Editing Leonhard Euler (section)

{{Short description|Swiss mathematician (1707–1783)}}
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{{Infobox scientist
| name              = Leonhard Euler
| image             = Leonhard Euler - Jakob Emanuel Handmann (Kunstmuseum Basel).jpg
| caption           = Portrait by [[Jakob Emanuel Handmann]], 1753
| birth_date        = {{birth date|df=y|1707|4|15}}
| birth_place       = [[Basel]], [[Swiss&nbsp;Confederacy]]
| death_date        = {{nowrap|{{death date and age|df=y|1783|9|18|1707|4|15}}}} {{awrap|{{bracket|[[Adoption of the Gregorian calendar#Adoption in Eastern Europe|OS]]: 7 September 1783}}}}
| death_place       = [[Saint Petersburg]], [[Russian&nbsp;Empire]]
| field             = {{hlist|[[Mathematics]]|[[Physics]]}}
| work_institutions = {{ubl|[[Imperial Russian Academy of Sciences]]|[[Prussian Academy of Sciences|Berlin Academy]]}}
| education         = [[University of Basel]] ([[MPhil]])
| thesis_title      = Dissertatio physica de sono (Physical dissertation on sound)
| thesis_url        = https://scholarlycommons.pacific.edu/euler-works/2/
| thesis_year       = 1726
| academic_advisors = [[Johann Bernoulli]]
| doctoral_students = [[Johann Hennert]]
| notable_students  = {{ubl|[[Nicolas Fuss]]|[[Stepan Rumovsky]]|[[Joseph-Louis Lagrange]] (epistolary correspondent){{efn|Euler is listed by an [[academic genealogy]] as the equivalent to the [[doctoral advisor]] of Lagrange.<ref name=mathg/>}}|[[Anders Johan Lexell]]}}
| known_for         = {{hlist|[[Contributions of Leonhard Euler to mathematics|Contributions]]|[[List of things named after Leonhard Euler|namesakes]]}}
| spouse            = {{plainlist|
* {{marriage|Katharina Gsell|1734|1773|end=died}}
* {{marriage|Salome Abigail Gsell|1776}}
}}
| children          = 13, including [[Johann Euler|Johann]]
| awards            = [[Fellow of the Royal Society|FRS]] (1747)
| signature         = Euler's signature.svg
}}
'''Leonhard Euler''' ({{IPAc-en|ˈ|ɔɪ|l|ər|audio=LL-Q1860 (eng)-Flame, not lame-Leonhard Euler.wav}} {{respell|OY|lər}};{{efn|The pronunciation {{IPAc-en|ˈ|juː|l|ər}} {{respell|YOO|lər}} is considered incorrect.<ref name="oxford"/><ref name="merriam"/><ref name="amer heritage"/><ref name="nets, puzzles"/>}} {{IPA|de-CH|ˈleːɔnhard ˈɔʏlər|lang}}; {{IPA|de|ˈleːɔnhaʁt ˈɔʏlɐ|lang|De-Leonhard Euler.ogg}}; 15 April 1707{{spaced ndash}}18 September 1783) was a Swiss [[polymath]] who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of [[graph theory]] and [[topology]] and made influential discoveries in many other branches of mathematics, such as [[analytic number theory]], [[complex analysis]], and [[infinitesimal calculus]]. He also introduced much of modern mathematical terminology and [[Mathematical notation|notation]], including the notion of a [[mathematical function]].{{sfn|Dunham|1999|p=17}} He is known for his work in [[mechanics]], [[fluid dynamics]], optics, astronomy, and music theory.<ref name=":0" /> Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory".<ref name=":02">{{Cite book |last=Debnath |first=Lokenath |author-link=Lokenath Debnath |url=https://books.google.com/books?id=HPRpDQAAQBAJ&pg=PA370 |title=The Legacy of Leonhard Euler: A Tricentennial Tribute |publisher=Imperial College Press |year=2010 |isbn=978-1-84816-525-0 |location=London |pages=370 |language=en}}</ref> He spent most of his adult life in [[Saint Petersburg]], Russia, and in [[Berlin]], then the capital of [[Kingdom of Prussia|Prussia]].

Euler is credited for popularizing the Greek letter <math>\pi</math> (lowercase [[Pi (letter)|pi]]) to denote [[Pi|the ratio of a circle's circumference to its diameter]], as well as first using the notation <math>f(x)</math> for the value of a function, the letter <math>i</math> to express the [[imaginary unit]] <math>\sqrt{-1}</math>, the Greek letter <math>\Sigma</math> (capital [[sigma]]) to express [[summation]]s, the Greek letter <math>\Delta</math> (capital [[Delta (letter)|delta]]) for [[finite difference]]s, and lowercase letters to represent the sides of a triangle while representing the angles as capital letters.<ref name="assad"/> He gave the current definition of the constant <math>e</math>, the base of the [[natural logarithm]], now known as [[Euler's number]].<ref name="britannica"/> Euler made contributions to [[applied mathematics]] and [[engineering]], such as his study of ships which helped navigation, his three volumes on optics contributed to the design of [[Microscope|microscopes]] and [[Telescope|telescopes]], and he studied the bending of beams and the critical load of columns.<ref name=":1" />

Euler is credited with being the first to develop [[graph theory]] (partly as a solution for the problem of the [[Seven Bridges of Königsberg]], which is also considered the first practical application of topology). He also became famous for, among many other accomplishments, solving several unsolved problems in number theory and analysis, including the famous [[Basel problem]]. Euler has also been credited for discovering that the sum of the numbers of vertices and faces minus the number of edges of a [[polyhedron]] equals 2, a number now commonly known as the [[Euler characteristic]]. In physics, Euler reformulated [[Isaac Newton]]'s [[Newton's laws of motion|laws of motion]] into [[Euler's laws of motion|new laws]] in his two-volume work ''[[Mechanica]]'' to better explain the motion of [[rigid bodies]]. He contributed to the study of [[Euler–Bernoulli beam theory|elastic deformations]] of solid objects. Euler formulated the [[Euler equations (fluid dynamics)|partial differential equations]] for the motion of [[Inviscid flow|inviscid fluid]],<ref name=":1" /> and laid the mathematical foundations of [[potential theory]].<ref name=":02" />

Euler is regarded as arguably the most prolific contributor in the history of mathematics and science, and the greatest mathematician of the 18th century.<ref>{{Cite book |last=Goldman |first=Jay R. |title=The Queen of Mathematics: A Historically Motivated Guide to Number Theory |date=1998 |publisher=A.K. Peters |isbn=978-1-56881-006-5 |location= |pages=24 |language=en}}</ref><ref name=":1">{{Cite journal |last=Debnath |first=Lokenath |author-link=Lokenath Debnath |date=2009-04-15 |title=The legacy of Leonhard Euler – a tricentennial tribute |url=http://www.tandfonline.com/doi/abs/10.1080/00207390802642237 |journal=International Journal of Mathematical Education in Science and Technology |language=en |volume=40 |issue=3 |pages=353–388 |doi=10.1080/00207390802642237 |issn=0020-739X}}</ref> His 866 publications and his correspondence are being collected in the ''[[Opera Omnia Leonhard Euler]]'' which, when completed, will consist of 81 ''[[quarto]]s''.<ref name="ivb"/><ref name="series ii done"/>{{sfn|Gautschi|2008|p=3}}  Several great mathematicians who worked after Euler's death have recognised his importance in the field: [[Pierre-Simon Laplace]] said, "Read Euler, read Euler, he is the master of us all";<ref name="Laplace"/>{{efn|name=fn2}} [[Carl Friedrich Gauss]] wrote: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it."<ref name="Grinstein" />{{efn|name=fn3|
Gauss wrote this in a letter to [[Paul Fuss]] dated September 11, 1849:{{r|fussletter}} "{{lang|de|Die besondere Herausgabe der kleinern Eulerschen Abhandlungen ist gewiß etwas höchst verdienstliches, [...] und das Studium aller Eulerschen Arbeiten doch stets die beste durch nichts anderes zu ersetzende Schule für die verschiedenen mathematischen Gebiete bleiben wird.}}" [The special publication of the smaller Euler treatises is certainly something highly deserving, [...] and the study of all Euler's works will always remain the best school for the various mathematical fields, which cannot be replaced by anything else.]}}