him private Saturday tutoring sessions.
Paul Euler wanted his son to become a pastor, but Johann Bernoulli intervened, convincing the father that Leonhard was destined for mathematical greatness.
Euler could recite the entire Aeneid from memory, noting which line appeared on each page of his edition. This extraordinary recall served him when he went blind later in life.
In 1727, at age 20, Euler joined the Imperial Russian Academy of Sciences in St. Petersburg, where Daniel Bernoulli was already working. He would stay for 14 years.
At 19, Euler won an accessit (honorable mention) from the Paris Academy for his analysis of mast placement on ships — remarkable given he had never seen a seagoing vessel.
Rose rapidly through the Academy. Lost sight in his right eye (1738), possibly from overwork or an infection. Published Mechanica (1736), reformulating Newtonian mechanics using analysis rather than geometry.
Invited by Frederick the Great to the Berlin Academy. Published Introductio in Analysin Infinitorum (1748), Institutiones Calculi Differentialis (1755), and hundreds of papers. Won the Paris Prize 12 times.
Returned to Russia under Catherine the Great. Lost sight in remaining eye (1771), becoming completely blind. Remarkably, his productivity increased — he dictated to assistants, producing nearly half his total output while blind.
On September 18, 1783, after calculating the orbit of Uranus and discussing the newly discovered planet with colleagues, Euler suffered a brain hemorrhage and died. Condorcet wrote: "He ceased to calculate and to live."
Euler worked during the Age of Enlightenment, when royal courts competed to attract scientific talent. The academies of St. Petersburg, Berlin, and Paris provided salaries, publication venues, and international prestige.
The calculus wars between Newton and Leibniz were fading, but their legacy shaped Euler's career. Working within the Leibnizian tradition he inherited from Johann Bernoulli, Euler systematized and extended calculus into the dominant language of physical science.
Euler navigated political upheavals: the turbulent Russian succession crises, Frederick the Great's wars, and the shifting alliances of European powers, all while maintaining an astonishing focus on mathematics.
Frederick the Great wanted a courtly wit; Euler was quiet and deeply religious. Frederick mockingly called him "my Cyclops" after he lost one eye. The friction eventually drove Euler back to Russia.
Catherine the Great received Euler warmly, providing a large house, a pension, and assistants. When his house burned, she had it rebuilt at state expense.
Euler maintained over 4,000 letters with hundreds of correspondents, including the Bernoullis, Goldbach, Lagrange, and d'Alembert.
Euler's formula e^(ix) = cos(x) + i*sin(x) unifies the exponential function with trigonometry through complex numbers. Setting x = pi yields the celebrated identity:
eiπ + 1 = 0
This single equation connects the five most fundamental constants in mathematics: e, i, π, 1, and 0.
Euler derived this in his 1748 Introductio in Analysin Infinitorum, treating sin and cos as infinite series and recognizing their connection to the complex exponential.
Euler recognized that e^x = 1 + x + x^2/2! + ..., cos(x) = 1 - x^2/2! + ..., and sin(x) = x - x^3/3! + ... could be unified by substituting ix for x in the exponential series, yielding e^(ix) = cos(x) + i sin(x).
Euler resolved a long debate with Leibniz and Johann Bernoulli about ln(-1). Using his formula: ln(-1) = i*pi. He showed logarithms of negative numbers are complex, settling a 50-year controversy.
Euler was the first to treat functions as the central objects of analysis, defining f(x) as any expression involving x. He introduced the notation f(x), systematized trigonometric functions, and defined e as the base of natural logarithms.
Euler astounded the mathematical world by proving the sum 1 + 1/4 + 1/9 + 1/16 + ... = pi^2/6. This result, which had stumped the Bernoullis, launched Euler's reputation and connected number theory to analysis.
"Who has not been amazed that the sum of the reciprocals of the squares of the positive integers equals exactly pi-squared over six?"
— Leonhard Euler, on solving the Basel problemIn 1736, Euler tackled a popular puzzle: can one walk through the city of Konigsberg (now Kaliningrad), crossing each of its seven bridges exactly once?
Euler proved it was impossible by abstracting the problem: he replaced landmasses with vertices and bridges with edges, creating what we now call a graph.
His key insight: such a walk (an Eulerian path) exists only if the graph has exactly 0 or 2 vertices of odd degree. Konigsberg had 4 odd-degree vertices, making the walk impossible.
This paper launched graph theory and topology as mathematical disciplines.
In 1750, Euler discovered a beautiful relationship for convex polyhedra:
V - E + F = 2
where V = vertices, E = edges, F = faces. This formula holds for any convex polyhedron and became the foundation of algebraic topology. The quantity V - E + F is now called the Euler characteristic, a topological invariant.
Euler's formula generalizes: for a connected planar graph, V - E + F = 2 (where F counts regions). For surfaces of genus g (with g "holes"), V - E + F = 2 - 2g. This connects combinatorics to the deep structure of spaces.
Euler proved that the sum over all integers equals the product over all primes: Sum(1/n^s) = Product(1/(1-p^(-s))). This bridge between additive and multiplicative number theory was the seed of analytic number theory and the Riemann zeta function.
phi(n) counts integers less than n that are coprime to n. Euler generalized Fermat's little theorem: a^phi(n) is congruent to 1 (mod n). This result is the foundation of RSA cryptography today.
Euler systematized the Bernoullis' ad hoc methods into a general framework. The Euler-Lagrange equation, derived in 1744, gives necessary conditions for extremals of functionals — the foundation of classical mechanics and field theory.
Euler introduced: e for the natural base, i for sqrt(-1), f(x) for functions, the sigma notation for sums, pi for the circle ratio, and the standard trigonometric abbreviations sin, cos, tan. He literally created the language of mathematics.
Calculate specific examples exhaustively
Detect patterns across examples
Formulate bold general claims
Seek rigorous demonstration
Euler was the supreme calculator. He computed effortlessly with divergent series, infinite products, and continued fractions, often arriving at correct results through formal manipulations that later required centuries to rigorize. His motto was essentially: "Calculate, and the patterns will reveal themselves."
Euler's sheer volume was itself a methodology. By working simultaneously across analysis, number theory, mechanics, optics, astronomy, and music theory, he discovered connections invisible to specialists. His 866 publications (with more found posthumously) created a web of interconnection that defined modern mathematics.
Euler and Goldbach exchanged 196 letters over 35 years (1729-1764). It was in this correspondence that Goldbach stated his famous conjecture: every even number greater than 2 is the sum of two primes.
When young Lagrange sent Euler his work on the calculus of variations, Euler delayed his own publication to give Lagrange priority — an extraordinary act of scientific generosity.
Euler and d'Alembert clashed over the vibrating string problem. D'Alembert insisted solutions must be "continuous" (analytic) functions; Euler argued for broader function classes including curves with corners. This debate, unresolved in their lifetimes, anticipated the modern theory of distributions.
Frederick the Great found Euler socially awkward and preferred the witty Voltaire and the urbane d'Alembert. Euler was passed over for the presidency of the Berlin Academy in favor of Maupertuis, and later d'Alembert was offered the post (he declined). Frederick's disdain eventually drove Euler back to Russia.
Euler lost his right eye around 1738 and became completely blind in 1771 after a failed cataract operation. Yet he produced roughly 400 papers while blind — dictating to scribes, computing in his head with breathtaking speed.
Euler supported Maupertuis's principle of least action, becoming entangled in the bitter dispute with Samuel Konig. Voltaire's satirical attacks on Maupertuis embarrassed the entire Berlin Academy.
Later critics (Abel, Cauchy) noted Euler's cavalier treatment of convergence. Yet his formal manipulations were almost always correct — his intuition outran the rigor of his era.
Euler's product formula for the zeta function opened the door to Riemann's hypothesis, Dirichlet's theorem on primes in arithmetic progressions, and the entire field of analytic number theory.
From the Konigsberg bridges to modern internet routing, social network analysis, and circuit design. Eulerian paths and circuits remain fundamental concepts in discrete mathematics.
The Euler characteristic chi = V - E + F generalizes to higher dimensions and arbitrary topological spaces. It is a cornerstone of algebraic topology, connecting to homology and the Gauss-Bonnet theorem.
Euler's equations for inviscid fluid flow (1757) remain the foundation of aerodynamics. The Navier-Stokes equations extend them, and the million-dollar Millennium Prize problem concerns their solutions.
RSA encryption relies directly on Euler's totient theorem. Every secure internet transaction uses mathematics Euler developed in the 18th century.
Euler's formula e^(ix) = cos(x) + i sin(x) is the engine of Fourier analysis, which underpins digital audio, image compression, telecommunications, and quantum mechanics.
Euler's formula for column buckling (critical load = pi^2 EI / L^2) is still used by engineers worldwide. Every tall building and bridge design accounts for Euler buckling of compression members.
Euler developed perturbation methods for computing planetary orbits, created lunar tables used for maritime navigation, and formulated the three-body problem that continues to challenge mathematicians today.
AC circuit analysis uses Euler's formula to represent sinusoidal voltages and currents as complex exponentials. Impedance, phasors, and transfer functions all rely on e^(iwt) representation.
The Schrodinger equation's solutions are built from complex exponentials. Euler's identity connects the wave-like and particle-like descriptions of quantum systems.
William Dunham — A tour through Euler's most brilliant proofs, made accessible without sacrificing mathematical substance. The best single introduction.
David Richeson — The story of V - E + F = 2 and its journey from Euler's polyhedron formula to the heart of modern topology.
Paul Nahin — Deep dive into e^(i*pi) + 1 = 0 and its ramifications in physics and engineering. Technical but rewarding.
Ronald Calinger — The definitive modern biography, covering Euler's scientific work within its full political and cultural context.
Howard Eves — Excellent chapter on Euler situating his work within the broader development of 18th-century mathematics.
Euler (Blanton translation) — Euler's 1748 masterpiece in English. Still readable and astonishing for the range and depth of its content.
"Read Euler, read Euler, he is the master of us all."
— Pierre-Simon LaplaceLeonhard Euler (1707–1783) — The Master of Us All