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Leonhard Euler

2007 Schools Wikipedia Selection. Related subjects: Mathematicians

   CAPTION: Leonhard Euler

   Portrait by Johann Georg Brucker
   Portrait by Johann Georg Brucker
      Born     April 15, 1707
               Basel, Switzerland
      Died     September 18, 1783
               St Petersburg, Russia
    Residence  Prussia (25 years), Russia (31 years), Switzerland (20 years)
      Field    Mathematics, Optics, Astronomy
   Institution Imperial Russian Academy of Sciences, Berlin Academy
   Alma Mater  University of Basel
    Known for  Mathematical analysis, number theory, graph theory
    Religion   Calvinist

   Leonhard Euler (pronounced Oiler; IPA [ˈɔʏlɐ]) ( Basel, Switzerland,
   April 15, 1707 – St Petersburg, Russia, September 18, 1783) was a Swiss
   mathematician and physicist.

   Euler made important discoveries in fields as diverse as calculus,
   number theory, and topology. He also introduced much of the modern
   mathematical terminology and notation, particularly for mathematical
   analysis, such as the notion of a mathematical function. He is also
   renowned for his work in mechanics, optics, and astronomy.

   Euler is considered to be the preeminent mathematician of the 18th
   century and one of the greatest of all time. He is also one of the most
   prolific; his collected works fill 60–80 quarto volumes. A statement
   attributed to Pierre-Simon Laplace expresses Euler's influence on
   mathematics: "Read Euler, read Euler, he is a master for us all".

   Euler was featured on the sixth series of the Swiss 10- franc banknote
   and on numerous Swiss, German, and Russian postage stamps. The asteroid
   2002 Euler was named in his honour.

Biography

Childhood

   Swiss 10 Franc banknote honoring Euler, the most successful Swiss
   mathematician in history.
   Enlarge
   Swiss 10 Franc banknote honoring Euler, the most successful Swiss
   mathematician in history.

   Euler’s parents were Paul Euler, a pastor of the Reformed Church, and
   Marguerite Brucker, a pastor's daughter. He had two younger sisters
   named Anna Maria and Maria Magdalena. Soon after the birth of Leonhard,
   the Eulers moved from Basel to the town of Riehen, where Euler spent
   most of his childhood. Paul Euler was a family friend of the
   Bernoullis, and Johann Bernoulli, who was then regarded as Europe's
   foremost mathematician, would eventually be an important influence on
   the young Leonhard. His early formal education started 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 in 1723,
   received a masters of philosophy degree with a dissertation that
   compared the philosophies of Descartes and Newton. At this time, he was
   receiving Saturday afternoon lessons from Johann Bernoulli who quickly
   discovered his new pupil's incredible talent for mathematics.

   Euler was at this point studying theology, Greek, and Hebrew at his
   father's urging, in order to become a pastor. Johann Bernoulli
   intervened, and convinced Paul Euler that Leonhard was destined to
   become a great mathematician. In 1726, Euler completed his Ph.D.
   dissertation on the propagation of sound and 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 won second
   place, losing only to Pierre Bouguer—a man now known as "the father of
   naval architecture". Euler, however, would eventually win the coveted
   annual prize twelve times in his career.

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.
   In July 1726, Nicolas died of appendicitis after spending 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 he unsuccessfully applied for a physics
   professorship at the University of Basel.
   1957 stamp of the former Soviet Union commemorating the 250th birthday
   of Euler.
   Enlarge
   1957 stamp of the former Soviet Union commemorating the 250th birthday
   of Euler.

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

   The Academy at St. Petersburg, established by Peter the Great, 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 Euler and his colleagues.

   Conditions improved slightly upon the death of Peter II, and Euler
   swiftly rose through the ranks in the academy and was made professor of
   physics in 1731. Two years later, Daniel Bernoulli, who was fed up with
   the censorship and hostility he faced at St. Petersburg, left for
   Basel. Euler succeeded him as the head of the mathematics department.

   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.

Berlin

   Stamp of the former German Democratic Republic honoring Euler on the
   200th anniversary of his death. In the middle, it is showing his
   polyhedral formula.
   Enlarge
   Stamp of the former German Democratic Republic honoring Euler on the
   200th anniversary of his death. In the middle, it is showing his
   polyhedral formula.

   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 and lived 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 Introductio in analysin
   infinitorum, a text on functions published in 1748 and the
   Institutiones calculi differentialis, a work on differential calculus.

   In addition, Euler was asked to tutor the Princess of Anhalt-Dessau,
   Frederick's niece. He 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 his 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.

   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 him as
   unsophisticated especially in comparison to the circle of philosophers
   the German king brought to the Academy. Voltaire was among those in
   Frederick's employ, and the Frenchman enjoyed a favored position in the
   king's social circle. Euler, a simple religious man and a hard worker,
   was in many ways the direct opposite of Voltaire. Euler had very
   limited training in rhetoric and tended to debate matters that he knew
   little about, making him a frequent target of Voltaire's wit. Frederick
   also expressed disappointment with Euler's practical engineering
   abilities:

     I wanted to have a water jet in my garden: Euler calculated the
     force of the wheels necessary to raise the water to a reservoir,
     from where it should fall back through channels, finally spurting
     out in Sans Souci. My mill was carried out geometrically and could
     not raise a mouthful of water closer than fifty paces to the
     reservoir. Vanity of vanities! Vanity of geometry!

Eyesight deterioration

   A 1753 portrait by Emanuel Handmann. This portrayal suggests problems
   of the right eyelid and that Euler is perhaps suffering from
   strabismus. The left eye appears healthy, as it was a later cataract
   that destroyed it.
   Enlarge
   A 1753 portrait by Emanuel Handmann. This portrayal suggests problems
   of the right eyelid and that Euler is perhaps suffering from
   strabismus. The left eye appears healthy, as it was a later cataract
   that destroyed it.

   Euler's eyesight worsened throughout his mathematical career. Three
   years after suffering a near-fatal fever in 1735 he became nearly blind
   in his right eye, but Euler rather blamed his condition on the
   painstaking work on cartography he performed 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 a few weeks after its discovery. Even so, his condition
   appeared to have little effect on his productivity, as he compensated
   for it with his mental calculation skills and photographic memory. For
   example, Euler could repeat the Aeneid of Virgil from beginning to end
   without hesitation, and for every page in the edition he used could
   indicate which line was the first and which the last.

Return to Russia

   Euler's grave at the Alexander Nevsky Monastery.
   Enlarge
   Euler's grave at the Alexander Nevsky Monastery.

   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 and spent the rest of his life in Russia.
   His second stay in the country was marred by tragedy. A 1771 fire in
   St. Petersburg cost him his home and almost his life. In 1773, he lost
   his wife of 40 years. Euler would eventually remarry three years later.

   On September 18, 1783, Euler passed away after suffering a brain
   hemorrhage and was buried in the Alexander Nevsky Monastery. His eulogy
   was written for the French Academy by the French mathematician and
   philosopher Marquis de Condorcet, and an account of his life, with a
   list of his works, by Nikolaus von Fuss, Euler's son-in-law and the
   secretary of the Imperial Academy of St. Petersburg. Condorcet
   commented,

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

Contributions to mathematics

   Euler worked in almost all areas of mathematics: geometry, calculus,
   trigonometry, algebra, and number theory, not to mention continuum
   physics, lunar theory and other areas of physics. His importance in the
   history of mathematics cannot be overstated: if printed, his works,
   many of which are of fundamental interest, would occupy between 60 and
   80 quarto volumes and Euler's name is associated with an impressive
   number of topics. The 20th century Hungarian mathematician Paul Erdős
   is perhaps the only other mathematician who could be considered to be
   as prolific.

Mathematical notation

   Euler introduced and popularized several notational conventions through
   his numerous and widely circulated textbooks. Most notably, he
   introduced the concept of a function and was the first to write f(x) to
   denote the function f applied to the argument x. He also introduced the
   modern notation for the trigonometric functions, the letter e for the
   base of the natural logarithm (now also known as Euler's number), the
   Greek letter Σ for summations and the letter i to denote the imaginary
   unit. The use of the Greek letter π to denote the ratio of a circle's
   circumference to its diameter was also popularized by Euler, although
   it did not originate with him.

Analysis

   The development of calculus was at the forefront of 18th century
   mathematical research, and the Bernoullis—family friends of Euler—were
   responsible for much of the early progress in the field. Thanks to
   their influence, studying calculus naturally became the major focus of
   Euler's work. While some of Euler's proofs may not have been acceptable
   under modern standards of rigour, his ideas led to many great advances.

   He 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. His daring (and, by modern standards,
   technically incorrect) use of power series enabled him to solve the
   famous Basel problem in 1735:

          \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}

   A geometric interpretation of Euler's formula
   Enlarge
   A geometric interpretation of Euler's formula

   Euler introduced the use of the exponential function and logarithms in
   analytic proofs. He discovered ways to express various logarithmic
   functions in terms of power series, and successfully defined logarithms
   for negative and complex numbers, thus greatly expanding the scope
   where logarithms could be applied in mathematics. He also defined the
   exponential function for complex numbers and discovered its relation to
   the trigonometric functions. For any real number Φ, Euler's formula
   states that the complex exponential function satisfies

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

   A special case of the above formula is known as Euler's identity,

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

   called "the most remarkable formula in mathematics" by Richard Feynman,
   for its single uses of the notions of addition, multiplication,
   exponentiation, and equality, and the single uses of the important
   constants 0, 1, e, i, and π.

   In addition, Euler elaborated the theory of higher transcendental
   functions by introducing the gamma function and introduced a new method
   for solving quartic equations. He also found a way to calculate
   integrals with complex limits, foreshadowing the development of modern
   complex analysis, and 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 mathematics
   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 led to the
   development of the prime number theorem.

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 his early work on number theory was based on the works of Pierre de
   Fermat, and developed some of Fermat's ideas while 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
   Lagrange's four-square theorem. He also invented the totient function
   φ(n) which assigns to a positive integer n the number of positive
   integers less than n and coprime to n. Using properties of this
   function he was able to generalize Fermat's little theorem to what
   would become known as Euler's theorem. He further contributed
   significantly to the understanding of perfect numbers, which had
   fascinated mathematicians since 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 his ideas paved the way for Carl Friedrich Gauss.

Graph theory

   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 a problem known as the seven bridges of
   Königsberg. 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. It is not; and
   therefore not an Eulerian circuit. This solution is considered to be
   the first theorem of graph theory and planar graph theory. Euler also
   introduced the notion now known as the Euler characteristic of a space
   and a formula relating the number of edges, vertices, and faces of a
   convex polyhedron with this constant. The study and generalization of
   this formula, specifically by Cauchy and L'Huillier, is at the origin
   of topology.

Applied mathematics

   Some of Euler's greatest successes were in using analytic methods to
   solve real world problems, describing numerous applications of
   Bernoulli's numbers, Fourier series, Venn diagrams, Euler numbers, e
   and π constants, continued fractions and integrals. He integrated
   Leibniz's differential calculus with Newton's method of fluxions, and
   developed tools that made it easier to apply calculus to physical
   problems. He made great strides in improving the 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. He 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 part of his work, however, did not receive wide
   attention and was once described as too mathematical for musicians and
   too musical for mathematicians.

Physics and astronomy

   Euler helped develop the Euler-Bernoulli beam equation, which became a
   cornerstone of engineering. Aside from successfully applying his
   analytic tools to problems in classical mechanics, Euler also applied
   these techniques to celestial problems. His work in astronomy was
   recognized by a number of Paris Academy Prizes over the course of his
   career. His accomplishments include determining with great accuracy the
   orbits of comets and other celestial bodies, understanding the nature
   of comets, and calculating the parallax of the sun. His calculations
   also contributed to the development of accurate longitude tables

   In addition, Euler made important contributions in optics. He disagreed
   with Newton's corpuscular theory of light in the Opticks, which was
   then the prevailing theory. His 1740's papers on optics helped ensure
   that the wave theory of light proposed by Christian Huygens would
   become the dominant mode of thought, at least until the development of
   the quantum theory of light.

Logic

   He is also credited with using closed curves to illustrate syllogistic
   reasoning [1768].These diagrams have become known as Euler diagrams

Philosophy and religious beliefs

   Euler and his friend Daniel Bernoulli were opponents 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; he went so far as to label Wolff's ideas as "heathen
   and atheistic".

   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).
   These works present Euler as a staunch Christian and a biblical
   literalist (for example, the Rettung was primarily an argument for the
   divine inspiration of scripture).

   There is a famous anecdote inspired by Euler's arguments with secular
   philosophers over religion, which is set during Euler's second stint at
   the St. Petersburg academy. The French philosopher Denis Diderot was
   visiting Russia on Catherine the Great's invitation. However, 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: he 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 were gibberish (or so the story says),
   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. However amusing the anecdote may be, it is
   almost certainly false, given that Diderot was actually a capable
   mathematician.
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