# Leonhard Euler

## Pioneer in Algebra and Number Theory

Leonhard Euler was born in Basel, Switzerland, on April 15, 1707. His father, a Calvinist clergyman, had studied mathematics with JAKOB BERNOULLI, one of nine members of a multigenerational family of distinguished mathematicians and physicists. While a teenager, Euler studied mathematics at the University of Basel with Jakob’s brother, JOHANN BERNOULLI. Euler received his master’s degree when he was 17 and, with the Bernoulli family’s encouragement, he changed his career from the ministry to mathematics.

In 1727 Johann Bernoulli’s son Daniel helped Euler obtain a modest stipend in Russia, teaching in the medical section of the St. Petersburg Academy: When Daniel returned to Switzerland six years later, Euler was chosen to replace him as the Academy’s leading mathematician. He produced ideas, calculations, and proofs at an amazing pace. In 1735 he lost the sight in one eye after he worked nonstop for three days to obtain a result that his peers estimated would require months to calculate and years to evaluate. In 1736 he wrote Mechanics, the first book entirely devoted to that subject

Euler returned to Germany; where he was to remain for 25 years, in 1741 at the invitation of King Frederick the Great. There, he joined the Berlin Academy. In 1744 he identified the numerous solutions of polynomial algebraic equations consisting of powers of x. These algebraic solutions included whole numbers, fractions, negatives, irrationals, and complex numbers. Beyond them, however, he described transcendental numbers that are never outcomes of an algebraic equation: exponents, logarithms, trigonometric functions, etc. He also introduced standardized mathematical symbols including summation notation and e for the base of the natural logarithm.

Euler contributed significantly to number theory. In particular, he solved and extended several theorems presented by sixteenth century mathematician PIERRE DE FERMAT.

Euler applied mathematical techniques to the theory of lunar motion and what is called the threebody problem, which refers to an imperfect understanding of the interaction of the movement of the Sun, moon, and Earth. Using ISAAC NEWTON’s gravitational theory; Euler was able to provide an approximation for the three bodies.

After 25 productive years in Germany; Euler accepted a more rewarding and lucrative position in Russia in 1766. Even though he was completely blind by then and had to rely on his students and his children to write his papers for him, he spent the last 17 years of his life thinking and composing ideas about mathematics. The more than 70 volumes of Opera Omnia contains nearly 900 smaller books and original papers. Euler worked until he died suddenly on September 18, 1783, in St. Petersburg, Russia.

**Leonhard Euler’s Legacy**

Euler’s legacy pervades nearly every area of mathematics—from number theory and geometry to algebra and trigonometry—and his practical application of mathematics had direct influence on navigation at sea.

His work on proving Fermat’s last theorem (there are no whole number solutions for the equation xn + yn = zn when n is greater than 2) provided a major clue to that problem for mathematicians of his generation and aided in its eventual solution over 200 years later. Euler’s work was advanced in the nineteenth century by KARL GAUSS, who is considered to be the father of modern numbertheory methodology:

Evidence of the lasting effects of his originality infuse every area of mathematics. Euler’s many contributions are memorialized in mathematics texts by references to Euler’s theorem, Euler’s coefficients, Euler’s proofs, Euler’s constant, Euler’s integrals, Eulerian circuits, Euler’s transcendental functions, and Euler lines and angles.

Euler’s immediate influence on mathematics extended beyond what he invented. He wrote textbooks on calculus that were used by generations of students and professionals alike. The clarity of his explanations continues to inspire and educate students of mathematics. Euler’s standardization of mathematical notations and his introduction of symbols have been adopted worldwide.

Although most of Euler’s work was theoretical, some of it had practical applications. Most importantly, his application of Newton’s gravitational theory to describe mutual effects of the motion of the Sun, moon, and Earth led to lunar tables that the British Navy used to help determine latitudinal position of ships at sea.

Leonhard Euler – 1707-1783