Euler may be the most influential mathematician who ever lived (though some would make him second to Euclid); he ranks #77 on Michael Hart's famous list of the Most Influential Persons in History. His colleagues called him "Analysis Incarnate." Laplace, famous for denying credit to fellow mathematicians, once said "Read Euler: he is our master in everything." His notations and methods in many areas are in use to this day, Euler was the most prolific mathematician in history and is also widely regarded as the best algorist of all time. Just as Archimedes extended Euclid's geometry to marvelous heights, so Euler took marvelous advantage of the analysis of Newton and Leibniz. He gave the world modern trigonometry. Along with Lagrange he pioneered the calculus of variations. He was also supreme at discrete mathematics, inventing graph theory and generating functions. Euler was also a major figure in number theory, proving the divergence of the sum of prime reciprocals, finding both the largest then-known prime and the largest then-known perfect number, proving e to be irrational, proving that all even perfect numbers must have the Mersenne number form that Euclid had discovered 2000 years earlier, and much more. Euler was also first to prove several interesting theorems of geometry, including facts about the 9-point Feuerbach circle; relationships among a triangle's altitudes, medians, and circumscribing and inscribing circles; and an expression for a tetrahedron's area in terms of its sides. Euler was first to explore topology, proving theorems about the Euler characteristic. He also made several important advances in physics, e.g. extending Newton's Laws of Motion to rotating rigid bodies. On a lighter note, Euler constructed a particularly "magical" magic square.
Euler combined his brilliance with phenomenal concentration. He developed the first method to estimate the Moon's orbit (the three-body problem which had stumped Newton), and he settled an arithmetic dispute involving 50 decimal places of a long convergent series. Both these feats were accomplished when he was totally blind. (About this he said "Now I will have less distraction.") François Arago said that "Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind."
Four of the most important constant symbols in mathematics (π, e, i = √-1, and γ = 0.57721566...) were all introduced or popularized by Euler. He did important work with Riemann's zeta function ζ(s) = ∑ k-s (although it was not then known with that name or notation). As a young student of the Bernoulli family, Euler discovered the stiking identity π2/6 = ζ(2) This catapulted Euler to instant fame, since the right-side infinite sum (1 + 1/4 + 1/9 + 1/16 + ...) was a famous problem of the time. Among many other famous and important identities, Euler proved the Pentagonal Number Theorem (a beautiful little result which has inspired a variety of discoveries), and the Euler Product Formula ζ(s) = ∏(1-p-s)-1 where the right-side product is taken over all primes p. His most famous identity (which Richard Feynman called an "almost astounding ... jewel") unifies the trigonometric and exponential functions:
ei x = cos x + i sin x.
Some of Euler's greatest formulae can be combined into curious-looking formulae for π: π2 = - log2(-1) = 6 ∏p∈Prime(1-p-2)-1/2
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