Leonhard Euler

Euler redirects here; another notable person named "Euler" is Carl Euler.

Leonhard Euler (pronounced oiler) (IPA ) (April 15, 1707 Basel, Switzerland - September 18, 1783 St Petersburg, Russia) was a Swiss mathematician and physicist. He is considered to be the preeminent mathematician of the 18th century and one of the greatest of all time; he is certainly among the most prolific, with collected works filling over 70 volumes.

Euler developed many important concepts and proved numerous lasting theorems in diverse areas of mathematics, from calculus to number theory to topology. In the course of this work, he introduced much of modern mathematical terminology, defining the concept of a function, and its notation, such as sin, cos, and tan for the trigonometric functions.

Biography

Euler’s parents were Paul Euler and Marguerite Brucker. Paul Euler was a Protestant pastor and wanted his son to follow in his footsteps. Although he was born in Basel, Switzerland, he spent most of his childhood in Riehen, a neighboring town where his father preached as a Lutheran minister. As Euler grew up, he became increasingly interested in mathematics and was educated by a friend of the family, Johann Bernoulli.

At his father's request, he studied theology, Hebrew, and Greek at the University of Basel, though he did not find them as interesting as mathematics. Euler was going to become a pastor when Bernoulli intervened. He convinced the father that his son was destined to become a great mathematician instead. Euler graduated from the University of Basel in 1726. Here he studied and reconstructed works of many famous mathematicians, including Varignon, Descartes, Newton, Galileo, van Schooten, Hermann, Taylor, Wallis, Jacob Bernoulli and of course Johann Bernoulli.

In 1727, he entered the Paris Academy Prize Problem competition, where the problem that year was to find the best way to place the masts on a ship. He received honorable mention, highly commendable for someone only twenty years old. However, Euler was not satisfied; he would go on to win the contest twelve years in a row.

Euler was offered a position teaching applications of mathematics at the St. Petersburg Academy. In November 1726, he accepted the post, but did not travel to Russia until the following spring. During this period, Euler applied unsuccessfully for a post at the University of Basel. On April 5, 1727, Euler left Basel for St. Petersburg. In 1730, he became a professor of physics. When Bernoulli returned to Basel in 1733, Euler was promoted to the senior mathematics chair.

On January 7, 1734, he married Katharina Gsell, daughter of a painter from the Academy Gymnasium. They had thirteen children, of whom only five survived childhood. He was married twice, his second wife being a half-sister of his first. Several of his children also attained distinction.

Euler began to suffer health problems in 1735. He had a severe fever that almost killed him. By 1740, he could not see with his right eye. Surgery fixed this temporarily, but his eye eventually failed again. New surgery in 1771 caused blindness in the other eye as well.

Concerned about continuing turmoil in Russia, Euler debated whether to stay in St. Petersburg or not. Frederick the Great of Prussia offered him a post at the Berlin Academy, which he accepted. He left St. Petersburg on June 19, 1741, but would return. He spent twenty-five years in Berlin, where he wrote over 380 articles. However, he left after Frederick placed d'Alembert into the mathematics position and made Euler President. Euler could not work with d’Alembert and returned to St. Petersburg, where he spent the rest of his life.

On September 18, 1783, he suffered a brain hemorrhage and died. His elegy was written for the French Academy by the Marquis de Condorcet, and an account of his life, with a list of his works, by von Fuss, Euler's son-in-law and the secretary of the Imperial Academy of St. Petersburg. The mathematician and philosopher Marquis de Condorcet commented,

"...il cessa de calculer et de vivre," (he ceased to calculate and to live).

Interests and output

Euler worked in almost all areas of mathematics: geometry, calculus, trigonometry, algebra, and number theory. He studied continuum mechanics, the lunar theory, and much more.

Euler's knowledge was more general than might have been expected in one who had pursued mathematics and astronomy with such ardor. He made considerable progress in medicine, botany and chemistry. He was also an excellent historian, and read much literature. He was endowed with an uncommon memory and seemed to retain every idea obtained by reading or meditation. He could repeat the Aeneid of Virgil in its entirety without hesitation, and indicate the first and last line of every page of the edition which he used.

Euler's works, if printed, would occupy between 60 and 80 quarto volumes. It has been estimated that it would take eight hours of work per day for 50 years to copy it all by hand. A project by the Swiss Academy of Sciences begun in 1907, the 200th anniversary of Euler's birth, to publish a complete collection of his works remains ongoing almost a hundred years later. To date, all of his published works have been republished, and about a quarter of his correspondence. Plans are underway to publish his notebooks and personal notes as well, which may take another 20 years. It was reported by Legendre that often he would write down a complete mathematical proof between the first and the second call for supper though this story must be second hand, if not apocryphal. Though they corresponded extensively, Euler and Legendre never met.

Discoveries

Euler's work touched upon so many fields that he is often the earliest written reference on a given matter. Physicists and mathematicians often jest that often times a discovery or theorem is named after the "first person after Euler to discover it". A list of his fundamental discoveries is bound to be incomplete -- he can be said to have founded elementary analysis, graph theory, and many of the physical applications of mathematics now fundamental to civil, mechanical, electrical and aeronautical engineering. So the following examples are just an incomplete sampling.

Euler was the first to publish formulas with the constant e (also known as Euler's constant), and showed the usefulness, consistency, and simplicity of defining the exponent of an imaginary number by means of the Euler's formula

e^{i \theta} = \cos\theta + i\sin\theta \,.

which establishes the central role of the exponential function in elementary analysis, where virtually all functions are either variations of the exponential function or polynomials. This formula was called "the most remarkable formula in mathematics" by Richard Feynman (Lectures on Physics, p.I-22-10). Euler's identity is a special case of this:

e^{i \pi} +1 = 0 \,.

Euler discovered quadratic reciprocity and proved that all even perfect numbers must be of Euclid's form. He investigated primitive roots, found new large primes, and deduced the infinitude of the primes from the divergence of the harmonic series. This was the first breakthrough in this area in 2000 years, heralding the birth of the analytic number theory. His work on factoring whole numbers over the complexes marked the beginning of the algebraic number theory. Amicable numbers had been known for 2000 years before Euler, and in all that time only 3 pairs were discovered. Euler found 59 more.

With Daniel Bernoulli, Euler developed the Euler-Bernoulli beam equation that allows the calculation of stress in beams. Euler also deduced the Euler equations, a set of laws of motion in fluid dynamics, formally identical to the Navier-Stokes equations, explaining, among other phenomena, the propagation of shock waves.

Leonhard Euler:

Elaborated the theory of higher transcendental functions by introducing the gamma function and the gamma density functions.
Introduced a new method for solving 4th degree polynomials.
Proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and made distinct contributions to the Lagrange's four-square theorem.
Made contributions to combinatorics, the calculus of variations and difference equations.
Created the theory of hypergeometric series, q-series and the analytic theory of continued fractions.
Solved a multitude of diophantine equations. Introduced and studied the hyperbolic trigonometric functions.
Calculated integrals with complex limits, which led (via Cauchy) to contour integration and complex analysis.
Discovered the addition theorem for elliptic integrals.
Invented the calculus of variations, including its most well-known result, the Euler-Lagrange equation.
Proved the binomial theorem for binomials with real number exponents.
Described numerous applications of Bernoulli's numbers, Fourier series, Venn diagrams, Euler numbers, e and pi constants, continued fractions and integrals.
Discovered the infinite product and partial fraction representations of the trigonometric functions.
Explicated logarithms of negative numbers.
Integrated Leibniz's differential calculus with Newton's method of fluxions. Pioneered applications of calculus to physics.
Co-discovered the Euler-Maclaurin formula which facilitates calculation of integrals, sums, and series.
Published substantial contributions to the theory of differential equations.
Defined a series of approximations which are used in computational mechanics. The most useful of these approximations is known as the Euler's method.
Euler is frequently misrepresented as having created the Latin square, which likely inspired Howard Garns' number puzzle SuDoku. However, Greco-Latin squares are several thousand years old, used frequently on sepulchres and graves as a talisman, and were exhaustively enumerated by Arabic numerologists from orders three through nine in the Jabirean Corpus a thousand years before Euler was born. Euler was simply responsible for a revival in their popularity.
In number theory, Euler invented the totient function. The totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. For example, φ(8) = 4 since the four numbers 1, 3, 5 and 7 are coprime to 8. With this function Euler was able to generalize Fermat's little theorem to Euler theorem.

1735: Euler reaffirmed his scientific reputation by solving the long-standing Basel problem:

\zeta(2) \ = \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}

where

\zeta(s)

is the Riemann zeta function and also described how to evaluate the zeta function at any positive even number.

1735: Euler defined the Euler-Mascheroni constant useful for solution of differential equations:

\gamma = \lim_{n \rightarrow \infty } \left( 1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} - \ln(n) \right).

Euler Proved that the sum of the reciprocals of the primes diverges.

In geometry and algebraic topology, there is a relationship (also called the Euler's Formula) which relates the number of edges, vertices, and faces of a convex polyhedron. Given such a polyhedron, the sum of the vertices and the faces is always the number of edges plus two. i.e.: F + V = E + 2. The theorem also applies to any planar graph. For nonplanar graphs, there is a generalization: If the graph can be embedded in a manifold M, then F - E + V = χ(M), where χ is the Euler characteristic of the manifold, a constant which is invariant under continuous deformations. The Euler characteristic of a simply-connected manifold such as a sphere or a plane is 2. A generalization of Euler's formula for arbitrary planar graphs exists: F - E + V - C = 1, where C is the number of components in the graph.

1736: Euler solved, or rather proved insoluble, a problem known as the seven bridges of Königsberg, publishing a paper Solutio problematis ad geometriam situs pertinentis which was the earliest application of graph theory or topology.
1739: Euler wrote Tentamen novae theoriae musicae, which was an attempt to combine mathematics and music; someone commented upon it that "for musicians it was too advanced in its mathematics and for mathematicians it was too musical."

Distinctions

The asteroid 2002 Euler is named in his honor.
Euler was ranked number 77 on Michael H. Hart's list of the most influential figures in history.
For nearly two decades (1979 to 1996), Euler was featured on a Swiss banknote. http://mypage.bluewin.ch/a-z/jke/CH-Noten/bn01090.htm

Quotes

"Lisez Euler, lisez Euler, c'est notre maitre a tous." (Read Euler, read Euler, he is the master of us all). attributed to —Pierre-Simon Laplace though this is quite probably apocryphal, apparently originating with the 19th century commentator Guido Libri.

Works

The works which Euler published separately are:

Dissertatio physica de sono (Dissertation on the physics of sound) (Basel, 1727, in quarto)
Mechanica, sive motus scientia analytice; expasita (St Petersburg, 1736, in 2 vols. quarto)
Ennleitung in die Arithmetik (ibid., 1738, in 2 vols. octavo), in German and Russian
Tentamen novae theoriae musicae (ibid. 1739, in quarto)
Methodus inveniendi lineas curvas, maximi minimive proprietate gaudentes (Lausanne, 1744, in quarto)
Theoria motuum planetarum et cometarum (Berlin, 1744, in quarto)
Beantwortung, &c., or Answers to Different Questions respecting Comets (ibid., 1744, in octavo)
Neue Grundsatze, c., or New Principles of Artillery, translated from the English of Benjamin Robins, with notes and illustrations (ibid., 1745, in octavo)
Opuscula varii argumenti (ibid., 1746-1751, in 3 vols. quarto)
Novae et carrectae tabulae ad loco lunae computanda (ibid., 1746, in quarto)
Tabulae astronomicae solis et lunae (ibid., quarto)
Gedanken, &c., or Thoughts on the Elements of Bodies (ibid. quarto)
Rettung der gall-lichen Offenbarung, &c., Defence of Divine Revelation against Free-thinkers (ibid., 1747, in 4t0)
Introductio in analysin infinitorum (Introduction to the analysis of the infinites)(Lausanne, 1748, in 2 vols. 4t0)
Scientia navalis, seu tractatus de construendis ac dirigendis navibus (St Petersburg, 1749, in 2 vols. quarto)
Theoria motus lunae (Berlin, 1753, in quarto)
Dissertatio de principio mininiae actionis, ' una cum examine objectionum cl. prof. Koenigii (ibid., 1753, in octavo)
Institutiones calculi differentialis, cum ejus usu in analysi Intuitorum ac doctrina serierum (ibid., 1755, in 410)
Constructio lentium objectivarum, &c. (St Petersburg, 1762, in quarto)
Theoria motus corporum solidoruni seu rigidorum (Rostock, 1765, in quarto)
Institutiones,calculi integralis (St Petersburg, 1768-1770, in 3 vols. quarto)
Lettres a une Princesse d'Allernagne sur quelques sujets de physique et de philosophie (St Petersburg, 1768-1772, in 3 vols. octavo)
Anleitung zur Algebra, or Elements of Algebra (ibid., 1770, in octavo); Dioptrica (ibid., 1767-1771, in 3 vols. quarto)
Theoria motuum lunge nova methodo pertr.arctata (ibid., 1772, in quarto)
Novae tabulae lunares (ibid., in octavo); La théorie complete de la construction et de la manteuvre des vaisseaux (ibid., .1773, in octavo)
Eclaircissements svr etablissements en favour taut des veuves que des marts, without a date
Opuscula analytica (St Petersburg, 1783-1785, in 2 vols. quarto). See Rudio, Leonhard Euler (Basel, 1884).

External links

Biography

Interests and output

Discoveries

Distinctions

Quotes

Works

Further reading

See also

External links

Gourt Home::Gourt :: Leonhard Euler

Biography

Interests and output

Discoveries

Distinctions

Quotes

Works

Further reading

See also

External links