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{{Infobox Scientist
|box_width         = 300px
|name              = Leonhard Euler
|image             = Leonhard Euler 2.jpg
|caption           = Portrait by [[Emanuel Handmann]] 1756(?)
|birth_date        = {{birth date|df=yes|1707|4|15}}
|birth_place       = [[Basel]], [[Switzerland]]
|death_date        = {{death date and age|1783|9|18|1707|4|15|df=y}}<br/><small><nowiki>[</nowiki>[[Old Style and New Style dates|OS]]: 7 September 1783<nowiki>]</nowiki></small>
|death_place       = [[St. Petersburg]], [[Russia]]
|residence         = [[Kingdom of Prussia|Prussia]], [[Russian Empire|Russia]]</br> [[Old Swiss Confederacy|Switzerland]]
|citizenship       = 
|nationality       = [[Old Swiss Confederacy|Swiss]]
|ethnicity         = 
|field             = [[Mathematician]] and [[Physicist]]
|work_institutions = [[Russian Academy of Sciences|Imperial Russian Academy of Sciences]]</br> [[Prussian Academy of Sciences|Berlin Academy]]
|alma_mater        = [[University of Basel]]
|doctoral_advisor  = [[Johann Bernoulli]]
|doctoral_students = [[Joseph Louis Lagrange]]
|known_for         = [[List of topics named after Leonhard Euler|See full list]]
|author_abbrev_bot = 
|author_abbrev_zoo = 
|influences        = 
|influenced        = 
|prizes            = 
|religion          = [[Calvinism|Calvinist]]<ref>{{cite book|title=Scientists of Faith|author=Dan Graves|location=Grand Rapids, MI|year=1996|publisher=Kregel Resources|pages=85–86}}</ref><ref>{{cite book|title=Men of Mathematics, Vol. 1|author=E. T. Bell|location=London|year=1953|publisher=Penguin|pages=155}}</ref>
|footnotes         = He is the father of the mathematician [[Johann Euler]]</br>He is listed by academic genealogy authorities as the equivalent to the doctoral advisor of Joseph Louis Lagrange.
|signature         = Euler's signature.svg
}}

'''Leonhard Paul Euler''' (15 April 1707 – 18 September 1783) was a pioneering [[Swiss (people)|Swiss]] [[mathematician]] and [[physicist]] who spent most of his life in [[Russia]] and [[Germany]].  His surname is {{pron-en|ˈɔɪlər}} {{respell|OY|lər}} in English<ref>"Euler", [[Oxford English Dictionary]], second edition, Oxford University Press, 1989.</ref><ref>[http://www.merriam-webster.com/dictionary/Euler "Euler"], [[Webster's Dictionary|Merriam–Webster's Online Dictionary]], 2009.</ref><ref>[http://www.bartleby.com/61/71/E0237100.html "Euler, Leonhard"], [[The American Heritage Dictionary of the English Language]], fourth edition, Houghton Mifflin Company, Boston, 2000.</ref><ref>{{cite book|title=Nets, Puzzles, and Postmen: An Exploration of Mathematical Connections|author=Peter M. Higgins|year=2007|publisher=Oxford University Press|pages=43}}</ref> ({{IPA-de|ˈɔʏlɐ}}); the common English pronunciation {{IPA-en|ˈjuːlər|}} {{respell|EW|lər}} is incorrect.<ref>{{cite web| url=http://binomial.csuhayward.edu/EulerNote3.html| title =Pascal's Triangle From Top To Bottom - the binomial coefficient website| dateformat =dmy| accessdate =22 April 2009| author=Matthew Hubbard and Tom Roby}}</ref> <!-- Yes, this section doesn't exactly align with WP:MOS and isn't referenced, but please see the "English pronunciation" section of the talk page before changing it.  Thanks. -->

Euler made important discoveries in fields as diverse as [[calculus]] and [[graph theory]]. He also introduced much of the modern mathematical terminology and notation, particularly for [[mathematical analysis]], such as the notion of a [[function (mathematics)|mathematical function]].<ref name="function">{{cite book| last = Dunham| first = William | authorlink=William Dunham (mathematician) | title = Euler: The Master of Us All| year = 1999| publisher =The Mathematical Association of America | pages = 17}}</ref> He is also renowned for his work in [[mechanics]], [[fluid dynamics]], [[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 (text)|quarto]] volumes.<ref name="volumes">{{cite journal|last = Finkel|first = B.F.|year = 1897|title = Biography- Leonard Euler|journal = The American Mathematical Monthly| volume = 4| issue = 12| pages = 300|doi = 10.2307/2968971}}</ref> A statement attributed to [[Pierre-Simon Laplace]] expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master [i.e., teacher] of us all."<ref name="Laplace">{{cite book| last = Dunham| first = William| title = Euler: The Master of Us All| year = 1999 | publisher =The Mathematical Association of America | pages = xiii | quote=Lisez Euler, lisez Euler, c'est notre maître à tous.| nopp = true}}</ref> 

Euler was  featured on the sixth series of the Swiss 10-[[Swiss franc|franc]] banknote and on numerous Swiss, German, and Russian [[postage stamp]]s. The [[asteroid]] [[2002 Euler]] was named in his honor. He is also commemorated by the [[Lutheran Church]] on their [[Calendar of Saints (Lutheran)|Calendar of Saints]] on 24 May - he was a devout Christian (and believer in [[biblical inerrancy]]) who wrote [[apologetics]] and argued forcefully against the prominent atheists of his time.{{Citation needed|date=July 2009}}

== Life ==
<!-- The biography could use more correlation with his mathematical activities. When was his most prolific period and discoveries, and how did in fit in with his general life? -->

===Early years===
[[Image:Euler-10 Swiss Franc banknote (front).jpg|thumb|Old Swiss 10 Franc banknote honoring Euler]] 
Euler was born in [[Basel, Switzerland|Basel]] to 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 friend of the [[Bernoulli family]]&mdash;[[Johann Bernoulli]], who was then regarded as Europe's foremost [[mathematician]], would eventually be the most important influence on young Leonhard.  Euler's 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 his [[M.Phil]] with a dissertation that compared the philosophies of [[René Descartes|Descartes]] and [[Isaac Newton|Newton]]. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics.<ref name="childhood">{{cite book |last= James |first= Ioan |title= Remarkable Mathematicians: From Euler to von Neumann |publisher= Cambridge |year= 2002|pages=2 |isbn= 0-521-52094-0}}</ref> Euler was at this point studying [[theology]], [[Greek language|Greek]], and [[Hebrew language|Hebrew]] at his father's urging, in order to become a pastor, but Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed his Ph.D. dissertation on the [[Speed of sound|propagation of sound]] with the title ''De Sono''<ref>{{PDFlink|[http://www.17centurymaths.com/contents/euler/e002tr.pdf Translation of Euler's Ph.D in English by Ian Bruce]|232&nbsp;[[Kibibyte|KiB]]<!-- application/pdf, 237913 bytes -->}}</ref> and in 1727, he entered the ''[[French Academy of Sciences|Paris Academy]] Prize Problem'' competition, where the problem that year was to find the best way to place the [[Mast (sailing)|mast]]s on a ship. He won second place, losing only to [[Pierre Bouguer]]&mdash;who is now known as "the father of naval architecture". Euler subsequently won this coveted annual prize twelve times in his career.<ref name="prize">{{cite journal| author = Calinger, Ronald | year = 1996| title = Leonhard Euler: The First St. Petersburg Years (1727–1741)| journal = Historia Mathematica| volume = 23| issue = 2| pages = 156 | doi = 10.1006/hmat.1996.0015}}</ref>

===St. Petersburg===
Around this time Johann Bernoulli's two sons, [[Daniel Bernoulli|Daniel]] and [[Nicolaus II Bernoulli|Nicolas]], were working at the [[Russian Academy of Sciences|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 had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.<ref name="stpetersburg">{{cite journal| author = Calinger, Ronald | year = 1996| title = Leonhard Euler: The First St. Petersburg Years (1727–1741)| journal = Historia Mathematica| volume = 23| issue = 2| pages = 125 | doi = 10.1006/hmat.1996.0015}}</ref>

[[Image:Euler-USSR-1957-stamp.jpg|thumb|1957 stamp of the former [[Soviet Union]] commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician and academician, Leonhard Euler.]]
Euler arrived in the Russian capital on 17 May 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 language|Russian]] and settled into life in St Petersburg. He also took on an additional job as a medic in the [[Russian Navy]].<ref name="medic">{{cite journal| author = Calinger, Ronald | year = 1996| title = Leonhard Euler: The First St. Petersburg Years (1727–1741)| journal = Historia Mathematica| volume = 23| issue = 2| pages = 127 | doi = 10.1006/hmat.1996.0015}}</ref>

The Academy at St. Petersburg, established by [[Peter I of Russia|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.<ref name="prize">{{cite journal| author = Calinger, Ronald | year = 1996| title = Leonhard Euler: The First St. Petersburg Years (1727–1741)| journal = Historia Mathematica| volume = 23| issue = 2| pages = 124 | doi = 10.1006/hmat.1996.0015}}</ref> 

The Academy's benefactress, [[Catherine I of Russia|Catherine I]], who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility then gained power upon the ascension of the twelve-year-old [[Peter II of Russia|Peter II]]. The nobility were suspicious of the academy's foreign scientists, and thus cut funding and caused 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.<ref name="promotion">{{cite journal| author = Calinger, Ronald | year = 1996| title = Leonhard Euler: The First St. Petersburg Years (1727–1741)| journal = Historia Mathematica| volume = 23| issue = 2| pages = 128–129 | doi = 10.1006/hmat.1996.0015}}</ref> 

On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of [[Georg Gsell]], a painter from the Academy Gymnasium.<ref>{{Cite book | first1=I.R. | last1=Gekker | first2=A.A. | last2=Euler | contribution=Leonhard Euler's family and descendants | editor1-first=N.N. | editor1-last=Bogoliubov | editor2-first=G.K. | editor2-last=Mikhaĭlov | editor3-first=A.P. |editor3-last=Yushkevich | title=Euler and modern science | publisher=Mathematical Association of America | year=2007 | isbn=088385564X }}, p. 402.</ref> The young couple bought a house by the [[Neva River]].  Of their thirteen children, only five survived childhood.<ref name="wife">{{cite web| url=http://www-history.mcs.st-and.ac.uk/~history/Extras/Euler_Fuss_Eulogy.html| title = Eulogy of Euler by Fuss| dateformat =dmy| accessdate =30 August 2006| last = Fuss| first = Nicolas}}</ref>

===Berlin===
[[Image:Euler GDR stamp.jpg|thumb|left|Stamp of the former [[German Democratic Republic]] honoring Euler on the 200th anniversary of his death. In the middle, it shows his [[Planar graph#Euler's formula|polyhedral formula]] <math>V-E+F=2</math>.]]

Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the ''[[Prussian Academy of Sciences|Berlin Academy]]'', which he had been offered by [[Frederick the Great of Prussia]].  He lived for 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]]'',<ref>{{cite web| title = E212 -- Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum|publisher=Dartmouth|url=http://www.math.dartmouth.edu/~euler/pages/E212.html}}</ref> published in 1755 on [[differential calculus]].<ref name="Friedrich">{{cite book| last = Dunham| first = William | title = Euler: The Master of Us All | year = 1999| publisher =The Mathematical Association of America | pages = xxiv–xxv| nopp = true }}</ref> In 1755, he was elected a foreign member of the [[Royal Swedish Academy of Sciences]].

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 entitled ''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 insights into Euler's personality and religious beliefs. This book became more widely read than any of his mathematical works, and it was published 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.<ref name="Friedrich"/>

Despite Euler's immense contribution to the Academy's prestige, he was eventually forced to leave Berlin. This was partly because of a conflict of personality with Frederick, who came to regard Euler 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 prominent position in the king's social circle. Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes. He was in many ways the direct opposite of Voltaire. Euler had limited training in [[rhetoric]], and tended to debate matters that he knew little about, making him a frequent target of Voltaire's wit.<ref name="Friedrich"/> [[Frederick II of Prussia|Frederick]] also expressed disappointment with Euler's practical engineering abilities: 

{{cquote|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 [[Sanssouci]]. 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!<ref>{{cite book | title=Letters of Voltaire and Frederick the Great, Letter H 7434, 25 January 1778 | author=[[Frederick II of Prussia]] | translator=[[Richard Aldington]] | publisher=Brentano's | location=New York | year=1927 }}</ref>}}

[[Image:Leonhard Euler.jpg|thumb|A 1753 portrait by [[Emanuel Handmann]]. This portrayal suggests problems of the right eyelid, and possible [[strabismus]]. The left eye appears healthy; it was later affected by a cataract.<ref name="blind"> {{cite journal|author = Calinger, Ronald | year = 1996| title = Leonhard Euler: The First St. Petersburg Years (1727–1741)| journal = Historia Mathematica| volume = 23| issue = 2| pages = 154–155 | doi = 10.1006/hmat.1996.0015}}</ref>]]

===Eyesight deterioration===
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 in 1766. Even so, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and [[Eidetic memory|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 could indicate which line was the first and which the last. With the aid of his scribes, Euler's productivity on many areas of study actually increased. He produced on average one mathematical paper every week in the year 1775.<ref name="volumes"/>

===Return to Russia===
[[Image:Euler Grave at Alexander Nevsky Monastry.jpg|thumb|Euler's grave at the [[Alexander Nevsky Lavra]]]] 
The situation in Russia had improved greatly since the accession to the throne of [[Catherine II of Russia|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 fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife of 40 years. Three years after his wife's death Euler married her half sister, Salome Abigail Gsell (1723–1794).<ref>{{Cite book | first1=I.R. | last1=Gekker | first2=A.A. | last2=Euler | contribution=Leonhard Euler's family and descendants | editor1-first=N.N. | editor1-last=Bogoliubov | editor2-first=G.K. | editor2-last=Mikhaĭlov | editor3-first=A.P. |editor3-last=Yushkevich | title=Euler and modern science | publisher=Mathematical Association of America | year=2007 | isbn=088385564X }}, p. 405.</ref> This marriage would last until his death.  

On 18 September 1783, Euler died in St. Petersburg after suffering a [[brain hemorrhage]], and was buried with his wife in the Smolensk Lutheran Cemetery on [[Vasilievsky Island]] (the Soviets destroyed the cemetery after transferring Euler's remains to the Orthodox [[Alexander Nevsky Lavra]]).  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 [[Russian Academy of Sciences|Imperial Academy of St. Petersburg]]. Condorcet commented,

{{cquote|''…il cessa de calculer et de vivre'' — … he ceased to calculate and to live.<ref name=condorcet>{{cite web| url = http://www.math.dartmouth.edu/~euler/historica/condorcet.html| title = Eulogy of Euler - Condorcet| dateformat =dmy| accessdate =30 August 2006| author =Marquis de Condorcet}}</ref>}}

==Contributions to mathematics==

{{E (mathematical constant)}}

Euler worked in almost all areas of mathematics: [[geometry]], [[calculus]], [[trigonometry]], [[algebra]], and [[number theory]], as well as [[continuum physics]], [[lunar theory]] and other areas of [[physics]]. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 [[Quarto (text)|quarto]] volumes.<ref name="volumes"/> Euler's name is associated with a [[List of topics named after Leonhard Euler|large number of topics]].

===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 (mathematics)|function]]<ref name="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 [[Sigma|Σ]] for summations and the letter <math>i</math> to denote the [[imaginary unit]].<ref name=Boyer>{{cite book|title = A History of Mathematics|last= Boyer|first=Carl B.|coauthors= Uta C. Merzbach|publisher= [[John Wiley & Sons]]|isbn= 0-471-54397-7|pages = 439–445|year = 1991}}</ref> The use of the Greek letter [[pi (letter)|π]] to denote the [[pi|ratio of a circle's circumference to its diameter]] was also popularized by Euler, although it did not originate with him.<ref name="pi">{{cite web| url = http://www.stephenwolfram.com/publications/talks/mathml/mathml2.html| title = Mathematical Notation: Past and Future| accessmonth = August| accessyear=2006| last = Wolfram| first = Stephen}}</ref>

===Analysis===
The development of [[calculus]] was at the forefront of 18th century mathematical research, and the [[Bernoulli family|Bernoulli]]s—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of [[mathematical rigor|mathematical rigour]],<ref name = "Basel"/> his ideas led to many great advances.
Euler is well-known in [[Mathematical analysis|analysis]] for his frequent use and development of [[power series]], the expression of functions as sums of infinitely many terms, such as

:<math>e^x = \sum_{n=0}^\infty {x^n \over n!} = \lim_{n \to \infty}\left(\frac{1}{0!} + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}\right).</math>

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:<ref name="Basel">{{cite book| last = Wanner| first = Gerhard| coauthors = Harrier, Ernst | title = Analysis by its history| edition = 1st| year = 2005| month = March| publisher = Springer| pages = 62}}</ref>

:<math>\sum_{n=1}^\infty {1 \over n^2} = \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}.</math>

[[Image:Euler's formula.svg|thumb|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 using power series, and he successfully defined logarithms for negative and [[complex number]]s, thus greatly expanding the scope of mathematical applications of logarithms.<ref name=Boyer>{{cite book|title = A History of Mathematics|last= Boyer|first=Carl B.|coauthors= Merzbach, Uta C. |publisher= [[John Wiley & Sons]]|isbn= 0-471-54397-7|pages = 439–445|year = 1991}}</ref> He also defined the exponential function for complex numbers, and discovered its relation to the [[trigonometric function]]s. For any [[real number]] [[φ]], [[Euler's formula]] states that the [[Exponential function#On the complex plane|complex exponential]] function satisfies

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

A special case of the above formula is known as [[Euler's identity]],
:<math>e^{i \pi} +1 = 0 \, </math>
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.<ref name="Feynman">
{{cite book |last= Feynman|first= Richard|title= The Feynman Lectures on Physics: Volume I |origmonth= June|pages=10 |chapter= Chapter 22: Algebra |year=1970}}</ref>  In 1988, readers of the ''[[Mathematical Intelligencer]]'' voted it "the Most Beautiful Mathematical Formula Ever".<ref name=MathInt/> In total, Euler was responsible for three of the top five formulae in that poll.<ref name=MathInt>{{cite journal | last= Wells | first= David | year= 1990 | title = Are these the most beautiful? | journal = Mathematical Intelligencer | volume = 12 | issue = 3 | pages= 37–41 | doi= 10.1007/BF03024015 }}<br>{{cite journal | last= Wells | first= David | year= 1988 | title = Which is the most beautiful? | journal = Mathematical Intelligencer | volume = 10 | issue = 4 | pages= 30–31 | doi= 10.1007/BF03023741 }}<br> See also: {{cite url | url = http://www.maa.org/mathtourist/mathtourist_03_12_07.html | title = The Mathematical Tourist | accessmonth = March | accessyear = 2008 | last = Peterson | first = Ivars }}</ref>

[[De Moivre's formula]] is a direct consequence of [[Euler's formula]].

In addition, Euler elaborated the theory of higher [[transcendental function]]s by introducing the [[gamma function]] and introduced a new method for solving [[quartic equation]]s. 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 best-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 functions|hyperbolic trigonometric functions]] and the analytic theory of [[generalized continued fraction|continued fractions]]. For example, he proved the [[infinitude of primes]] using the divergence of the [[Harmonic series (mathematics)|harmonic series]], and he 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]].<ref name="analysis">{{cite book| last = Dunham| first = William| title = Euler: The Master of Us All | year = 1999| publisher =The Mathematical Association of America | chapter = 3,4 }}</ref>

=== Number theory ===
Euler's interest in number theory can be traced to the influence of [[Christian Goldbach]], his friend in the St. Petersburg Academy.  A lot of Euler's early work on number theory was based on the works of [[Pierre de Fermat]].  Euler developed some of Fermat's ideas, and disproved some of his conjectures.

Euler linked the nature of prime distribution with ideas in analysis.  He proved that [[Proof that the sum of the reciprocals of the primes diverges|the sum of the reciprocals of the primes diverges]].  In doing so, he discovered the connection between the Riemann zeta function and the prime numbers; this is known as the [[Proof of the Euler product formula for the Riemann zeta function|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 he made distinct contributions to [[Lagrange's four-square theorem]].  He also invented the [[totient function]] φ(''n'') which is the number of positive integers less than or equal to the integer ''n'' that are [[coprime]] to ''n''.  Using properties of this function, he generalized Fermat's little theorem to what is now known as [[Euler's theorem]].  He contributed significantly to the theory of [[perfect numbers]], which had fascinated mathematicians since [[Euclid]].  Euler also made progress toward the [[prime number theorem]], and he conjectured the law of [[quadratic reciprocity]].  The two concepts are regarded as fundamental theorems of number theory, and his ideas paved the way for the work of [[Carl Friedrich Gauss]].<ref name="numbertheory">{{cite book| last = Dunham| first = William| title = Euler: The Master of Us All | year = 1999| publisher =The Mathematical Association of America | chapter = 1,4}}</ref>

By 1772 Euler had proved that 2<sup>31</sup>&nbsp;&minus;&nbsp;1 = [[2147483647|2,147,483,647]] is a [[Mersenne prime]]. It may have remained the [[largest known prime]] until 1867.<ref>Caldwell, Chris. [http://primes.utm.edu/notes/by_year.html ''The largest known prime by year'']</ref>

===Geometry===
* [[Euler line]]
* [[Euler's circle]]
* [[Affine geometry]]

===Graph theory===
{{Seealso|Seven Bridges of Königsberg}}
[[Image:Konigsberg bridges.png|frame|right|Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges.]]
In 1736, Euler solved the problem known as the [[Seven Bridges of Königsberg]].<ref name="bridge">{{cite journal| last = Alexanderson| first = Gerald| year = 2006| month = July| title = Euler and Königsberg's bridges: a historical view| journal = Bulletin of the American Mathematical Society| doi = 10.1090/S0273-0979-06-01130-X| volume = 43| pages = 567}}</ref> The city of [[Königsberg]], [[Kingdom of Prussia|Prussia]] was set on the [[Pregel]] River, and included two large islands which were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not: there is no [[Eulerian path|Eulerian circuit]]. This solution is considered to be the first theorem of [[graph theory]], specifically of [[planar graph]] theory.<ref name="bridge"/>

Euler also discovered the [[Planar_graph#Euler.27s_formula|formula]] ''V''&nbsp;&minus;&nbsp;''E''&nbsp;+&nbsp;''F''&nbsp;=&nbsp;2 relating the number of vertices, edges, and faces of a convex [[polyhedron]]<ref>{{cite book|title=Polyhedra|author=Peter R. Cromwell|location=Cambridge|year=1997|publisher=Cambridge University Press|pages=189–190}}</ref>, and hence of a [[planar graph]] (for a planar graph, ''V''&nbsp;&minus;&nbsp;''E''&nbsp;+&nbsp;''F''&nbsp;=&nbsp;1).  The constant in this formula is now known as the [[Euler characteristic]] for the graph (or other mathematical object), and is related to the [[genus (mathematics)|genus]] of the object.<ref>{{cite book|title=Algorithmic Graph Theory|author=Alan Gibbons|location=Cambridge|year=1985|publisher=Cambridge University Press|pages=72}}</ref> The study and generalization of this formula, specifically by [[Augustin Louis Cauchy|Cauchy]]<ref name="Cauchy">{{cite journal|author=Cauchy, A.L.|year=1813|title=Recherche sur les polyèdres—premier mémoire|journal=Journal de l'Ecole Polytechnique|volume= 9 (Cahier 16)|pages=66–86}}</ref> and [[Simon Antoine Jean L'Huillier|L'Huillier]],<ref name="Lhuillier">{{cite journal|author=L'Huillier, S.-A.-J.|title=Mémoire sur la polyèdrométrie|journal=Annales de Mathématiques|volume=3|year=1861|pages=169–189}}</ref> is at the origin of [[topology]].

===Applied mathematics===
Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the [[Bernoulli numbers]], [[Fourier series]], [[Venn diagrams]], [[Euler numbers]], the constants [[E (mathematical constant)|e]] and [[pi|π]], continued fractions and integrals. He integrated [[Gottfried Leibniz|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]]: 

:<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>

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 incorporate [[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.<ref name="music">{{cite journal| author = Calinger, Ronald | year = 1996| title = Leonhard Euler: The First St. Petersburg Years (1727–1741)| journal = Historia Mathematica| volume = 23| issue = 2| pages = 144–145 | doi = 10.1006/hmat.1996.0015}}</ref>

=== Physics and astronomy ===
{{Classical mechanics|cTopic=Scientists}}
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 [[solar parallax|parallax]] of the sun.  His calculations also contributed to the development of accurate [[Lunar_distance (navigation)#History|longitude tables]].<ref>Youschkevitch, A P; Biography in ''Dictionary of Scientific Biography'' (New York 1970–1990).</ref>

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 1740s 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 [[wave-particle duality|quantum theory of light]].<ref name="optics">{{cite journal
| author = Home, R.W.
| year = 1988
| title = Leonhard Euler's 'Anti-Newtonian' Theory of Light
| journal = Annals of Science
| volume = 45
| issue = 5
| pages = 521–533 | doi = 10.1080/00033798800200371
}}</ref>

===Logic===
He is also credited with using [[closed curve]]s to illustrate [[syllogism|syllogistic]] reasoning (1768). These diagrams have become known as [[Euler diagram]]s.<ref name="logic"> Baron, M. E.; A Note on The Historical Development of Logic Diagrams. The Mathematical Gazette: The Journal of the Mathematical Association. Vol LIII, no. 383 May 1969.</ref>

==Personal philosophy and religious beliefs==
Euler and his friend [[Daniel Bernoulli]] were opponents of [[Gottfried Leibniz|Leibniz's]] [[monadism]] and the philosophy of [[Christian Wolff (philosopher)|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".<ref name="wolff">{{cite journal| author = Calinger, Ronald  | year = 1996| title = Leonhard Euler: The First St. Petersburg Years (1727–1741)| journal = Historia Mathematica| volume = 23| issue = 2| pages = 153–154 | doi = 10.1006/hmat.1996.0015}}</ref>

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 show that Euler was a devout [[Christian]] who believed the Bible to be inspired; the ''Rettung'' was primarily an argument for the [[Biblical inspiration|divine inspiration of scripture]].<ref name="theology">{{cite journal| last = Euler| first = Leonhard | editor = Orell-Fussli| year = 1960| title = Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister| journal = Leonhardi Euleri Opera Omnia (series 3)| volume = 12 }}</ref>

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, <math>\frac{a+b^n}{z}=x</math>, hence God exists—reply!". Diderot, to whom (says the story) 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. However amusing the anecdote may be, it is apocryphal, given that Diderot was a capable mathematician who had published mathematical treatises.<ref name="diderot">{{cite journal| last = Brown | first = B.H.| year = 1942| month = May| title = The Euler-Diderot Anecdote| journal =The American Mathematical Monthly| volume = 49| issue = 5| pages = 302–303| doi = 10.2307/2303096}}; {{cite journal| last = Gillings | first = R.J.| year = 1954| month = February| title = The So-Called Euler-Diderot Incident| journal =The American Mathematical Monthly| volume = 61| issue = 2| pages = 77–80| doi = 10.2307/2307789}}</ref>

==Selected bibliography==
[[Image:Methodus inveniendi - Leonhard Euler - 1744.jpg|thumb|The cover page of Euler's ''Methodus inveniendi lineas curvas''.]]
Euler has an [[Contributions of Leonhard Euler to mathematics#Works|extensive bibliography]] but his best known books include:
*''[http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ Elements of Algebra]''.  This elementary algebra text starts with a discussion of the nature of numbers and gives a comprehensive introduction to algebra, including formulae for solutions of polynomial equations.
*''[[Introductio in analysin infinitorum]]'' (1748). 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).
*Two influential textbooks on calculus: ''[[Institutiones calculi differentialis]]'' (1755) and ''[[Institutionum calculi integralis]]'' (1768–1770).
*''Lettres à une Princesse d'Allemagne'' (Letters to a German Princess) (1768–1772). Available [http://perso.club-internet.fr/nielrowclub-internet.fr/nielrowbooks/euler.tif online] (in French). English translation, with notes, and a life of Euler, available online from [[Google Books]]: [http://books.google.com/books?vid=09-Fi9xi6pUzqBOnQzlnRS&id=hAm5VsEeu1EC&printsec=titlepage&dq=%22Leonhard+Euler%22 Volume 1], [http://books.google.com/books?vid=OCLC00826569&id=CZLPNtEnFRcC&printsec=titlepage&dq=%22Leonhard+Euler%22 Volume 2]
*''Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti'' (1744). The Latin title translates as ''a method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense''.<ref>[http://math.dartmouth.edu/~euler/pages/E065.html E65 — Methodus… entry at Euler Archives]</ref>

A definitive collection of Euler's works, entitled ''Opera Omnia'', has been published since 1911 by the [http://www.leonhard-euler.ch/ Euler Commission] of the [[Swiss Academy of Sciences]].

== See also ==
* [[List of topics named after Leonhard Euler]]
* [[Leonhard Euler Telescope]]

==References and notes==
{{reflist|2}}

== Further reading ==
*''Lexikon der Naturwissenschaftler'', 2000. Heidelberg: Spektrum Akademischer Verlag.
*Demidov, S.S., 2005, "Treatise on the differential calculus" in [[Ivor Grattan-Guinness|Grattan-Guinness, I.]], ed., ''Landmark Writings in Western Mathematics''. Elsevier: 191&ndash;98.
*[[William Dunham (mathematician)|Dunham, William]] (1999) ''Euler: The Master of Us All'', Washington: Mathematical Association of America. ISBN 0883853280
*Fraser, Craig G., 2005, "Leonhard Euler's 1744 book on the calculus of variations" in [[Ivor Grattan-Guinness|Grattan-Guinness, I.]], ed., ''Landmark Writings in Western Mathematics''. Elsevier: 168&ndash;80.
*Gladyshev, Georgi, P. (2007) “[http://www.ceser.res.in/ijamas/cont/2007/ams-n07-cont.html  Leonhard Euler’s methods and ideas live on in the thermodynamic hierarchical theory of biological evolution,]” ''International Journal of Applied Mathematics & Statistics'' (IJAMAS) 11 (N07), Special Issue on [[Leonhard Paul Euler]]’s: Mathematical Topics and Applications (M. T. A.).
*{{cite journal | title= Leonhard Euler: his life, the man, and his works | author= W. Gautschi | year= 2008 | journal= SIAM Review | volume = 50 | issue= 1 | pages=3–33 |	doi= 10.1137/070702710 }}
*Heimpell, Hermann, Theodor Heuss, Benno Reifenberg (editors). 1956. ''Die großen Deutschen'', volume 2, Berlin: Ullstein Verlag.
*Krus, D.J. (2001) "[http://www.visualstatistics.net/Statistics/Euler/Euler.htm Is the normal distribution due to Gauss? Euler, his family of gamma functions, and their place in the history of statistics,]" ''Quality and Quantity: International Journal of Methodology,'' 35: 445&ndash;46.
*Nahin, Paul (2006) ''Dr. Euler's Fabulous Formula'', New Jersey: Princeton, ISBN 978-06-9111-822-2
*Reich, Karin, 2005, " 'Introduction' to analysis" in [[Ivor Grattan-Guinness|Grattan-Guinness, I.]], ed., ''Landmark Writings in Western Mathematics''. Elsevier: 181&ndash;90.	
*Richeson, David S. (2008) ''Euler's Gem: The Polyhedron Formula and the Birth of Topology''. Princeton University Press.
*Sandifer, Edward C. (2007), ''The Early Mathematics of Leonhard Euler'', Mathematical Association of America. IBSN 0883855593
*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
*Thiele, Rüdiger. (2005). The mathematics and science of Leonhard Euler, in ''Mathematics and the Historian's Craft: The Kenneth O. May Lectures'', G. Van Brummelen and M. Kinyon (eds.), CMS Books in Mathematics, Springer Verlag. ISBN 0-387-25284-3.
*{{cite journal | year = 1983 | month = November | title = A Tribute to Leohnard Euler 1707&ndash;1783  | journal = [[Mathematics Magazine]]  | volume = 56 | issue = 5}}

== External links ==
{{sisterlinks|s=Author:Leonhard Euler}}
*{{ScienceWorldBiography | urlname=Euler | title=Euler, Leonhard (1707–1783)}}
*[http://www.britannica.com/eb/article-9033216/Leonhard-Euler Encyclopedia Britannica article]
*{{MathGenealogy|id=38586}}
* [http://www.maa.org/news/howeulerdidit.html How Euler did it] contains columns explaining how Euler solved various problems
* [http://www.eulerarchive.org/ Euler Archive]
* [http://www.leonhard-euler.ch/ Euler Committee of the Swiss Academy of Sciences]
* [http://www-history.mcs.st-andrews.ac.uk/References/Euler.html References for Leonhard Euler]
* [http://www.euler-2007.ch/en/index.htm Euler Tercentenary 2007]
* [http://www.eulersociety.org/ The Euler Society]
* [http://www.pdmi.ras.ru/EIMI/2007/AG/ Leonhard Euler Congress 2007]—St. Petersburg, Russia
* [http://www.projecteuler.net Project Euler]
* [http://www.math.dartmouth.edu/~euler/historica/family-tree.html Euler Family Tree]
* [http://friedrich.uni-trier.de/oeuvres/20/219/ Euler's Correspondence with Frederick the Great, King of Prussia]
*[http://www.gresham.ac.uk/event.asp?PageId=45&EventId=518 "Euler - 300th anniversary lecture"], given by Robin Wilson at [[Gresham College]], 9 May 2007 (can download as video or audio files)
*{{MacTutor Biography|id=Euler}}
{{featured article}}

{{Persondata
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|SHORT DESCRIPTION=Mathematician
|DATE OF BIRTH={{birth date|df=yes|1707|4|15}}
|PLACE OF BIRTH=[[Basel]], [[Switzerland]]
|DATE OF DEATH={{death date|df=yes|1783|9|18}}
|PLACE OF DEATH=[[St Petersburg]], [[Russia]]
}}
{{DEFAULTSORT:Euler, Leonhard}}
[[Category:1707 births]]
[[Category:1783 deaths]]
[[Category:18th-century Latin writers]]
[[Category:18th-century mathematicians]]
[[Category:Blind people]]
[[Category:Calculating prodigies]]
[[Category:Fluid dynamicists]]
[[Category:Latin squares]]
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[[Category:Members of the Prussian Academy of Sciences]]
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[[Category:Number theorists]]
[[Category:People celebrated in the Lutheran liturgical calendar]]
[[Category:People from Basel-City]]
[[Category:Religion and science]]
[[Category:Saint Petersburg State University faculty]]
[[Category:Swiss mathematicians]]
[[Category:Swiss music theorists]]
[[Category:Swiss physicists]]
[[Category:University of Basel alumni]]

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