Leonhard Euler

Euler redirects here. See also List of topics named after Leonhard Euler. Another notable person named "Euler" is Carl Euler.

Leonhard Euler (pronounced [ˈɔʏlɐ], that is, Oiler) (Basel, Switzerland, April 15, 1707 – September 18, 1783 in 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 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

Childhood

Euler’s parents were Paul Euler and Marguerite Brucker. He had two younger sisters named Anna Maria and Maria Magdalena. Paul Euler was a pastor of the Reformed Church, and Marguerite was also a pastor's daughter. Soon after their son Leonhard was born, the Eulers moved from Basel to the town of Riehen, where Leonhard Euler would spend most of his childhood. Paul Euler was a family friend of the Bernoullis, and Johann Bernoulli in particular would eventually be an important influence on the young Leonhard. Leonhard Euler's early formal education would start in Basel, where he was sent to live with his maternal grandmother. At the age of thirteen he matriculated at the University of Basel. He then graduated in 1722 at age fifteen. After graduation, Euler earned a masters of arts in philosophy degree two years later. At this time, Euler received Saturday afternoon lessons from Johann Bernoulli, who was then regarded as Europe's foremost mathematician. Johann quickly discovered his new pupil's incredible talent for mathematics ^[1].

Euler was at this point studying Theology, Greek and Hebrew on his father's urging, in the hopes of becoming a pastor. Johann Bernoulli intervened, and managed to convince Paul Euler that his son Leonhard was destined to become a great mathematician instead. In 1727, Leonhard Euler 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 won second place, losing only to Pierre Bouguer- a man known as "the father of naval architecture".

St. Petersburg

Around this time Johann Bernoulli's two sons, Daniel and Nicolas were working at the Imperial Russian Academy of Sciences in St Petersburg. Nicolas died of appendicitis after a year in Russia, and when Daniel assumed his brother's position in the mathematics/physics divison he recommended that the post in physiology that he vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg. In the interim Euler unsuccesfully applied for a physics professorship at the University of Basel.

Euler arrived in the Russian capital on May 17, 1727. He was immediately promoted from his junior post in the medical department of the academy to a position in the mathematics department. Euler lodged with Daniel Bernoulli, and the two often worked in close collaboration. Euler mastered Russian and settled into life in St Petersburg, even taking on, for a time an additional job as medic in the Russian Navy.

The Academy at St. Petersburg was established by Peter the Great and was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler: the academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy so as to lessen the faculty's teaching burden, and the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions.

However, the Academy's benefactress, Catherine I, who had attempted to continue the progressive policies of her late husband, died shortly before Euler's arrival. The Russian nobility then gained power upon the ascension of the twelve-year-old Peter II. The nobility were suspicious of the academy's foreign scientists, and thus cut funding and caused numerous other difficulties for the faculty.

Conditions improved slightly upon the death of Peter II, and Euler swiftly rose through the ranks in the acedemy. In 1731 Euler was made professor of physics. In 1733 Daniel Bernoulli returned to Basel, fed up at the censorship and hostility he faced at St. Petersburg. Euler succeeded Daniel as the head of the mathematics department at the academy.

On January 7, 1734, he married Katharina Gsell, daughter of a painter from the Academy Gymnasium. The young couple bought a house by the River Neva, and 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.

Berlin

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. In Berlin, he published the two works which he would be most renowned for: the Introduction in analysin infinitorum, a text on functions published in 1748 and the Institutiones calculi differntialis, a work on differential calculus.

In addition, Euler was asked to tutor the Princess of Anhalt Dessau, Frederick's niece. Euler wrote over 200 letters to her, which were later compiled into a best-selling volume, titled the Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess. This work contained Euler's exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insight on Euler's personality and religious beliefs. This book ended up being more widely read than any of Euler's mathematical works, and was published all across Europe and in the United States. The popularity of the Letters testifies to Euler's ability to communicate scientific matters effectively to a lay audience, a rare ability for a research scientist.

However, despite Euler's immense contribution to the Academy's prestige, he was soon forced to leave Berlin. This was caused in part by a personality conflict with Frederick. Frederick eventually came to regard Euler as unsophisticated, especially in comparison to the witty philosophers Frederick hired in the Academy, the great Volataire among them.

On September 18, 1783, he suffered a brain hemorrhage and died. His eulogy 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.^{[citation needed]}

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 (occasionally called Euler's number), 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\,.

(Note: The constant e is not to be confused with γ, the Euler-Mascheroni constant, which is itself sometimes called Euler's constant.)

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.

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)
Einleitung 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)
- Additamentum II (its English translation)
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)
Exposé concernant l’examen de la lettre de M. de Leibnitz (1752, its English translation)
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).

References

^ James, Ioan (2002). Remarkable Mathematicians: From Euler to von Neumann. Cambridge. p. 2. ISBN 0-521-52094-0.

External links

O'Connor, John J.; Robertson, Edmund F., "Leonhard Euler", MacTutor History of Mathematics Archive, University of St Andrews
Leonhard Euler on the 10 Swiss Franc banknote.
Euler Archive
Biography of Leonhard Euler
Leonhard Euler at the Mathematics Genealogy Project
Weisstein, Eric Wolfgang (ed.). "Euler, Leonhard (1707-1783)". ScienceWorld.

[childhood-1] James, Ioan (2002). Remarkable Mathematicians: From Euler to von Neumann. Cambridge. p. 2. ISBN 0-521-52094-0.

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