Leonhard Euler: The Master of Us All

The Pastor's Son from Basel

Leonhard Euler was born on 15 April 1707 in Basel, Switzerland. His father, Paul Euler, was a Calvinist pastor who had studied mathematics under Jakob Bernoulli. His mother, Marguerite Brucker, came from a family of scholars. The family moved to the nearby town of Riehen when Euler was an infant, and it was there that he received his earliest education.

Paul Euler intended his son to follow him into the ministry. He sent young Leonhard to the University of Basel at the age of thirteen, where the boy completed his Master's degree by fifteen. At Basel, Euler had the extraordinary good fortune to study under Johann Bernoulli — then the leading mathematician in Europe — who recognized Euler's talent and gave him private lessons on Saturday afternoons.

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St. Petersburg and Berlin

In 1727, at the age of twenty, Euler accepted a position at the newly founded Imperial Russian Academy of Sciences in St. Petersburg, joining Daniel and Nicolaus Bernoulli. He would remain in Russia until 1741, producing a torrent of mathematical and scientific work.

In 1741, Euler moved to the Berlin Academy at the invitation of Frederick the Great of Prussia. He spent twenty-five productive years in Berlin, during which he published over 380 papers and several major books. His relationship with Frederick was uneasy — the king preferred French literary culture and reportedly found Euler's quiet, deeply religious personality unsophisticated.

In 1766, Euler returned to St. Petersburg at the invitation of Catherine the Great, where he remained for the rest of his life. By this time, he was almost completely blind — he had lost sight in his right eye in 1738 and his left eye deteriorated in the 1760s. Yet his blindness scarcely diminished his output. He dictated his work to assistants, holding the mathematics in his prodigious memory.

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The Scope of Euler's Work

Euler's collected works, the Opera Omnia, fill over 80 quarto volumes. He published approximately 866 papers and books during his lifetime, and new works continued to appear for decades after his death. It has been estimated that Euler accounts for roughly one-third of all mathematical and scientific output of the eighteenth century.

His contributions span virtually every area of mathematics known in his day, and he created several new ones.

Analysis and Calculus

Euler essentially created the field of mathematical analysis as a systematic discipline. His textbooks — Introductio in analysin infinitorum (1748), Institutiones calculi differentialis (1755), and Institutiones calculi integralis (1768–1770) — established the framework and notation that we still use.

He introduced or popularized the notation:

• $f(x)$ for a function of $x$
• $e$ for the base of the natural logarithm
• $\pi$ for the ratio of circumference to diameter
• $i$ for $\sqrt{-1}$
• $\sum$ for summation
• $\Delta$ for finite differences

Euler's Identity

Euler's most celebrated formula connects the five most important constants in mathematics:

This is a special case of Euler's formula:

$e^{ix} = \cos x + i \sin x$

which provides the fundamental link between the exponential function and trigonometry, and is essential in every area of mathematics and physics.

Number Theory

Euler made deep contributions to number theory, inspired by the work of Fermat. He proved Fermat's little theorem: if $p$ is prime and $\gcd(a, p) = 1$, then

$a^{p-1} \equiv 1 \pmod{p}$

He generalized this with the Euler totient function $\varphi(n)$, which counts the number of integers from $1$ to $n$ that are coprime to $n$:

$a^{\varphi(n)} \equiv 1 \pmod{n} \quad \text{for } \gcd(a, n) = 1$

Euler also introduced the Euler product formula for the Riemann zeta function:

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

This identity connects the additive structure of the integers with the multiplicative structure of the primes and is the starting point for analytic number theory.

The Basel Problem

One of Euler's earliest triumphs was his 1735 solution to the Basel problem, which had been open for nearly a century: find the exact value of

$\sum_{n=1}^{\infty} \frac{1}{n^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots$

Euler showed, astonishingly, that the answer is:

$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$

He later computed $\zeta(2k)$ for all positive integers $k$, expressing them in terms of Bernoulli numbers $B_{2k}$:

$\zeta(2k) = (-1)^{k+1} \frac{(2\pi)^{2k} B_{2k}}{2(2k)!}$

Graph Theory and the Königsberg Bridges

In 1736, Euler solved the famous Seven Bridges of Königsberg problem, asking whether one could walk through the city crossing each of its seven bridges exactly once. Euler proved this was impossible by abstracting the problem into what we now call a graph — a set of vertices and edges — and showing that a necessary condition for such a walk is that every vertex have even degree.

This paper is regarded as the founding document of graph theory and topology. Euler also discovered the polyhedron formula for convex polyhedra:

$V - E + F = 2$

where $V$ is the number of vertices, $E$ the number of edges, and $F$ the number of faces. This formula is one of the first results in topology.

Differential Equations and Mechanics

Euler developed much of the theory of ordinary differential equations. He introduced the method now called Euler's method for numerical approximation, the Euler–Lagrange equation in the calculus of variations:

$\frac{\partial L}{\partial y} - \frac{d}{dt} \frac{\partial L}{\partial y'} = 0$

and made fundamental contributions to fluid mechanics through the Euler equations:

$\frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla)\mathbf{v} = -\frac{1}{\rho}\nabla p + \mathbf{g}$

describing the motion of an inviscid fluid.

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Personal Life and Character

Euler married Katharina Gsell in 1734, and they had thirteen children, of whom five survived to adulthood. After Katharina's death in 1773, he married her half-sister Salome Abigail Gsell. Despite his fame, Euler was known for his modesty, kindness, and deep religious faith.

He was remarkably productive under any circumstances. The story is told that he could do mathematics while holding a child on his lap with another playing at his feet. His mental calculation abilities were legendary — he could compute long sums and products in his head even after losing his sight.

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Death

On 18 September 1783, Euler spent the afternoon calculating the orbit of Uranus and discussing the recently discovered planet with friends. That evening, while playing with a grandchild, he suffered a brain hemorrhage. He said "I am dying" and lost consciousness. He died a few hours later, at the age of 76.

As the mathematician Condorcet wrote: "He ceased to calculate, and he ceased to live."

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Legacy

Euler's influence on mathematics is immeasurable. An extraordinary number of theorems, formulas, constants, and methods bear his name — the Forbes list of "Euler's contributions" would be one of the longest in the history of science. His clear, expository writing style set the standard for mathematical communication.

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References

• Euler, L., Introductio in analysin infinitorum, 1748. Translated by J.D. Blanton, Springer, 1988.
• Dunham, W., Euler: The Master of Us All, MAA, 1999.
• Calinger, R., Leonhard Euler: Mathematical Genius in the Enlightenment, Princeton University Press, 2016.
• Sandifer, C.E., How Euler Did It, MAA, 2007.
• Wikipedia — Leonhard Euler
• MacTutor — Leonhard Euler
• The Euler Archive