Leonhard Euler

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{{Infobox Celebrity | name = Leonhard Euler | image = Leonhard Euler.jpg

| caption =
Leonhard Euler (1707-1783) by Emanuel Handmann

| birth_date = April 15, 1707 | birth_place = Basel, Switzerland | death_date = September 18, 1783 | death_place = St Petersburg, Russia | occupation = Mathematician and physicist | salary = | networth = | spouse = | website = | footnotes = }} Leonhard Euler (IPA Template:IPA) (April 15, 1707 to September 18, 1783) was a Swiss mathematician and physicist. He is considered to be one of the greatest mathematicians of all time. Euler was the first to use the term "function" to describe an expression involving various arguments; i.e., y = f(x). Also he introduced lasting notation for common geometric functions such as sine, cosine, and tangent.

He was born and educated in Basel. He showed an early talent for mathematics and languages. He worked as a professor of mathematics in St. Petersburg in Russia and later in Berlin. He was possibly the most prolific mathematician of them all. His collected work fills over 70 volumes. He dominated 18th-century mathematics and deduced many consequences of the newly invented calculus. He was almost completely blind for the last 17 years of his life, during which time he produced almost half of his writings and fathered most of his many children.

Contents

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Biography

Leonhard Euler's father Paul Euler had considerable attainments as a mathematician. He was a {Calvinistic//Lutheran} pastor of the village of {Riehen//Riechen} near Basel, Switzerland.

  • 15th of April 1707: Leonhard Euler was born in Basel.
  • He was raised in {Riehen//Riechen} by his mother Margret Brucker and father Paul Euler, a Lutheran minister. Although in his childhood he exhibited great mathematical talents, and he had preliminary instructions in mathematics from his father, but his father wanted him to study theology.
  • 1720: Leonhard Euler began his studies at the University of Basel. There geometry soon became his favourite study. There he met Jean Bernoulli and his sons Daniel Bernoulli and Nicolaus II Bernoulli, who noticed Euler's skills in mathematics. Paul Euler had attended Jakob Bernoulli's mathematical lectures and when Daniel and Nikolaus asked Paul Euler to allow Leonhard Euler to study mathematics, Paul Euler finally agreed.
  • 1723: Euler took his degree as master of arts. Then he applied himself, at his father's desire, to studying theology and the Oriental languages with the view of entering the church. But with his father's consent he soon returned to geometry as his principal pursuit.
  • 1725: Jean and Daniel Bernoulli moved to St. Petersburg in Russia.
  • Euler applied himself to the study of physiology, to which he made a happy application of his mathematical knowledge; and he also attended the medical lectures at Basel. While he was engaged in physiological researches, he composed a dissertation on the nature and propagation of sound.
  • 1727: Euler answered a prize question concerning the masting of ships. The French Academy of Sciences adjudged his answer as having the second rank.
  • 1727, on the invitation of Catherine I, Euler moved to St Petersburg, and was made an associate of the Academy of Sciences.
  • 1730: He became professor of physics.
  • 1733: He succeeded Daniel Bernoulli in the chair of mathematics. When he began his new career he enriched the academical collection with many memoirs, which excited a noble emulation between him and the Bernoullis, though this did not in any way affect their friendship. It was at this time that he carried the integral calculus to a higher degree of perfection, invented the calculation of sines, and reduced analytical operations to pure mathematics.
  • 1733: Euler married Katharina Gsell, the daughter of the director of the academy of arts. They had thirteen children, of whom only three sons and two daughters survived. The descendants of his children occupied prominent positions in the 19th century Russia.
  • 1735: Euler solved in 3 days a problem proposed by the academy, for whose solution several eminent mathematicians had demanded the space of some months. But the effort threw him into a fever which endangered his life and deprived him of the use of his right eye.
  • 1735: Euler lost much of his vision due to observing the sun without filtering out the excessive radiation. Despite this handicap, Euler continued to be productive, perhaps due to his extraordinary powers of memory and mental calculation. It is reported that once he let his assistant calculate a series to 17 summands and noticed an error in the 50th digit.
  • 1736: Euler was the first to publish a systematic introduction to mechanics in Mechanica sive motus scientia analytice exposita (= "Mechanics or motion explained with analytical science" (that is, calculus).
  • 1738: The Academy of Sciences at Paris adjudged the prize to his memoir on the nature and properties of fire.
  • 1740: His treatise on the tides shared the prize with those of Colin Maclaurin and Daniel Bernoulli - a higher honour than if he had carried it away from inferior rivals.
  • While in Russia, Euler participated in an extensive astronomical project by the Russian Academy of Sciences on the passage of Venus over the Sun. During this time he had to become a Cossack to ensure safe passage of scientific equipment and collaboration of local authorities.
  • 1741: Euler accepted the invitation of Frederick the Great to Berlin, where he was made a member of the Academy of Sciences and professor of mathematics. He enriched the last volume of the Melanges or Miscellanies of Berlin with five memoirs, and these were followed, with an astonishing rapidity, by a great number of important researches, which are scattered throughout the annual memoirs of the Prussian Academy. At the same time he continued his philosophical contributions to the Academy of St Petersburg.
  • 1742: The Academy of St Petersburg granted him a pension.
  • 1760: The respect in which he was held by the Russians was strikingly shown, when a farm he occupied near Charlottenburg was pillaged by the invading Russian army. On finding that the farm belonged to Euler, the general immediately ordered compensation to be paid, and the empress Elizabeth sent an extra sum of 4000 crowns.
  • 1766: Euler with difficulty obtained permission from the king of Prussia to return to St Petersburg. This was for various reasons, including lack of support from the King. Soon after his return to St Petersburg, a cataract formed in his left eye, which ultimately deprived him almost entirely of sight.
  • 1770: Euler published Anleitung zur Algebra, a work which, though purely elementary, displays the mathematical genius of its author, and is still reckoned one of the best works of its class. He dictated to his servant, a tailor's apprentice, who was absolutely devoid of mathematical knowledge, but had helped Euler much in his handicap caused by blindness.
  • 1768 to 1772: Euler published his Lettres a une princesse d'Allemagne sur quelques sujets de physique et de philosophie (3 vols.). They were written at the request of the princess of Anhalt-Dessau, and contain an admirably clear exposition of the principal facts of mechanics, optics, acoustics and physical astronomy. Theory, however, is frequently unsoundly applied in it, and Euler's strength lay rather in pure than in applied mathematics.
  • 1755: Euler had been elected a foreign member of the Academy of Sciences at Paris.
  • 1770 and 1772: The Academy of Sciences at Paris proposed two prize-questions to obtain a more perfect theory of the moon's motion. The Academy of Sciences's prize was adjudged to three of his memoirs Concerning the Inequalities in the Motions of the Planets. Euler, assisted by his eldest son Johann Albert, was a competitor for these prizes, and obtained both. In the second memoir he reserved for further consideration several inequalities of the moon's motion, which he could not determine in his first theory because the method he then used had engaged him in complicated calculations.
  • 1771: Euler's house burnt, destroying the greater part of his property. His manuscripts were fortunately preserved. His life was only saved by the courage of a native of Basel, Peter Grimmon, who carried him out of the burning house.
  • 1772: Euler published his Theoria motuum lunae. That was a review of his whole theory which he made helped by his son and W. L. Krafft and A. J. Lexell. Instead of confining himself, as before, to the fruitless integration of three differential equations of the second degree, which are furnished by mathematical principles, he reduced them to the three co-ordinates which determine the place of the moon; and he divided into classes all the inequalities of "that planet, as far as they depend either on the elongation of the sun and moon, or upon the eccentricity, or the parallax, or the inclination of the lunar orbit". This task were immensely harder becauset Euler was virtually blind, and had to carry all the elaborate computations it involved in his memory.

Some time after this an operation restored Euler's sight; but a too harsh use of the recovered faculty, and some carelessness on the part of the surgeons, brought about a relapse. With the assistance of his sons, and of Krafft and Lexell, however, he continued his labours, neither the loss of his sight nor the infirmities of an advanced age being sufficient to check his activity. Having engaged to furnish the Academy of St Petersburg with memoirs enough to complete its Acta for 20 years after his death, he in seven years transmitted to the academy above 70 memoirs, and left above 200 more, which were revised and completed by another hand.

Euler's knowledge was more general than might have been expected in one who had pursued with such unremitting ardour mathematics and astronomy as his favourite studies. He had made very considerable progress in medical, botanical and chemical science, and he was an excellent classical scholar, and extensively, read in general literature. He was much indebted to an uncommon memory, which seemed to retain every idea that was conveyed to it, either from reading or meditation. He could repeat the Aeneid of Virgil from the beginning to the end without hesitation, and indicate the first and last line of every page of the edition which he used. Euler's constitution was uncommonly vigorous, and his general health was always good. He was enabled to continue his labours to the very close of his life. His last subject of investigation was the motion of balloons, and the last subject on which he conversed was the newly discovered planet Herschel (Uranus).

Euler's genius was great and his industry still greater. His works, if printed in their completeness, would occupy from 60 to 80 quarto volumes: He was simple and upright in his character, and had a strong religious faith. He was twice married; his second wife being a half-sister of his first, and he had a numerous family, several of whom attained to distinction. His elogy was written for the French Academy by the marquis de Condorcet, and an account of his life, with a list of his works, was written by Von Fuss, the secretary to the Imperial Academy of St Petersburg.

  • 18th September 1783: Euler died of apoplexy while amusing himself at tea with one of his grandchildren.

It has been estimated that it would take eight hours of work per day for 50 years to copy all his works by hand. It was not until 1910 that a complete collection of his works was published. It was told by Legendre that often he would write down a complete mathematical proof between the first and the second call for supper.

Euler was a deeply religious Calvinist throughout his life, however the frequently quoted remark

"Sir, eiπ + 1 = 0, hence God exists; reply!"

with which Euler reportedly challenged Denis Diderot at the court of Catherine the Great, is likely not true. When Euler died, the mathematician and philosopher Marquis de Condorcet commented,

"...il cessa de calculer et de vivre,"

he ceased to calculate and ceased to live.

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Discoveries

Euler, with Daniel Bernoulli, 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 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.

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. Discovered the hyperbolic geometric 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's 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.
  • Created the Latin square, which likely inspired Howard Garns' number puzzle SuDoku.
  • 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.
  • 1735: Euler reaffirmed his scientific reputation by solving the long-standing Basel problem:
<math>\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}</math>,

where <math>\zeta(s)</math> is the Riemann zeta function and also described how to evaluate the zeta function at any positive even number. Euler also showed the usefulness, consistency, and simplicity of defining the exponent of an imaginary number by means of the Euler's formula

<math> e^{i \theta} = \cos\theta + i\sin\theta \,.</math>

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

<math>e^{i \pi} +1 = 0 \,.</math>
<math>\gamma = \lim_{n \rightarrow \infty } \left( 1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} - \ln(n) \right).</math>

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 - E + V = 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."
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Distinctions

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Quotes

Template:Wikiquote

  • "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.
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Works

The works which Euler published separately are:

  • Dissertatio physica de song (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 limas curvas, maximi minimive proprictate 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 it analysin infinitorum (Lausanne, 1748, in 2 vols. 4t0)
  • Scientia navalis, seu tractatus de construendis ac dirigendis navi bus (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)
  • Lett', es a une Princesse d'Allernagne sur quelques sujets de physique it de philosophic (St Petersburg, 1768-1772, in 3 vols. octavo)
  • Anleitung zur Algebra, or Introduction to 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)
  • M. Cantor, Geschichte der 1lfathematik.
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Further reading

  • Euler Leonhardt : "Lettres à une Princesse d'Allemagne"; free book at http://www.bookmine.org ;
  • Euler, Leonhard (1748). Introductio in analysin infinitorum. English translation Introduction to Analysis of the Infinite by John Blanton (Book I, ISBN 0387968245, Springer-Verlag 1988; Book II, ISBN 0387971327, Springer-Verlag 1989).
  • Dunham, William (1999). Euler: The Master of Us All, Washington: Mathematical Association of America. ISBN 0-88385-328-0.
  • Heimpell, Hermann, Theodor Heuss, Benno Reifenberg (editors). 1956. Die großen Deutschen, volume 2, Berlin: Ullstein Verlag.
  • Krus, D.J. (2001) Is normal distribution due to Karl Gauss? Euler, his family of gamma functions, and place in history of statistics. Quality and Quantity: International Journal of Methodology, 35, 445-446.(Request reprint).
  • Simmons, J. (1996). The giant book of scientists: The 100 greatest minds of all time, Sydney: The Book Company.
  • Singh, Simon. (2000). Fermats letzter Satz, Munich: Deutscher Taschenbuch Verlag.
  • Lexikon der Naturwissenschaftler, Spektrum Akademischer Verlag Heidelberg, 2000.
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See also

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External links

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