Leonhard Euler

Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced "oiler", not "yooler") was a Swiss mathematician and physicist.

Born and educated in Switzerland, he worked as a professor of mathematics in Saint Petersburg, later in Berlin, and then returned to Saint Petersburg. He is considered to be the most prolific mathematician of all time. He dominated the eighteenth century mathematics and deduced many consequences of the then new calculus. He was completely blind for the last seventeen years of his life, during which time he produced almost half of his total output.

Euler was deeply religious throughout his life. The widely told anecdote that Euler challenged Denis Diderot at the court of Catherine the Great with "Sir, (a+bⁿ)/n = x; hence God exists, reply!" is however false.

Discoveries

He is the physicist who, with Daniel Bernoulli, established the law that the torque on a thin elastic beam is proportional to a measure of the elasticity of the material and the moment of inertia of a cross section, about an axis through the center of mass and perpendicular to the plane of the couple.

He also deduced a set of laws of motion in fluid dynamics from Newton's laws of motion that state:

1. The force acting on a small element of a fluid is equal to the rate of change of its momentum.
2. The torque acting on a small element of a fluid is equal to the rate of change of its angular momentum.

In mathematics, he made important contributions to number theory as well as to the theory of differential equations. His contribution to analysis, for example, came through his synthesis of Leibniz's differential calculus with Newton's method of fluxions.

He established his fame early on by solving a long-standing problem:

$\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots ={\frac {\pi ^{2}}{6}}$

He also showed that for all real numbers x,

e^ix = cos(x) + i sin(x)

This is Euler's formula, which establishes the central role of the exponential function. In essence, all functions studied in elementary analysis are either variations of the exponential function or they are polynomials. The most remarkable formula in the world is an easy consequence.

In 1735, he defined the Euler-Mascheroni constant useful for differential equations:

$\gamma =\lim _{n\rightarrow \infty }\left(1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}...+{\frac {1}{n}}-\log(n)\right)$

He is a co-discoverer of the Euler-Maclaurin formula which is an extremely useful tool for calculation of difficult integrals, sums and series.

Euler wrote Tentamen novae theoriae musicae in 1739 which was an attempt to combine mathematics and music; a biography comments that the work was "for musicians too advanced in its mathematics and for mathematicians too musical".

In economics, he showed that if each factor of production is paid the value of its marginal product, then (under constant returns to scale) the total income and output will be completely exhausted.

In geometry and algebraic topology, there is a relationship called Euler's Formula which relates the number of edges, vertices, and faces of a convex solid with planar faces. Given such a polyhedron, the sum of the vertices and the faces is always the number of edges plus two. i.e.: V + F = 2 + E.

In 1736 Euler solved a problem known as the seven bridges of Königsberg, publishing a paper Solutio problematis ad geometriam situs pertinentis which may be the earliest usage of graph theory or topology.

External Links:

• MacTutor biography of Euler: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Euler.html

See also:

• Mathematician, Physicist, Mathematical constants, Complex number, Euler's conjecture, Euler number