Leonhard Euler

Leonhard Euler (1707–1783) was the most productive mathematician in history. His name attaches to so many theorems, formulas, and constants that entire textbooks can be written without using the name twice. He worked in every branch of mathematics then known, and his notation — $e$ , $i$ , $\pi$ , $f(x)$ , $\Sigma$ for summation — became the standard vocabulary of the field.

Life

Euler was born in Basel, Switzerland, in 1707. He studied under Johann Bernoulli, who saw his talent and persuaded Euler’s father to let him pursue mathematics rather than theology. At 20 he joined the newly founded Saint Petersburg Academy of Sciences. He later moved to Berlin at the invitation of Frederick the Great (1741–1766), then returned to Saint Petersburg under Catherine the Great, where he spent the last seventeen years of his life.

Euler married twice and fathered 13 children, only five of whom survived infancy. He was devoutly religious — a Calvinist — and a devoted family man.

Blindness

In 1738, at age 31, Euler lost sight in his right eye, probably from a fever. In 1766, after unsuccessful cataract surgery, he went fully blind. Remarkably, this had little effect on his productivity. He dictated mathematics to his sons and secretaries and continued to produce about one paper a week for the rest of his life. Many of his greatest works date from this period.

He died in 1783. The French mathematician and philosopher Condorcet, in his eulogy, said: “Euler ceased to live and to calculate.”

Contributions

Analysis

Euler was the first mathematician to treat analysis — the calculus of Newton and Leibniz — as a systematic discipline. His Introductio in analysin infinitorum (1748) is the founding textbook of the field. In it he introduced and popularized:

The constant $e \approx 2.71828$ as the base of the natural logarithm
The notation $f(x)$ for a function of a variable
Euler’s formula $e^{ix} = \cos x + i \sin x$ , of which the famous identity $e^{i\pi} + 1 = 0$ is a special case

Number theory

Euler made enormous contributions to number theory. He proved Fermat’s Little Theorem, introduced the totient function $\phi(n)$ , and discovered the connection between the Riemann zeta function and the prime numbers (the Euler product formula):

$\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}$

Graph theory

In 1736, Euler solved the Seven Bridges of Königsberg problem: can you walk across all seven bridges of Königsberg exactly once, returning to your starting point? Euler proved it was impossible, and in doing so created the entire field of graph theory — the mathematics of networks, now central to computer science, biology, and social science.

Geometry

Euler discovered the polyhedron formula $V - E + F = 2$ , the first theorem of topology, and numerous facts about triangles (the Euler line, the Euler circle).

Physics

Euler’s work in mechanics, fluid dynamics, and optics is foundational. The Euler equations of fluid mechanics are still the starting point for aeronautics and meteorology. The Navier-Stokes equations — one of the Millennium Prize Problems — are a direct generalization.

Legacy

Euler effectively modernized the notation of mathematics. Before him, even basic concepts were written in cumbersome verbal or idiosyncratic symbolic forms. After him, mathematical writing became recognizably contemporary.

His output is so vast that mathematicians still regularly discover unpublished Euler results that anticipate later rediscoveries. As the physicist C.G.J. Jacobi put it: “Read Euler, read Euler, he is the master of us all.”

Known for

Euler's identity
Euler's polyhedron formula
Founding graph theory
Extensive work in number theory and analysis

Frequently asked

How productive was Euler really?

He published around 850 papers and books during his lifetime. His collected works (Opera Omnia) still aren't complete and run to over 80 volumes. Pierre-Simon Laplace is reported to have said: 'Read Euler, read Euler, he is the master of us all.'

Did Euler really work while blind?

Yes. He lost sight in his right eye at 31 and in his left eye at 59, but remained mathematically productive until his death at 76. He is said to have produced about half of his total output after going fully blind, dictating to his children and secretaries.

What are some things named after Euler?

So many that 'not named after Euler' is sometimes used as a lighthearted criterion for crediting a result. Just a partial list: Euler's identity, Euler's formula (multiple!), Euler's polyhedron formula, Euler's totient function, the Euler-Lagrange equations, the Euler-Maclaurin formula, Eulerian paths, Euler's line, Euler's method…