**Lived 1707 – 1783.**

Leonhard Euler was one of the greatest mathematicians in history: not only did he produce outstanding mathematics, he produced it at an outrageous rate, publishing more than any other single mathematician before or after him. If quality of output multiplied by quantity of output is equal to greatness, then Euler is the greatest mathematician ever.

His work encompassed all of mathematics, nearly all of physics, and a significant part of astronomy. An astonishing fraction of the total research in mathematics and the physical sciences carried out between 1730 and 1780 was solely Euler’s work. Although he went completely blind at the age of 64, his remarkable mathematical flow continued without interruption – in fact, it increased.

### Beginnings

Leonhard Euler (pronounced “oiler”) was born on April 15, 1707 in the city of Basel, Switzerland. His mother was Marguerite Brucker and his father was Paul Euler. Leonhard was the eldest of their four children.

Leonhard’s mother was a pastor’s daughter. His father was a pastor in a Calvinist church. Leonhard’s father had studied some mathematics at university while taking his theology degree, and had become a friend of the Bernoulli family, famed for their mathematical prowess.

Leonhard grew up in the town of Riehen, about 5 miles (7 km) from Basel, very close to Switzerland’s borders with both France and Germany. His school in Riehen was not especially good, so his father gave him extra mathematics lessons – Leonhard was interested enough in the subject to get his own mathematics books to work through.

Like Carl Friedrich Gauss, born 70 years after him, Leonhard showed unusual talents in mental arithmetic, performing difficult calculations in his head. He also had a phenomenal memory – when prompted, he could readily recite long poems, and his knowledge of facts, scientific, and mathematical data was encyclopedic.

### Becoming a Mathematician

At the age of just 13, in 1720, Leonhard enrolled at the University of Basel, graduating three years later as a Master of Philosophy. His thesis analyzed the philosophical works of two mathematical geniuses, Isaac Newton and René Descartes.

His father hoped Leonhard would follow in his footsteps to become a Calvinist pastor, but private lessons with Johann Bernoulli revealed Leonhard was blessed with astonishing mathematical talents. Bernoulli was probably the world’s greatest practicing mathematician at the time. He was also a good friend of Leonhard’s father, and the two discussed Leonhard’s remarkable aptitude for mathematics. The result of their discussion was that Leonhard was allowed to spend three further years studying under Bernoulli’s guidance. So Leonhard followed a mathematical rather than a spiritual calling. Nevertheless, he remained devoutly religious throughout his life.

### The Russian Academy

Leonhard Euler tried hard for a professorship at the University of Basel, but was unsuccessful.

On May 17, 1727, he arrived in Russia’s capital city Saint Petersburg, where his friend Daniel Bernoulli was employed as a mathematician at the Imperial Russian Academy of Sciences.

Daniel’s brother Nicolaus had died in Russia the previous year of fever, and Daniel was now homesick, wishing he had Swiss colleagues. Euler was sent to Saint Petersburg by Johann Bernoulli, who was Daniel’s father. Euler arrived bearing gifts – supplies of tea, coffee, and brandy that Daniel had been longing for.

With an unusual talent for learning languages, Euler quickly mastered Russian. Beginning work as a physiology researcher, he was soon working in the more appropriate field of physics, becoming a professor of physics in 1730.

In 1733, Bernoulli returned to Switzerland and Euler, now age 26, replaced him as senior chair of mathematics. Soon Euler also became head of Geography.

Geography work might have been at least partly responsible for Euler’s eyesight problems, which began in 1738, when he was 31. He badly strained his eyes working on mapping Russia, and in 1740 he lost his right eye.

In Saint Petersburg Euler became great friends with Christian Goldbach, famous for the still unproven Goldbach Conjecture, which says that every even number greater than 2 can be expressed as the sum of two prime numbers.

## Leonhard Euler, Mathematics, and Science

Euler worked brilliantly in all branches of the mathematical and the physical sciences. He authored over 800 papers and books in these fields. In fact, his work adds up to an impressive fraction of all the scientific research carried out in the 1700s. Clifford Truesdell, a physicist and historian of science observed that:

It’s clear, therefore, that any brief summary such as this one can present only a few drops from the great ocean of Euler’s achievements.

### The Basel Problem

Euler’s first really big discovery in mathematics came in 1735, when he solved the Basel Problem, which had defeated the efforts of the best mathematicians for decades. The problem was to find the exact value of summing the reciprocals of squared whole numbers to infinity. (Mathematicians today would describe the problem as finding the zeta function of 2.)

Each successive term in the series is smaller than its predecessor and mathematicians already knew that the sum would converge to a particular value, but nobody had been able to find that value exactly.

Euler solved the Basel Problem by proving that, when the number of terms grows infinite, the series converges to equal:

This discovery made Euler a star in the mathematical world.

### Mechanica

Euler took his next step to superstardom in 1737, when he published *Mechanica*, a great leap forward in the mathematics of movement, made possible by Euler’s own innovations in mathematical analysis. Euler would have described analysis as the mathematics of the infinite and the infinitesimal. Today we could describe analysis roughly and somewhat incompletely as sophisticated calculus dealing with limits and continuity.

In *Mechanica* Euler utilized analysis to express the discoveries Isaac Newton had presented 50 years earlier in *Principia* in a more mathematically refined and useful way.

### Euler’s Laws of Motion

After *Mechanica*, Euler continued to work on the laws of motion. Whereas Newton’s laws applied to point-sized particles, Euler’s derived new laws that could be applied to rigid bodies with real dimensions, devised the principles of linear momentum and angular momentum, and derived the familiar differential equations of motion for rigid bodies, which today are described as Newtonian equations.

### Analysis of the Infinite

In 1748, Euler released the book *Introductio in analysin infinitorum*, which, like most of his output, he wrote in simple and elegant Latin. Its title in English is *Introduction to the Analysis of the Infinite*. It is probably the greatest modern textbook in all of mathematics. It deals in great detail with analysis, studying mathematical functions through infinite processes, particularly infinite series, which were something of a specialty of Euler’s.

### Foundations of Differential Calculus

Euler made a great number of other contributions to calculus, the mathematics that reigns supreme in the physical sciences – his greatest was *Institutiones calculi differentialis*, or *Foundations of Differential Calculus*, published in 1755, presenting among other things differential calculus of functions. Euler’s book is the basis of all future work in the field.

### The Language of Mathematics

Euler introduced or popularized many of the mathematical terms we are familiar with today.

- He popularized using the Greek letter π for the mathematical constant given by the ratio of a circle’s circumference to its diameter, which to six significant figures is 3.14159
- He assigned the letter e – nowadays often called Euler’s number – to represent the vital mathematical constant whose value to six significant figures is 2.71828
- He introduced the notation f(x) to mean function of x
- He introduced the letter i to represent √-1
- He discovered the equality many mathematicians believe is the most beautiful ever discovered – the Euler Identity, which relates the five most important numbers in mathematics – three of which Euler named or popularized the name of:
- The Euler Identity arose as a specific example of Euler’s success in defining the exponential function for complex numbers and his discovery of its relationship with the trigonometric functions.

### Euler’s Polyhedral Formula

Euler’s Polyhedral Formula, one of the first great discoveries in topology, comes second only to the Euler Identity for mathematical beauty. The formula applies to convex polyhedrons, which are shapes with straight edges and flat faces. For a shape with V vertices, E edges, and F faces, the formula says:

For example, a cube has 6 vertices, 12 edges and 8 vertices. Of course, when you put these numbers into Euler’s formula, you get the expected answer of 2.

Euler’s formula works not just for cubes, but for all convex polyhedrons.

### Popularizer of Science

Euler did not only write for the intellectual elite. His two-volume work *Letters to a German Princess, On Different Subjects in Physics and Philosophy* was one of the first popular science books. Published in 1768 and 1774, it was read throughout Europe and North America. The book was a compilation of more than 200 letters Euler sent between 1760-1762 to Friederike Charlotte, when he tutored her on behalf of her uncle, Frederick the Great.

Euler addressed many of the common questions about the world he had heard asked, such as:

- How fast does sound travel?
- Is air held by gravity?
- Why does it get cold on high mountains?

A religious man, he also considered philosophical problems, such as:

- The real destination of man
- Moral and physical evil
- The union between the soul and the body

### Some Personal Details

**Marriage and Family**

In January 1734, after becoming senior professor of mathematics at the Russian Academy of Sciences, Euler married Katharina Gsell, whose father was a Swiss artist working at the Academy. Only five of their thirteen children survived their infant years. Euler loved spending time with his children, and found some of his best ideas came to him while cradling a baby in his arms. Every evening he would spend time with his family reading and discussing the Bible.

**Berlin, Frederick the (not so) Great, and Russian Respect**

After 14 years in Saint Petersburg, Euler grew worried about the growing political turmoil in Russia and the sheer number of people being executed. It seemed to him that *anyone* in Russia who said *anything* could be hanged for it.

He was glad to accept a generous offer from Frederick the Great of Prussia to move to his capital city, Berlin, to work in its Academy of Sciences. In 1741, Euler began the first of what would be 25 years in Berlin.

Although Euler was undoubtedly a mathematical superstar, his quiet, unassuming demeanor began to annoy Frederick, who expected people at his Academy to engage in witty, sparkling intellectual conversations.

On the credit side, Frederick was a significant champion of Enlightenment values. He was also, however, something of a poseur, who refused to speak German, communicating exclusively in French. He became great friends with the French philosopher Voltaire – they fell out eventually – and the pair would mock the scholarly Euler for his simple, pious ways. Frederick taunted Euler for having lost an eye, calling him ‘Cyclops.’

In contrast, the Russians, even though Euler had abandoned Saint Petersburg, continued to communicate with him, pay him a salary – with which Euler bought books for the Russian Academy – and treat him with the greatest respect.

In 1760, the Russian army invaded Prussia and entered Berlin. Russian soldiers pillaged Euler’s country estate, where his mother, wife, sister-in-law, and his children lived. When he learned of this, the Russian commander personally paid Euler for the damage. Later, Empress Elizabeth of Russia sent Euler a very large sum of money as further compensation for the trouble he had been put to.

**Saint Petersburg again and Blindness**

In 1766, the Saint Petersburg Academy invited Euler to return, offering him an enormous salary and offering his sons jobs too. The political situation in Russia was now secure, so age 59, Euler gladly left Berlin and returned to Russia.

Sadly, his second and final spell there was difficult.

In 1771, he went completely blind, but he was undaunted. Assisted by his:

- phenomenal memory,
- mathematical inventiveness,
- ability to process complex mathematical procedures in his head, and
- his mathematician son Johann, who transcribed his words,

Euler’s mathematical output actually increased!

However, to add to his woes in 1771, his home burned to the ground, and the blind Euler was saved by a servant carrying him to safety.

### The End

Euler’s wife Katharina died in late 1773. In 1776, Euler married Katharina’s sister-in-law Salome Abigail Gsell.

Leonhard Euler died, age 76, from a brain hemorrhage on September 18, 1783 in Saint Petersburg, Russia. He was buried beside Katharina at the Smolensk Lutheran Cemetery on Dekabristov Island, close to Saint Petersburg. He had continued working productively to the end.

He was survived by Salome and three sons from his marriage to Katharina. His eldest son Johann became an illustrious astronomer and mathematician, his second son Karl became a renowned physician, and his third son Christoph became a military officer and amateur astronomer.

**Author of this page: The Doc**

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**Further Reading**

Alfred Hooper

Makers of Mathematics

Faber and Faber Ltd., 1961

Clifford Truesdell

An Idiot’s Fugitive Essays on Science: Methods, Criticism, Training, Circumstances

Springer-Verlag, 1984

C. Edward Sandifer

The Early Mathematics of Leonhard Euler

MAA, 2007

**Creative Commons**

Image of Clifford Truesdell is licensed under the Creative Commons Attribution-Share Alike 2.0 Germany license by Mathematisches Forschungsinstitut Oberwolfach.