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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 also listed on the Guinness Book of Records as the most prolific, with collected works filling between 60 and 80 quarto volumes.[1]

Euler developed many important concepts and proved numerous mathematical theorems in fields encompassing calculus to number theory to topology. In the course of this work he introduced the fundamentally important concept of a mathematical function,[2] as well as much of modern mathematical terminology.

The measure of his influence can be expressed by this quote often attributed to Pierre-Simon Laplace: "Lisez Euler, lisez Euler, c'est notre maître à tous." (Read Euler, read Euler, he is the master of us all).[3]

Contents

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 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. 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, and he graduated two years later. After graduation, Euler earned a masters of arts in philosophy. At this time, Euler was receiving 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.[4]

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". Euler, however would eventually win the coveted annual prize twelve times.[5]

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 division 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 unsuccessfully applied for a physics professorship at the University of Basel.[6]

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.[7]

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.[5]

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 academy. In 1731 Euler was made professor of physics. In 1733 when Daniel Bernoulli returned to Basel fed up at the censorship and hostility he faced at St. Petersburg, Euler succeeded him as the head of the mathematics department.[8]

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.[9]

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.[10]

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 dedicated research scientist.[10]

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 came to regard Euler as unsophisticated especially in comparison to the circle of philosophers the German King brought to the Academy. The great Voltaire was among those in Frederick's employ, and the Frenchman enjoyed a favored position in the king's social circle. Euler, a devout religious man and a hard worker was in many ways the direct opposite of Voltaire. He usually ended up looking like a fool when engaging in philosophical discussions, and Euler had an unfortuante tendency to debate matters that he knew little about. Having had no real training in philosophy, Euler often found himself at the sharp end of Voltaire's wit.[10]

Eyesight deterioration

A 1753 portrait by Emanuel Handmann. Note the problems of the right eyelid and that Euler is clearly suffering from strabismus. The left eye looks healthy, as it was a later cataract that destroyed it rather than gradual deterioration.
A 1753 portrait by Emanuel Handmann. Note the problems of the right eyelid and that Euler is clearly suffering from strabismus. The left eye looks healthy, as it was a later cataract that destroyed it rather than gradual deterioration.[11]

Euler suffered problems with his eyesight throughout his mathematical career. He had suffered a near-fatal fever in 1735 and three years later became nearly blind in his right eye. Euler rather blamed his condition on the painstaking work on cartography he had to perform for the St. Petersburg Academy. Euler's sight in that eye worsened throughout his stay in Germany, so much so that Frederick referred to him as "Cyclops". Euler later suffered a cataract in his good left eye, rendering him almost totally blind. Even after he left Berlin, his condition appeared not to affect his productivity at all. This was chiefly due to Euler's great skill at mental calculation and photographic memory. Euler could repeat the Aeneid of Virgil from beginning to end without hestitation, and indicate the first and last line of every page in the edition he used.[12]

Return to Russia

The situation in Russia had improved greatly since the ascension of Catherine the Great, and in 1766, Euler accepted an invitation to return to the St. Petersburg Academy. Euler would live the rest of his life in Russia.

Euler had to overcome several tragedies in his second stay. A fire in St. Petersburg cost him his home and almost his life. In 1773, he lost his wife of 40 years, and remarried three years later.

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 live and to calculate).[13]

Philosophy and religious beliefs

Euler, along with his friend Daniel Bernoulli was an opponent of Leibniz's monadism and the philosophy of Christian Wolff. Euler insisted that knowledge is founded in part on the basis of precise quantitative laws, something that monadism and Wolffian science were unable to provide. Euler's religious leanings might also have had a bearing on his dislike of the doctrine, going so far as to label Wolff's ideas as "heathen and atheistic".[14]

Much of what is known of Euler's religious beliefs can be deduced from his Letters to a German Princess and an earlier work, Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister (Defense of the Divine Revelation against the Objections of the Freethinkers). Euler was a staunch Christian and a biblical literalist (The Rettung was primarily an argument for the divine inspiration of scripture).[15]

There is a famous anecdote inspired by Euler's arguments with secular philosophers over religon. During Euler's second stint at the St. Petersburg academy, the philosopher Diderot visited Russia on Catherine the Great's invitation. The empress was alarmed that the philosopher's arguments for atheism were influencing members of her court, and so Euler was asked to confront the Frenchman. Diderot was later informed that a learned mathematician had produced a proof of the existence of God: Diderot agreed to view the proof as it was presented in court. Euler appeared, advanced toward Diderot and in a tone of perfect conviction announced "Sir, \begin{matrix}\frac{a+b^n}{n}=x\end{matrix}, hence God exists- reply!". Diderot, to whom all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. Embarrassed, he asked to leave Russia, a request that was graciously granted by the Empress. This anecdote is almost certainly false, least of all due to the fact that Diderot was actually a capable mathematician.[16]

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 works, if printed, would occupy between 60 and 80 quarto volumes.[12]

Mathematical notation

Euler introduced much of the mathematical notation in use today, such as the notation f(x) to describe a function and the modern notation for the trigonometric functions. Euler was the first to use the letter e for the base of the natural logarithm, now also known as Euler's number. He is also credited for inventing the notation i to denote \sqrt{-1}. The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized (though not invented) by Euler.[17]

Complex analysis

Euler made important contributions to complex analysis. He discovered what is now known as Euler's formula, that for any real number φ, the complex exponential function satisfies

e^{i\phi} = \cos \phi + i\sin \phi \!.

This has been 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 \,.

This identity is notable in that it encapsulates the relationship between e, π, i, 1 and 0, fundamentally important constants in mathematics.

Analysis

The development of calculus was at the forefront of 18th century mathematical research, and the Bernoullis, family friends of Euler were responsible for a lot of early progress in the field. Understanding the infinite was naturally the major focus of Euler's research. While some of Euler's proofs may not have been acceptable under modern standards of rigor, his ideas were responsible for many great advances. First of all, Euler introduced the concept of a function,[2] and introduced the use of the exponential function and, more importantly logarithms in analytic proofs.

Euler is well known in analysis for his frequent use and development of power series: that is, the expression of functions as sums of infinitely many terms, such as

e = \sum_{n=0}^\infty {1 \over n!} = \lim_{n \to \infty}\left(\frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \cdots + \frac{1}{n!}\right)

Notably, Euler discovered the power series expansions for e and the inverse tangent function. Euler's daring (and, by modern standards, technically incorrect) use of power series enabled him to solve the famous Basel problem in 1735:[18]

\lim_{n \to \infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}\right) = \frac{\pi ^2}{6}

In addition, Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving 4th degree polynomials. He also found a way to calculate integrals with complex limits, foreshadowing the development of complex analysis. Euler invented the calculus of variations including its most well-known result, the Euler-Lagrange equation.

Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of matehmatics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions. For example, he proved the infinitude of primes using the divergence of the harmonic series, and used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area would suggest the prime number theorem.[19]

Number theory

Euler's great interest in number theory can be traced to the influence of his friend in the St. Peterburg Academy, Christian Goldbach. A lot of Euler's early work on number theory was based on the works of Pierre de Fermat, and he was responsible for rigorizing some of Fermat's ideas, as well as disproving some of his more outlandish conjectures.

One focus of Euler's work was to link the nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between Riemann zeta function and prime numbers, known as the Euler product formula for the Riemann zeta function.

Euler 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. Euler also 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 what would become known as Euler's theorem. Euler also made significant contributions to the understanding of perfect numbers, a concept introduced by Euclid. Euler made progress toward the prime number theorem and conjectured the law of quadratic reciprocity. The two concepts are regarded as the fundamental theorems of number theory, and Euler's ideas paved the way for Carl Friedrich Gauss.[20]

Graph theory and Topology

See also: Seven Bridges of Königsberg
Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregolya and the bridges.
Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregolya and the bridges.

In 1736 Euler solved, or rather proved insoluble, a problem known as the seven bridges of Königsberg.[21] The city of Königsberg, Prussia (now Kaliningrad, Russia) is set on the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The question is whether it is possible to walk with a route that crosses each bridge exactly once, and return to the starting point.

Euler's solution of the Königsberg bridge problem is considered to be the first theorem of graph theory. In addition, Euler's recognition that the key information was the number of bridges and the list of their endpoints (rather than their exact positions) presaged the development of topology.[21]

Euler also made contributions to the understanding of planar graphs. He introduced a formula governing the relationship between the number of edges, vertices, and faces of a convex polyhedron. Given such a polyhedron, the alternating sum of vertices, edges and faces equals a constant: V-E+F=2. This constant, χ, is the Euler characteristic of the plane. The study and generalization of this equation, specially by Cauchy [22] and Lhuillier [23], is at the origin of topology. Euler characteristic, which may be generalized to any topological space as the alternating sum of the Betti numbers, naturally arises from homology. In particular, it is equal to 2-2g for a closed oriented surface with genus g and to 2-k for a non-orientable surface with k crosscaps. This property led to the definition of rotation systems in topological graph theory.

Applied mathematics

Some of Euler's greatest successes were in applying analytic methods to real world problems, describing numerous applications of Bernoulli's numbers, Fourier series, Venn diagrams, Euler numbers, e and π constants, continued fractions and integrals. Euler integrated Leibniz's differential calculus with Newton's method of fluxions, and developed tools that made it easier to apply calculus to physical problems. In particular, Euler made great strides in improving numerical approximation of integrals, inventing what are now known as the Euler approximations. The most notable of these approximations are Euler's method and the Euler-Maclaurin formula. Euler also facilitated the use of differential equations, in particular introducing the Euler-Mascheroni constant: \gamma = \lim_{n \rightarrow \infty } \left( 1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} - \ln(n) \right).

One of Euler's more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually integrate musical theory as part of mathematics. This work wasn't very influential however. An unnamed source noted that for musicians it was too mathematical and for mathematicians it was too musical.[24]

Trivia

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)
  • 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).

See also

References

  1. ^ Guinness Book of Records: Most prolific mathematician. Retrieved on September 2006.
  2. ^ a b Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America, 17.
  3. ^ Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America, xiii.
  4. ^ James, Ioan (2002). Remarkable Mathematicians: From Euler to von Neumann. Cambridge, 2. ISBN 0-521-52094-0.
  5. ^ a b Ronald Calinger (1996). "Leonhard Euler: The First St. Petersburg Years (1727-1741)". Historia Mathematica 23 (2): 156.
  6. ^ Ronald Calinger (1996). "Leonhard Euler: The First St. Petersburg Years (1727-1741)". Historia Mathematica 23 (2): 125.
  7. ^ Ronald Calinger (1996). "Leonhard Euler: The First St. Petersburg Years (1727-1741)". Historia Mathematica 23 (2): 127.
  8. ^ Ronald Calinger (1996). "Leonhard Euler: The First St. Petersburg Years (1727-1741)". Historia Mathematica 23 (2): 128-129.
  9. ^ Fuss, Nicolas. Eulogy of Euler by Fuss. Retrieved on August 30, 2006.
  10. ^ a b c Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America, xxiv-xxv.
  11. ^ Ronald Calinger (1996). "Leonhard Euler: The First St. Petersburg Years (1727-1741)". Historia Mathematica 23 (2): 154-155.
  12. ^ a b Finkel, B.F. (1897). "Biography- Leonard Euler". The American Matehmatical Monthly 4 (12): 300.
  13. ^ Marquis de Condorcet. Eulogy of Euler - Condorcet. Retrieved on August 30, 2006.
  14. ^ Ronald Calinger (1996). "Leonhard Euler: The First St. Petersburg Years (1727-1741)". Historia Mathematica 23 (2): 153-154.
  15. ^ Euler, Leonhard (1960). "Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister". Leonhardi Euleri Opera Omnia (series 3) 12.
  16. ^ Brown, B.H. (May 1942). "The Euler-Diderot Anecdote". The American Mathematical Monthly 49 (5): 302-303.
  17. ^ Wolfram, Stephen. Mathematical Notation: Past and Future.
  18. ^ Wanner, Gerhard; Harrier, Ernst (March 2005). Analysis by its history, 1st, Springer, 62.
  19. ^ Dunham, William (1999). “3,4”, Euler: The Master of Us All. The Mathematical Association of America.
  20. ^ Dunham, William (1999). “1,4”, Euler: The Master of Us All. The Mathematical Association of America.
  21. ^ a b Alexanderson, Gerald (July 2006). "Euler and Königsberg's bridges: a historical view". Bulletin of the American Mathematical Society.
  22. ^ Cauchy, A.L. (1813). "Recherche sur les polyèdres - premier mémoire". Journal de l'Ecole Polytechnique 9 (Cahier 16): 66–86.
  23. ^ Lhuillier, S.-A.-J. (1861). "Mémoire sur la polyèdrométrie". Annales de Mathématiques 3: 169–189.
  24. ^ Ronald Calinger (1996). "Leonhard Euler: The First St. Petersburg Years (1727-1741)". Historia Mathematica 23 (2): 144-145.
  25. ^ Edwards, Harold (November 1983). "Euler and Quadratic Reciprocity". Mathematics Magazine 56 (5): 291.

Further reading

  • Euler Leonhard : "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 0-387-96824-5, Springer-Verlag 1988; Book II, ISBN 0-387-97132-7, 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. (1997). Fermat's last theorem, Fourth Estate: New York, ISBN 1-85702-669-1
  • Lexikon der Naturwissenschaftler, Spektrum Akademischer Verlag Heidelberg, 2000.

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