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{{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           = 1753 portrait
| 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}} {{respell|OY|lər}};{{efn|In English Euler's name is pronounced {{IPAc-en|ˈ|ɔɪ|l|ər|audio=LL-Q1860 (eng)-Flame, not lame-Leonhard Euler.wav}} {{respell|OY|lər}}; the pronunciation {{IPAc-en|ˈ|juː|l|ər}} {{respell|YOO|lər}} is considered incorrect.{{r|YOO-lər}} {{IPA|de-CH|small=no|ˈleːɔnhard ˈɔʏlər|lang}}. {{IPA|de|small=no|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 |page=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, which contributed to the design of [[Microscope|microscopes]] and [[Telescope|telescopes]]; and his studies of beam bending and column critical loads.<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]] that has no holes 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. |url=https://books.google.com/books?id=A0FZDwAAQBAJ&pg=PA24 |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= |page=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]]''.<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.]}}

==Early life==
Leonhard Euler was born in [[Basel]] on 15 April 1707 to Paul III Euler, a pastor of the [[Reformed Church]], and Marguerite (née Brucker), whose ancestors include a number of well-known scholars in the classics.{{sfn|Gautschi|2008|p=4}} He was the oldest of four children, with two younger sisters, Anna Maria and Maria Magdalena, and a younger brother, Johann Heinrich.{{sfn|Calinger|2016|p=11}}{{sfn|Gautschi|2008|p=4}} Soon after Leonhard's birth, the Eulers moved from Basel to [[Riehen]], Switzerland, where his father became pastor in the local church and Leonhard spent most of his childhood.{{sfn|Gautschi|2008|p=4}}

From a young age, Euler received schooling in mathematics from his father, who had taken courses from [[Jacob Bernoulli]] some years earlier at the [[University of Basel]]. Around the age of eight, Euler was sent to live at his maternal grandmother's house and enrolled in the Latin school in Basel. In addition, he received private tutoring from Johannes Burckhardt, a young theologian with a keen interest in mathematics.{{sfn|Gautschi|2008|p=4}}

In 1720, at age 13, Euler enrolled at the [[University of Basel]].<ref name=":0">{{Cite book |last=Debnath |first=Lokenath |title=The Legacy of Leonhard Euler: A Tricentennial Tribute |publisher=Imperial College Press |year=2010 |isbn=978-1-84816-525-0 |location=London |page=vii |language=en |author-link=Lokenath Debnath |url=https://archive.org/details/legacyofleonhard0000debn}}</ref> Attending university at such a young age was not unusual at the time.{{sfn|Gautschi|2008|p=4}} The course on elementary mathematics was given by [[Johann Bernoulli]], the younger brother of the deceased Jacob Bernoulli, who had taught Euler's father. Johann Bernoulli and Euler soon got to know each other better. Euler described Bernoulli in his autobiography:{{sfn|Gautschi|2008|p=5}}

<blockquote>the famous professor Johann Bernoulli [...] made it a special pleasure for himself to help me along in the mathematical sciences. Private lessons, however, he refused because of his busy schedule. However, he gave me a far more salutary advice, which consisted in myself getting a hold of some of the more difficult mathematical books and working through them with great diligence, and should I encounter some objections or difficulties, he offered me free access to him every Saturday afternoon, and he was gracious enough to comment on the collected difficulties, which was done with such a desired advantage that, when he resolved one of my objections, ten others at once disappeared, which certainly is the best method of making happy progress in the mathematical sciences.</blockquote>

During this time, Euler, backed by Bernoulli, obtained his father's consent to become a mathematician instead of a pastor.{{sfn|Calinger|1996|p=124}}<ref name="zum werk leonhard" />

In 1723, Euler received a [[Master of Philosophy]] with a dissertation that compared the philosophies of [[René Descartes]] and [[Isaac Newton]].{{sfn|Gautschi|2008|p=4}} Afterwards, he enrolled in the theological faculty of the University of Basel.<ref name="zum werk leonhard"/>

In 1726, Euler completed a dissertation on the [[Speed of sound|propagation of sound]] titled ''De Sono'',{{sfn|Calinger|2016|p=32}}<ref name="17cent"/> with which he unsuccessfully attempted to obtain a position at the University of Basel.{{sfn|Calinger|1996|p=125}} In 1727, he entered the [[French Academy of Sciences|Paris Academy]] prize competition (offered annually and later biennially by the academy beginning in 1720)<ref name=paris-acad/> for the first time. The problem posed that year was to find the best way to place the [[mast (sailing)|masts]] on a ship. [[Pierre Bouguer]], who became known as "the father of naval architecture", won and Euler took second place.{{sfn|Calinger|1996|p=156}} Over the years, Euler entered this competition 15 times,<ref name=paris-acad/> winning 12 of them.{{sfn|Calinger|1996|p=156}}

==Career==

=== First Saint Petersburg period (1727–1741) ===
[[File:Euler-USSR-1957-stamp.jpg|thumb|left|1957 [[Soviet Union]] stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.]]
Johann Bernoulli's two sons, [[Daniel Bernoulli|Daniel]] and [[Nicolaus II Bernoulli|Nicolaus]], entered into service at the [[Russian Academy of Sciences|Imperial Russian Academy of Sciences]] in [[Saint Petersburg]] in 1725, leaving Euler with the assurance they would recommend him to a post when one was available.{{sfn|Calinger|1996|p=125}} On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia.{{sfn|Calinger|1996|pp=121–166}}<ref name=mactutor>{{MacTutor|title=Nicolaus (II) Bernoulli|id=Bernoulli_Nicolaus(II)|mode=cs1 |access-date=2 July 2021}}</ref> 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.{{sfn|Calinger|1996|p=125}} In November 1726, Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.{{sfn|Calinger|1996|p=125}}

Euler arrived in Saint Petersburg in May 1727.{{sfn|Calinger|1996|p=125}}<ref name="zum werk leonhard" /> 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 worked in close collaboration.{{sfn|Calinger|1996|pp=126–127}} Euler mastered Russian, settled into life in Saint Petersburg and took on an additional job as a medic in the [[Imperial Russian Navy|Russian Navy]].{{sfn|Calinger|1996|p=127}}

The academy at Saint 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.{{sfn|Calinger|1996|p=156}} The academy's benefactress, [[Catherine I of Russia|Catherine I]], who had continued the progressive policies of her late husband, died before Euler's arrival to Saint Petersburg.{{sfn|Calinger|1996|p=126}} The Russian conservative nobility then gained power upon the ascension of the twelve-year-old [[Peter II of Russia|Peter II]].{{sfn|Calinger|1996|p=126}} The nobility, suspicious of the academy's foreign scientists, cut funding for Euler and his colleagues and prevented the entrance of foreign and non-aristocratic students into the Gymnasium and universities.{{sfn|Calinger|1996|p=126}}

Conditions improved slightly after the death of Peter II in 1730 and the German-influenced [[Anna of Russia]] assumed power.{{sfn|Calinger|1996|p=128}} Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731.{{sfn|Calinger|1996|p=128}} He also left the Russian Navy, refusing a promotion to [[Lieutenant (navy)|lieutenant]].{{sfn|Calinger|1996|p=128}} Two years later, Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department.{{sfn|Calinger|1996|pp=128–129}} In January 1734, he married Katharina Gsell (1707–1773), a daughter of [[Georg Gsell]].{{sfn|Gekker|Euler|2007|p=[https://books.google.com/books?id=Ta9bz1wv79AC&pg=PA402 402]}} [[Frederick the Great|Frederick II]] had made an attempt to recruit the services of Euler for his newly established [[Prussian Academy of Sciences|Berlin Academy]] in 1740, but Euler initially preferred to stay in St Petersburg.{{sfn|Calinger|1996|pp=157–158}} But after Empress Anna died and Frederick II agreed to pay 1600 [[Écu|ecus]] (the same as Euler earned in Russia) he agreed to move to Berlin. In 1741, he requested permission to leave for Berlin, arguing he was in need of a milder climate for his eyesight.{{sfn|Calinger|1996|pp=157–158}} The Russian academy gave its consent and would pay him 200 rubles per year as one of its active members.{{sfn|Calinger|1996|pp=157–158}}

=== Berlin period (1741–1766) ===
Concerned about the continuing turmoil in Russia, Euler left St. Petersburg in 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]].{{sfn|Gautschi|2008|p=7}} He lived for 25 years in [[Berlin]], where he wrote several hundred articles.<ref name="zum werk leonhard" /> In 1748 his text on functions called the ''[[Introductio in analysin infinitorum]]'' was published and in 1755 a text on [[differential calculus]] called the ''[[Institutiones calculi differentialis]]'' was published.<ref name="dartm" />{{sfn|Dunham|1999|pp=xxiv–xxv}} In 1755, he was elected a foreign member of the [[Royal Swedish Academy of Sciences]]<ref name=sten/> and of the [[French Academy of Sciences]].<ref name="volumes"/> Notable students of Euler in Berlin included [[Stepan Rumovsky]], later considered as the first Russian astronomer.<ref name="BEA"/><ref name="ClarkGolinski1999"/> In 1748 he declined an offer from the University of Basel to succeed the recently deceased Johann Bernoulli.<ref name="zum werk leonhard" /> In 1753 he bought a house in [[Charlottenburg]], in which he lived with his family and widowed mother.<ref name="zum 300 knobloch"/>{{sfn|Gautschi|2008|pp=8–9}}

Euler became the tutor for [[Friederike Charlotte of Brandenburg-Schwedt]], the Princess of [[Anhalt-Dessau]] and Frederick's niece. He wrote over 200 letters to her in the early 1760s, which were later compiled into a volume entitled ''[[Letters to a German Princess|Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess]]''.<ref name='Digital Copy of "Letters to a German Princess"'/> This work contained Euler's exposition on various subjects pertaining to physics and mathematics and offered valuable insights into Euler's personality and religious beliefs. It was translated into multiple languages, published across Europe and in the United States, and became more widely read than any of his mathematical works. 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.{{sfn|Dunham|1999|pp=xxiv–xxv}}

Despite Euler's immense contribution to the academy's prestige and having been put forward as a candidate for its presidency by [[Jean le Rond d'Alembert]], [[Frederick II of Prussia|Frederick II]] named himself as its president.{{sfn|Gautschi|2008|pp=8–9}} The Prussian king had a large circle of intellectuals in his court, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs. He was, in many ways, the polar opposite of [[Voltaire]], who enjoyed a high place of prestige at Frederick's court. Euler was not a skilled debater and often made it a point to argue subjects that he knew little about, making him the frequent target of Voltaire's wit.{{sfn|Dunham|1999|pp=xxiv–xxv}} Frederick also expressed disappointment with Euler's practical engineering abilities, stating:

{{blockquote|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 name=fredlett/>}}

However, the disappointment was almost surely unwarranted from a technical perspective. Euler's calculations look likely to be correct, even if Euler's interactions with Frederick and those constructing his fountain may have been dysfunctional.<ref name=Lynch/>

Throughout his stay in Berlin, Euler maintained a strong connection to the academy in St. Petersburg and also published 109 papers in Russia.<ref name="vucinich"/> He also assisted students from the St. Petersburg academy and at times accommodated Russian students in his house in Berlin.<ref name="vucinich" /> In 1760, with the [[Seven Years' War]] raging, Euler's farm in Charlottenburg was sacked by advancing Russian troops.<ref name="zum 300 knobloch" /> Upon learning of this event, [[Ivan Saltykov|General Ivan Petrovich Saltykov]] paid compensation for the damage caused to Euler's estate, with [[Empress Elizabeth]] of Russia later adding a further payment of 4000 rubles—an exorbitant amount at the time.<ref name="gindikin" /> Euler decided to leave Berlin in 1766 and return to Russia.{{sfn|Gautschi|2008|p=9}}

During his Berlin years (1741–1766), Euler was at the peak of his productivity. He wrote 380 works, 275 of which were published.<ref name="math at prussian"/> This included 125 memoirs in the Berlin Academy and over 100 memoirs sent to the [[Russian Academy of Sciences|St. Petersburg Academy]], which had retained him as a member and paid him an annual stipend. Euler's ''Introductio in Analysin Infinitorum'' was published in two parts in 1748. In addition to his own research, Euler supervised the library, the observatory, the botanical garden, and the publication of calendars and maps from which the academy derived income.<ref name="historian's craft"/> He was even involved in the design of the water fountains at [[Sanssouci]], the King's summer palace.<ref name="fountains"/>

=== Second Saint Petersburg period (1766–1783) ===
The political situation in Russia stabilized after [[Catherine the Great|Catherine the Great's]] accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Petersburg Academy. His conditions were quite exorbitant—a 3000 ruble annual salary, a pension for his wife, and the promise of high-ranking appointments for his sons. At the university he was assisted by his student [[Anders Johan Lexell]].<ref name="lexell's theorem"/> While living in St. Petersburg, a fire in 1771 destroyed his home.<ref name="thiele" />

==Personal life==
On 7 January 1734, Euler married Katharina Gsell, daughter of [[Georg Gsell]], a painter at the Academy Gymnasium in Saint Petersburg.{{sfn|Gekker|Euler|2007|p=[https://books.google.com/books?id=Ta9bz1wv79AC&pg=PA402 402]}} The couple bought a house by the [[Neva River]].

Of their 13 children, five survived childhood,<ref name="novaacta" /> three sons and two daughters.{{sfn|Calinger|1996|p=129}} Their first son was [[Johann Euler|Johann Albrecht Euler]], whose godfather was [[Christian Goldbach]].{{sfn|Calinger|1996|p=129}}

Three years after his wife's death in 1773,<ref name="thiele" /> Euler married her half-sister, Salome Abigail Gsell.{{sfn|Gekker|Euler|2007|p=[https://books.google.com/books?id=Ta9bz1wv79AC&pg=PA405 405]}} This marriage lasted until his death in 1783.

His brother Johann Heinrich settled in St. Petersburg in 1735 and was employed as a painter at the academy.{{sfn|Calinger|1996|pp=157–158}}

Early in his life, Euler memorized [[Virgil]]'s ''[[Aeneid]]'', and by old age, he could recite the poem and give the first and last sentence on each page of the edition from which he had learnt it.<ref>{{Cite web |last=Meade |first=Phil |date=1999-11-27 |title=Letter: Uncommon talent |url=https://www.newscientist.com/letter/mg16422147-400-uncommon-talent/ |access-date=2024-09-22 |website=www.newscientist.com |language=en-US}}</ref><ref>{{Cite book |last=Nahin |first=Paul J. |author-link=Paul J. Nahin |url=https://books.google.com/books?id=m3GYDwAAQBAJ&dq=aeneid+euler&pg=PA326 |title=Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills |publisher=Princeton University Press |year=2017 |isbn=978-0-691-17591-1 |series=Princeton Science Library |location=Princeton Oxford |page=326 |language=en}}</ref> Euler knew the first hundred prime numbers and could give each of their powers up to the sixth degree.<ref name=":2" /> 

Euler was known as a generous and kind person, not neurotic as seen in some geniuses, keeping his good-natured disposition even after becoming entirely blind.<ref name=":2">{{Cite news |last=Lynch |first=Peter |date=2021-01-21 |title=Euler: a mathematician without equal and an overall nice guy |url=https://www.irishtimes.com/news/science/euler-a-mathematician-without-equal-and-an-overall-nice-guy-1.4455424 |access-date=2024-12-09 |newspaper=The Irish Times}}</ref>

===Eyesight deterioration===
Euler's eyesight worsened throughout his mathematical career. In 1738, three years after nearly dying of fever,{{sfn|Gautschi|2008|p=6}} he became almost blind in his right eye. Euler blamed the [[cartography]] he performed for the St. Petersburg Academy for his condition,<ref name="blindness" /> but the cause of his blindness remains the subject of speculation.<ref name="righteye" /><ref name="bullock"/> Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick called him "[[Cyclopes|Cyclops]]". Euler said of his loss of vision, "Now I will have fewer distractions."<ref name="blindness" /> In 1766 a [[cataract]] in his left eye was discovered. Though [[Couching (ophthalmology)|couching of the cataract]] temporarily improved his vision, complications rendered him almost totally blind in the left eye as well.<ref name="volumes"/> His condition appeared to have little effect on his productivity. With the aid of his scribes, Euler's productivity in many areas of study increased;{{sfn|Gautschi|2008|pp=9–10}} in 1775, he produced, on average, one mathematical paper per week.<ref name="volumes" />

=== Death ===
[[File:Euler Grave at Alexander Nevsky Monastry.jpg|thumb|Euler's grave at the [[Alexander Nevsky Lavra|Alexander Nevsky Monastery]]]]

In St. Petersburg on 18 September 1783, after a lunch with his family, Euler was discussing the newly discovered planet [[Uranus]] and its orbit with [[Anders Johan Lexell]] when he collapsed and died of a [[brain hemorrhage]].<ref name="righteye" /> {{ill|Jacob von Staehlin|de}} wrote a short obituary for the [[Russian Academy of Sciences]] and Russian mathematician [[Nicolas Fuss]], one of Euler's disciples, wrote a more detailed eulogy,<ref name="novaacta" /> which he delivered at a memorial meeting. In his eulogy for the [[French Academy of Sciences|French Academy]], French mathematician and philosopher [[Marquis de Condorcet]] wrote:

{{blockquote|multiline=yes|{{lang|fr|...{{nbsp}}il cessa de calculer et de vivre.}}
...{{nbsp}}he ceased to calculate and to live.<ref name=condorcet/>}}

Euler was buried next to Katharina at the [[Smolensk Lutheran Cemetery]] on [[Vasilievsky Island]]. In 1837, the [[Russian Academy of Sciences]] installed a new monument, replacing his overgrown grave plaque. In 1957, to commemorate the 250th anniversary of his birth, his tomb was moved to the [[Lazarevskoe Cemetery]] at the [[Alexander Nevsky Monastery]].{{sfn|Calinger|2016|pp=530–536}}

==Contributions to science==
{{main|Contributions of Leonhard Euler to mathematics}}
{{E (mathematical constant)}}
Euler worked in almost all areas of mathematics, including [[geometry]], [[infinitesimal 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]]. Euler's work averages 800 pages a year from 1725 to 1783. He also wrote over 4500 letters and hundreds of manuscripts. It has been estimated that Leonhard Euler was the author of a quarter of the combined output in mathematics, physics, mechanics, astronomy, and navigation in the 18th century, while other researchers credit Euler for a third of the output in mathematics in that century.<ref name="assad"/>

===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]]{{sfn|Dunham|1999|p=17}} and was the first to write {{math|''f''(''x'')}} to denote the function {{math|''f''}} applied to the argument {{math|''x''}}. He also introduced the modern notation for the [[trigonometric functions]], the letter {{math|''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''}} to denote the [[imaginary unit]].<ref name=Boyer/> 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 originated with [[Welsh people|Welsh]] mathematician [[William Jones (mathematician)|William Jones]].<ref name=arndt/>

===Analysis===
The development of [[infinitesimal calculus]] was at the forefront of 18th-century mathematical research, and the [[Bernoulli family|Bernoullis]]—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"/> (in particular his reliance on the principle of the [[generality of algebra]]), 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,{{sfn|Ferraro|2008|p=155}} such as
<math display=block>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>

Euler's use of power series enabled him to solve the [[Basel problem]], finding the sum of the reciprocals of squares of every natural number, in 1735 (he provided a more elaborate argument in 1741). The Basel problem was originally posed by [[Pietro Mengoli]] in 1644, and by the 1730s was a famous open problem, popularized by [[Jacob Bernoulli]] and unsuccessfully attacked by many of the leading mathematicians of the time. Euler found that:<ref name ="Morris PhD thesis ">{{cite thesis |last=Morris |first=Imogen I. |title=Mechanising Euler's use of Infinitesimals in the Proof of the Basel Problem |degree=PhD |publisher=University of Edinburgh | date=24 October 2023 | doi=10.7488/ERA/3835 }}</ref>{{sfn|Dunham|1999}}<ref name="Basel"/>

<math display=block>\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>

Euler introduced the constant
<math display=block>\gamma = \lim_{n \rightarrow \infty } \left( 1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} - \ln(n) \right) \approx 0.5772,</math>
now known as [[Euler's constant]] or the Euler–Mascheroni constant, and studied its relationship with the [[harmonic series (mathematics)|harmonic series]], the [[gamma function]], and values of the [[Riemann zeta function]].<ref name=lagarias/>

[[File:Euler's formula.svg|thumb|A geometric interpretation of [[Euler's formula]]]]

Euler introduced the use of the [[exponential function]] and [[logarithms]] in [[analytic proof]]s. 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/> He also defined the exponential function for complex numbers and discovered its relation to the [[trigonometric function]]s. For any [[real number]] {{math|[[φ]]}} (taken to be radians), [[Euler's formula]] states that the [[Exponential function#On the complex plane|complex exponential]] function satisfies
<math display=block>e^{i\varphi} = \cos \varphi + i\sin \varphi</math>

which was called "the most remarkable formula in mathematics" by [[Richard Feynman]].<ref name="Feynman"/>

A special case of the above formula is known as [[Euler's identity]],
<math display=block>e^{i \pi} +1 = 0 </math>

Euler elaborated the theory of higher [[transcendental function]]s by introducing the [[gamma function]]{{sfn|Ferraro|2008|p=159}}<ref name=davis/> and introduced a new method for solving [[quartic equation]]s.<ref name=nickalls/> He found a way to calculate integrals with complex limits, foreshadowing the development of modern [[complex analysis]]. He invented the [[calculus of variations]] and formulated the [[Euler–Lagrange equation]] for reducing [[Mathematical optimization|optimization problems]] in this area to the solution of [[differential equation]]s.

Euler 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 [[Generalized hypergeometric series|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]].{{sfn|Dunham|1999|loc=Ch. 3, Ch. 4}}

===Number theory===
Euler's interest in number theory can be traced to the influence of [[Christian Goldbach]],{{sfn|Calinger|1996|p=130}} his friend in the St. Petersburg Academy.{{sfn|Gautschi|2008|p=6}} Much of Euler's early work on number theory was based on the work of [[Pierre de Fermat]]. Euler developed some of Fermat's ideas and disproved some of his conjectures, such as his conjecture that all numbers of the form <math display="inline">2^{2^n}+1</math> ([[Fermat numbers]]) are prime.{{sfn|Dunham|1999|p=7}}

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 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]].<ref name=patterson/>

Euler invented the [[totient function]] φ(''n''), 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]].<ref name=shiu/> He contributed significantly to the theory of [[perfect number]]s, which had fascinated mathematicians since [[Euclid]]. He proved that the relationship shown between even perfect numbers and [[Mersenne prime]]s (which he had earlier proved) was one-to-one, a result otherwise known as the [[Euclid–Euler theorem]].<ref name=stillwell/> Euler also conjectured the law of [[quadratic reciprocity]]. The concept is regarded as a fundamental theorem within number theory, and his ideas paved the way for the work of [[Carl Friedrich Gauss]], particularly ''[[Disquisitiones Arithmeticae]]''.{{sfn|Dunham|1999|loc=Ch. 1, Ch. 4}} By 1772 Euler had proved that 2<sup>31</sup>&nbsp;−&nbsp;1 = [[2147483647|2,147,483,647]] is a Mersenne prime. It may have remained the [[largest known prime]] until 1867.<ref name=caldwell/>

Euler also contributed major developments to the theory of [[partitions of an integer]].<ref name=hopwil/>

===Graph theory===
[[File:Konigsberg bridges.png|frame|right|Map of [[Königsberg]] in Euler's time showing the actual layout of the [[Seven Bridges of Königsberg|seven bridges]], highlighting the river [[Pregolya|Pregel]] and the bridges]]
In 1735, Euler presented a solution to the problem known as the [[Seven Bridges of Königsberg]].<ref name="bridge"/> The city of [[Königsberg]], [[Kingdom of Prussia|Prussia]] was set on the [[Pregolya|Pregel]] River, and included two large islands that 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. Euler showed that it is not possible: there is no [[Eulerian path]]. This solution is considered to be the first theorem of [[graph theory]].<ref name="bridge"/>

Euler also discovered the [[Planar graph#Euler's formula|formula]] <math>V - E + F = 2</math> relating the number of vertices, edges, and faces of a [[Convex polytope|convex polyhedron]],{{sfn|Richeson|2012}} and hence of a [[planar graph]]. 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 name=gibbons/> The study and generalization of this formula, specifically by [[Augustin-Louis Cauchy|Cauchy]]<ref name="Cauchy"/> and [[Simon Antoine Jean L'Huilier|L'Huilier]],<ref name="Lhuillier"/> is at the origin of [[topology]].{{sfn|Richeson|2012}}

===Physics, astronomy, and engineering===
{{Classical mechanics|cTopic=Scientists}}
Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the [[Bernoulli numbers]], [[Fourier series]], [[Euler number]]s, the constants {{math|[[E (mathematical constant)|e]]}} and [[pi|{{pi}}]], 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]]<ref name=butcher/> and the [[Euler–Maclaurin formula]].{{sfn|Calinger|2016|pp=96, 137}}{{sfn|Ferraro|2008|loc=Chapter 14: Euler's derivation of the Euler–Maclaurin summation formula|pages=171–180}}<ref name=mills/>

Euler helped develop the [[Euler–Bernoulli beam equation]], which became a cornerstone of engineering.<ref name=ojalvo/> Besides successfully applying his analytic tools to problems in [[classical mechanics]], Euler applied these techniques to celestial problems. His work in astronomy was recognized by multiple [[French Academy of Sciences|Paris Academy]] Prizes over the course of his career. His accomplishments include determining with great accuracy the [[orbit]]s of [[comet]]s and other celestial bodies, understanding the nature of comets, and calculating the [[solar parallax|parallax]] of the Sun. His calculations contributed to the development of accurate [[History of longitude|longitude tables]].<ref name=yousch/>

Euler made important contributions in [[optics]].<ref name=davidson/> He disagreed with Newton's [[corpuscular theory of light]],{{sfn|Calinger|1996|pp=152–153}} which was the prevailing theory of the time. His 1740s papers on optics helped ensure that the [[wave theory of light]] proposed by [[Christiaan 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"/>

In [[fluid dynamics]], Euler was the first to predict the phenomenon of [[cavitation]], in 1754, long before its first observation in the late 19th century, and the [[Euler number (physics)|Euler number]] used in fluid flow calculations comes from his related work on the efficiency of [[turbine]]s.{{r|li}} In 1757 he published an important set of equations for [[inviscid flow]] in [[fluid dynamics]], that are now known as the [[Euler equations (fluid dynamics)|Euler equations]].<ref name=euler2/>

Euler is well known in [[structural engineering]] for his formula giving [[Euler's critical load]], the critical [[buckling]] load of an ideal strut, which depends only on its length and [[Flexural rigidity|flexural stiffness]].{{sfn|Gautschi|2008|p=22}}

===Logic===
Euler is credited with using [[closed curve]]s to illustrate [[syllogism|syllogistic]] reasoning (1768). These diagrams have become known as [[Euler diagram]]s.<ref name=logic/>
[[File:Euler Diagram.svg|thumb|upright|An Euler diagram]]

An Euler diagram is a [[diagram]]matic means of representing [[Set (mathematics)|sets]] and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict [[Set (mathematics)|sets]]. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the [[element (mathematics)|elements]] of the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important; the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships ([[intersection (set theory)|intersection]], [[subset]], and [[Disjoint sets|disjointness]]). Curves whose interior zones do not intersect represent [[disjoint sets]]. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the [[intersection (set theory)|intersection]] of the sets). A curve that is contained completely within the interior zone of another represents a [[subset]] of it.

Euler diagrams (and their refinement to [[Venn diagram]]s) were incorporated as part of instruction in [[set theory]] as part of the [[new math]] movement in the 1960s.<ref name=lemanski/> Since then, they have come into wide use as a way of visualizing combinations of characteristics.<ref name=rodgers/>

===Demography===
In his 1760 paper ''A General Investigation into the Mortality and Multiplication of the Human Species''  Euler produced a model which showed how a population with constant fertility and mortality might grow geometrically using a difference equation. Under this geometric growth Euler also examined relationships among various demographic indices showing how they might be used to produce estimates when observations  were missing.  Three papers published around 150 years later  by [[Alfred J. Lotka]] (1907, 1911 (with F.R. Sharpe) and  1922) adopted a similar approach to  Euler's and produced their Stable Population Model. These marked the start of 20th century formal demographic modelling.<ref>Smith, D.P.  and N. Keyfitz, (2013) Mathematical Demography: Selected Papers, Monographs, DOI 10.1007/978-3-642-35858-6_1 Springer-Verlag   Demographic Research - Euler, L. (1760). 11. A general investigation into the mortality and multiplication of the human species. A General Investigation into the Mortality and Multiplication of the Human Species, Theoretical Population Biology 1: 307-314. Translated by Nathan and Beatrice Keyfitz</ref><ref>Newell, Colin. (1988) Methods and models in demography. Belhaven Press.</ref><ref>Inaba, Hisashi (2017) Chapter 1 The Stable Population Model in ''Age-structured population dynamics in demography and epidemiology''. Springer Singapore.</ref><ref>Lotka, A. J. (1907). Relation between birth rates and death rates. Science, 26(653), 21-22.</ref><ref>Sharpe, F. R., & Lotka, A. J. (1911). L. A problem in age-distribution. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 21(124), 435-438.</ref><ref>Lotka, A. J. (1922). The stability of the normal age distribution. Proceedings of the National Academy of Sciences, 8(11), 339-345.</ref>

===Music===
One of Euler's more unusual interests was the application of [[Music and mathematics|mathematical ideas in music]]. In 1739 he wrote the ''Tentamen novae theoriae musicae'' (''Attempt at a New Theory of Music''), 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.{{sfn|Calinger|1996|pp=144–145}} Even when dealing with music, Euler's approach is mainly mathematical,<ref name=pesic/> for instance, his introduction of [[binary logarithm]]s as a way of numerically describing the subdivision of [[octave]]s into fractional parts.<ref name=tegg/> His writings on music are not particularly numerous (a few hundred pages, in his total production of about thirty thousand pages), but they reflect an early preoccupation and one that remained with him throughout his life.<ref name=pesic/>

A first point of Euler's musical theory is the definition of "genres", i.e. of possible divisions of the octave using the prime numbers 3 and 5. Euler describes 18 such genres, with the general definition 2<sup>m</sup>A, where A is the "exponent" of the genre (i.e. the sum of the exponents of 3 and 5) and 2<sup>m</sup> (where "m is an indefinite number, small or large, so long as the sounds are perceptible"{{sfn|Euler|1739|p=115}}), expresses that the relation holds independently of the number of octaves concerned. The first genre, with A = 1, is the octave itself (or its duplicates); the second genre, 2<sup>m</sup>.3, is the octave divided by the fifth (fifth + fourth, C–G–C); the third genre is 2<sup>m</sup>.5, major third + minor sixth (C–E–C); the fourth is 2<sup>m</sup>.3<sup>2</sup>, two-fourths and a tone (C–F–B{{music|b}}–C); the fifth is 2<sup>m</sup>.3.5 (C–E–G–B–C); etc. Genres 12 (2<sup>m</sup>.3<sup>3</sup>.5), 13 (2<sup>m</sup>.3<sup>2</sup>.5<sup>2</sup>) and 14 (2<sup>m</sup>.3.5<sup>3</sup>) are corrected versions of the [[Genus (music)|diatonic, chromatic and enharmonic]], respectively, of the Ancients. Genre 18 (2<sup>m</sup>.3<sup>3</sup>.5<sup>2</sup>) is the "diatonico-chromatic", "used generally in all compositions",<ref name=emery/> and which turns out to be identical with the system described by [[Johann Mattheson]].<ref name=mattheson/> Euler later envisaged the possibility of describing genres including the prime number 7.<ref name=perret/>

Euler devised a specific graph, the ''Speculum musicum'',{{sfn|Euler|1739|p=147}}<ref name="de harmoniae"/> to illustrate the diatonico-chromatic genre, and discussed paths in this graph for specific intervals, recalling his interest in the Seven Bridges of Königsberg (see [[#Graph theory|above]]). The device drew renewed interest as the [[Tonnetz]] in [[Neo-Riemannian theory]] (see also [[Lattice (music)]]).<ref name=gollin/>

Euler further used the principle of the "exponent" to propose a derivation of the ''gradus suavitatis'' (degree of suavity, of agreeableness) of intervals and chords from their prime factors – one must keep in mind that he considered [[just intonation]], i.e. 1 and only the prime numbers 3 and 5.<ref name=lindley/> Formulas have been proposed extending this system to any number of prime numbers, e.g. in the form
<math display=block>\ ds=\sum_i\left( k_i\cdot p_i - k_i\right) + 1\ ,</math>
where {{math|''p''<sub>''i''</sub>}} are prime numbers and {{math|''k''<sub>''i''</sub>}} their exponents.<ref name=bailhache/>

==Personal philosophy and religious beliefs==
Euler was religious throughout his life.<ref name="zum werk leonhard" /> Much of what is known of his 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 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"/><ref name="Ho2" />

Euler opposed the concepts of [[Gottfried Leibniz|Leibniz's]] [[monadism]] and the philosophy of [[Christian Wolff (philosopher)|Christian Wolff]].{{sfn|Calinger|1996|p=123}} He 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 called Wolff's ideas "heathen and atheistic".<ref name="wolff"/>

There is a legend<ref name="diderot"/> 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. The Empress was alarmed that Diderot's arguments for atheism were influencing members of her court, and so Euler was asked to confront him. Diderot was 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 this [[Non sequitur (literary device)|non sequitur]]:

"Sir, <math>\frac{a+b^n}{n}=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 Catherine granted. However amusing the anecdote may be, it is [[wikt:apocryphal|apocryphal]], given that Diderot himself did research in mathematics.<ref name=persee/>
The legend was apparently first told by [[Dieudonné Thiébault]] with embellishment by [[Augustus De Morgan]].<ref name="diderot"/>

==Legacy==
{{Main|List of topics named after Leonhard Euler}}
===Recognition===
Euler is widely recognized as one of the greatest mathematicians of all time, and more likely than not the most prolific contributor to mathematics and science.<ref name=":1" /> Mathematician and physicist [[John von Neumann]] called Euler "the greatest [[virtuoso]] of the period".<ref name=":022">{{Cite book |last=Debnath |first=Lokenath |author-link=Lokenath Debnath |url=https://books.google.com/books?id=HPRpDQAAQBAJ&pg=PA56 |title=The Legacy of Leonhard Euler: A Tricentennial Tribute |publisher=Imperial College Press |year=2010 |isbn=978-1-84816-525-0 |location=London |page=56 |language=en}}</ref> Mathematician [[François Arago]] said, "Euler calculated without any apparent effort, just as men breathe and as eagles sustain themselves in air".<ref name=":3">{{Cite book |last=Davis |first=Donald M. |url=https://books.google.com/books?id=qn7rYiyS7mUC&pg=PA48 |title=The Nature and Power of Mathematics |date=2004 |publisher=Dover Publications |isbn=978-0-486-43896-2 |edition= |series= |location=Mineola, N.Y |page=48 |language=en |oclc=ocm56214613}}</ref> He is generally ranked right below [[Carl Friedrich Gauss]], [[Isaac Newton]], and [[Archimedes]] among the greatest mathematicians of all time,<ref name=":3" /> while some rank him as equal with them.{{sfn|Calinger|2016|p=ix}} Physicist and mathematician [[Henri Poincaré]] called Euler the "god of mathematics".{{sfn|Calinger|2016|p=241}} 

French mathematician [[André Weil]] noted that Euler stood above his contemporaries and more than anyone else was able to cement himself as the leading force of his era's mathematics:<ref name=":022" />{{blockquote|No mathematician ever attained such a position of undisputed leadership in all branches of mathematics, pure and applied, as Euler did for the best part of the eighteenth century.}}Swiss mathematician [[Nicolas Fuss]] noted Euler's extraordinary memory and breadth of knowledge, saying:<ref name=":02" />

{{blockquote|Knowledge that we call erudition was not inimical to him. He had read all the best Roman writers, knew perfectly the ancient history of mathematics, held in his memory the historical events of all times and peoples, and could without hesitation adduce by way of examples the most trifling of historical events. He knew more about medicine, botany, and chemistry than might be expected of someone who had not worked especially in those sciences.}}

===Commemorations===
[[File:Euler-10 Swiss Franc banknote (front).jpg|thumb|Euler portrait on the sixth series of the [[Swiss Franc|10 Franc]] banknote]]

[[File:CHF10 7 front horizontal.jpg|thumb|Euler portrait on the seventh series of the 10 Franc banknote]]

Euler was featured on both the sixth<ref name=swiss6/> and seventh<ref name=swiss7/> series of the Swiss 10-[[Swiss franc|franc]] banknote and on numerous Swiss, German, and Russian postage stamps. In 1782 he was elected a Foreign Honorary Member of the [[American Academy of Arts and Sciences]].<ref name="aaas" /> The [[asteroid]] [[2002 Euler]] was named in his honour.<ref name=schmadel/>

==Selected bibliography==
Euler has [[Contributions of Leonhard Euler to mathematics#Works|an extensive bibliography]]. His books include:
* ''[[Mechanica]]'' (1736)
* [https://www.loc.gov/item/04028085/ ''Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti'' (1744)]<ref name=fraser/> (''A method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense'')<ref name=dartm2/>
* ''[[Introductio in analysin infinitorum]]'' (1748)<ref name=reich/><ref name=ferraro/> (''Introduction to Analysis of the Infinite'')<ref name=revaninf/>
* ''[[Institutiones calculi differentialis]]'' (1755)<ref name=ferraro/><ref name=demidov/> (''Foundations of differential calculus'')
* ''[[Elements of Algebra|Vollständige Anleitung zur Algebra]]'' (1765)<ref name=ferraro/> (''Elements of Algebra'')
* ''[[Institutiones calculi integralis]]'' (1768–1770)<ref name=ferraro/> (''Foundations of integral calculus'')
* ''[[Letters to a German Princess]]'' (1768–1772){{sfn|Dunham|1999|pp=xxiv–xxv}}
* ''Dioptrica'', published in three volumes beginning in 1769<ref name=davidson/>

It took until 1830 for the bulk of Euler's posthumous works to be individually published,<ref name=kleinert/> with an additional batch of 61 unpublished works discovered by [[Paul Heinrich von Fuss]] (Euler's great-grandson and [[Nicolas Fuss]]'s son) and published as a collection in 1862.<ref name=kleinert/><ref name=fuss/> A chronological catalog of Euler's works was compiled by Swedish mathematician [[Gustaf Eneström]] and published from 1910 to 1913.{{sfn|Calinger|2016|pages=ix–x}} The catalog, known as the '''Eneström index''', numbers Euler's works from E1 to E866.<ref name=enestrom/> The Euler Archive was started at [[Dartmouth College]]<ref name=archive-start/> before moving to the [[Mathematical Association of America]]<ref name=archive-move/> and, most recently, to [[University of the Pacific (United States)|University of the Pacific]] in 2017.<ref name="euler archive"/>

In 1907, the [[Swiss Academies of Arts and Sciences|Swiss Academy of Sciences]] created the [[Euler Commission]] and charged it with the publication of Euler's complete works. After several delays in the 19th century,<ref name=kleinert/> the first volume of the ''[[Opera Omnia Leonhard Euler|Opera Omnia]]'', was published in 1911.<ref name="pluss"/> However, the discovery of new manuscripts continued to increase the magnitude of this project. Fortunately, the publication of Euler's Opera Omnia has made steady progress, with over 70 volumes (averaging 426 pages each) published by 2006 and 80 volumes published by 2022.<ref name="new look"/><ref name="series ii done"/><ref name="assad"/> These volumes are organized into four series. The first series compiles the works on analysis, algebra, and number theory; it consists of 29 volumes and numbers over 14,000 pages. The 31 volumes of Series II, amounting to 10,660 pages, contain the works on mechanics, astronomy, and engineering. Series III contains 12 volumes on physics. Series IV, which contains the massive amount of Euler's correspondence, unpublished manuscripts, and notes only began compilation in 1967. After publishing 8 print volumes in Series IV, the project decided in 2022 to publish its remaining projected volumes in Series IV in online format only.<ref name="series ii done"/><ref name="pluss" /><ref name="assad"/>

<gallery widths="200px" heights="200px">
File:Acta Eruditorum - II geometria, 1744 – BEIC 13411238.jpg|Illustration from ''Solutio problematis... a. 1743 propositi'' published in [[Acta Eruditorum]], 1744
File:Methodus inveniendi - Leonhard Euler - 1744.jpg|The title page of Euler's ''Methodus inveniendi lineas curvas''
File:Leonhard Euler World Map AD1760.jpg|Euler's 1760 world map
File:Euler Tab. Geogr. Africae 1753 UTA.jpg|Euler's 1753 map of Africa
</gallery>

==Notes==
{{notelist|refs=

{{efn|name=fn2|
The quote appeared in [[Gugliemo Libri]]'s review of a recently published collection of correspondence among eighteenth-century mathematicians: "{{lang|fr|...{{nbsp}}nous rappellerions que Laplace lui même,{{nbsp}}... ne cessait de répéter aux jeunes mathématiciens ces paroles mémorables que nous avons entendues de sa propre bouche : 'Lisez Euler, lisez Euler, c'est notre maître à tous.'}}{{-"}} [...{{nbsp}}we would recall that Laplace himself,{{nbsp}}... never ceased to repeat to young mathematicians these memorable words that we heard from his own mouth: 'Read Euler, read Euler, he is our master in everything.']<ref>{{cite journal|first1=Gugliemo|last1=Libri |author-link1=Guglielmo Libri Carucci dalla Sommaja |date=January 1846|title=Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIe siècle, ...|language=fr|trans-title=Mathematical and physical correspondence of some famous geometers of the eighteenth century, ...|journal=[[Journal des sçavans|Journal des Savants]] |url=http://gallica.bnf.fr/ark:/12148/bpt6k57253t/f52.image.langEN|page=51|access-date=7 April 2014}}</ref>}}

}}

==References==
{{Reflist|refs=

<ref name="YOO-lər">
{{cite encyclopedia|title=Euler|encyclopedia=[[Oxford English Dictionary]]|edition=2nd|publisher=[[Oxford University Press]]|year=1989}} {{pb}}
{{cite encyclopedia|url=http://www.merriam-webster.com/dictionary/Euler|title=Euler|encyclopedia=[[Webster's Dictionary|Merriam–Webster's Online Dictionary]]|year=2009|access-date=5 June 2009}} {{pb}}
{{cite encyclopedia|url=http://ahdictionary.com/word/search.html?q=Euler%2C+Leonhard&submit.x=40&submit.y=16|title=Euler, Leonhard|encyclopedia=[[The American Heritage Dictionary of the English Language]]|edition=5th|publisher=[[Houghton Mifflin Harcourt|Houghton Mifflin Company]]|location=Boston|year=2011|access-date=30 May 2013}} {{pb}}
{{cite book|title=Nets, Puzzles, and Postmen: An Exploration of Mathematical Connections|url=https://archive.org/details/netspuzzlespostm00higg|url-access=limited|last1=Higgins |first1=Peter M. |year=2007|publisher=[[Oxford University Press]]|page=[https://archive.org/details/netspuzzlespostm00higg/page/n51 43]|isbn=978-0-19-921842-4}}
</ref>

<ref name="17cent">{{cite book|first=Leonhard|last=Euler|via=Euler archive|url=https://scholarlycommons.pacific.edu/euler-works/2/|title=Dissertatio physica de sono|language=la|trans-title=Physical dissertation on sound|location=Basel|publisher=E. and J. R. Thurnisiorum|year=1727|access-date=6 June 2021}}<br />
Translated into English as<br />
{{cite web |url=http://www.17centurymaths.com/contents/euler/e002tr.pdf |first=Ian |last=Bruce |title=Euler's Dissertation De Sono: E002 |work=Some Mathematical Works of the 17th & 18th Centuries, including Newton's Principia, Euler's Mechanica, Introductio in Analysin, etc., translated mainly from Latin into English |access-date=12 June 2021 }}</ref>

<ref name=aaas>{{cite book |title=Members of the American Academy of Arts & Sciences, 1780–2017 |chapter=E |pages=164–179 |publisher=[[American Academy of Arts and Sciences]] |chapter-url=https://www.amacad.org/sites/default/files/academy/multimedia/pdfs/publications/bookofmembers/ChapterE.pdf |access-date=17 February 2019 }} Entry for Euler is on p. 177.</ref>

<ref name=archive-move>{{Cite magazine|url=http://digitaleditions.walsworthprintgroup.com/article/Euler+Archive+Moves+To+MAA+Website/757939/72837/article.html|title=Euler Archive Moves To MAA Website|magazine=[[MAA FOCUS]] |publisher=[[Mathematical Association of America]] |date=June–July 2011|first=Dominic|last=Klyve |access-date=9 January 2020}}</ref>

<ref name=archive-start>{{Cite news|url=http://www.dartmouth.edu/~vox/0607/0219/euler.html|archive-url=https://web.archive.org/web/20100528022155/http://www.dartmouth.edu/~vox/0607/0219/euler.html |archive-date=28 May 2010 |title=Dartmouth students build online archive of historic mathematician |newspaper=Vox of Dartmouth |first=Susan |last=Knapp |date=19 February 2007|publisher=[[Dartmouth College]]}}</ref>

<ref name=arndt>{{cite book | last1 = Arndt | first1 = Jörg | last2 = Haenel | first2 = Christoph | title = Pi Unleashed | publisher = [[Springer Science+Business Media|Springer-Verlag]] | year = 2006 | isbn = 978-3-540-66572-4 | page = 166 | url = https://books.google.com/books?id=QwwcmweJCDQC&pg=PA166 | access-date = 8 June 2021 }}</ref>

<ref name=Basel>{{cite book| last1 = Wanner| first1 = Gerhard | author-link = Gerhard Wanner|last2=Hairer | first2 = Ernst| author2-link = Ernst Hairer  | title = Analysis by its history| edition = 1st|date=2005| publisher = [[Springer Publishing]]| page = 63 | isbn = 978-0-387-77036-9}}</ref>

<ref name=blindness>{{cite book|first=Howard W.|last=Eves|author-link=Howard Eves|title=In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes, Quadrants III and IV|contribution=Euler's blindness|year=1969|publisher=Prindle, Weber, & Schmidt|page=48 |oclc=260534353}} Also quoted by {{harvtxt|Richeson|2012}}, [https://books.google.com/books?id=kv2EDwAAQBAJ&pg=PA17 p. 17], cited to Eves.</ref>

<ref name=Boyer>{{cite book|title = A History of Mathematics|last = Boyer|first = Carl B.|author-link=Carl Benjamin Boyer|author2 = Merzbach, Uta C.|author2-link = Uta Merzbach|publisher = [[John Wiley & Sons]]|isbn = 978-0-471-54397-8|pages = [https://archive.org/details/historyofmathema00boye/page/439 439–445]|year = 1991|url = https://archive.org/details/historyofmathema00boye/page/439}}</ref>

<ref name=bridge>{{cite journal| last = Alexanderson| first = Gerald|author-link=Gerald L. Alexanderson|date=July 2006| 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| page = 567| issue = 4| doi-access = free}}</ref>

<ref name=britannica>{{Cite encyclopedia|last=Boyer|first=Carl B|author-link=Carl Benjamin Boyer|date=1 June 2021|title=Leonhard Euler|url=https://www.britannica.com/biography/Leonhard-Euler|access-date=27 May 2021|encyclopedia=[[Encyclopædia Britannica|Encyclopedia Britannica]]}}</ref>

<ref name=bullock>{{cite journal
 | last1 = Bullock | first1 = John D.
 | last2 = Warwar | first2 = Ronald E.
 | last3 = Hawley | first3 = H. Bradford
 | date = April 2022
 | doi = 10.1080/26375451.2022.2052493
 | journal = [[British Journal for the History of Mathematics]]
 | pages = 24–42
 | title = Why was Leonhard Euler blind?| volume = 37
 | s2cid = 247868159
 }}</ref>

<ref name=caldwell>{{cite web|last1=Caldwell|first1=Chris |author-link1=PrimePages |url=http://primes.utm.edu/notes/by_year.html|title=The largest known prime by year|work=PrimePages|publisher=[[University of Tennessee at Martin]]|access-date=9 June 2021}}</ref>

<ref name=Cauchy>{{cite journal|last=Cauchy|first=A. L.|author-link=Augustin-Louis Cauchy|year=1813|title=Recherche sur les polyèdres – premier mémoire|journal=Journal de l'École polytechnique|volume=9 (Cahier 16)|pages=66–86|language=fr|url=https://books.google.com/books?id=4Wk_AAAAcAAJ&pg=PA68|access-date=10 June 2021}}</ref>

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<ref name=childhood>{{cite book |last= James |first= Ioan |author-link=Ioan James|title= Remarkable Mathematicians: From Euler to von Neumann |url= https://archive.org/details/remarkablemathem0000jame |url-access= registration |publisher= [[Cambridge University Press]] |year= 2002|page=[https://archive.org/details/remarkablemathem0000jame/page/2 2] |isbn= 978-0-521-52094-2}}</ref>
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<ref name=condorcet>{{cite web| url = http://www.math.dartmouth.edu/~euler/historica/condorcet.html| title = Eulogy of Euler&nbsp;– Condorcet| access-date = 30 August 2006| author = Marquis de Condorcet |author-link=Marquis de Condorcet }}</ref>

<ref name=dartm>{{cite journal|first1=Leonhard|last1=Euler |author-link1=Leonhard Euler |language=la|title=Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum|journal=Academiae Imperialis Scientiarum Petropolitanae|trans-title=Foundations of Differential Calculus, with Applications to Finite Analysis and Series|volume=1|url=https://scholarlycommons.pacific.edu/euler-works/212/|year=1787|pages=1–880|publisher=Petri Galeatii|via=Euler Archive|access-date=8 June 2021}}</ref>

<ref name=dartm2>{{cite book|url=https://scholarlycommons.pacific.edu/euler-works/65/|first=Leonhard|last=Euler|title=Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti|language=la|trans-title=A method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense|publisher=Bosquet|year=1744|via=Euler archive|access-date=8 June 2021}}</ref>

<ref name=diderot>See:
* {{cite journal| last = Brown | first = B. H.|date=May 1942| title = The Euler–Diderot anecdote| url = https://archive.org/details/sim_american-mathematical-monthly_1942-05_49_5/page/302 | journal =[[The American Mathematical Monthly]]| volume = 49| issue = 5| pages = 302–303| doi = 10.2307/2303096| jstor = 2303096}}
* {{cite journal| last = Gillings | first = R. J.|date=February 1954| title = The so-called Euler–Diderot incident| url = https://archive.org/details/sim_american-mathematical-monthly_1954-02_61_2/page/76 | journal =[[The American Mathematical Monthly]]| volume = 61| issue = 2| pages = 77–80| doi = 10.2307/2307789| jstor = 2307789}}
* {{cite book | title = A Concise History of Mathematics | edition = 3rd revised | last1 = Struik | first1 = Dirk J. | publisher = [[Dover Books]] | year = 1967 | page = [https://archive.org/details/concisehistoryof0000stru_m6j1/page/129 129] | author-link = Dirk Jan Struik | isbn = 978-0-486-60255-4 | url = https://archive.org/details/concisehistoryof0000stru_m6j1/page/129 }}</ref>

<ref name='Digital Copy of "Letters to a German Princess"'>{{cite book|last1=Euler|first1=Leonhard |author-link1= Leonhard Euler |title=Letters of Euler on Different Subjects of Physics and Philosophy, Addressed to a German Princess |title-link=Letters to a German Princess |translator-first=Henry|translator-last=Hunter|edition=2nd|year=1802|publisher=Murray and Highley|location=London}} [https://archive.org/details/letterseulertoa00eulegoog Archived via] Internet Archives</ref>

<ref name=enestrom>{{cite web|url=https://scholarlycommons.pacific.edu/euler-works/|title=The Eneström Index|work=Euler Archive|access-date=27 May 2021}}</ref>

<ref name=euler2>{{cite journal|last=Euler|first=Leonhard|title=Principes généraux de l'état d'équilibre d'un fluide|trans-title=General principles of the state of equilibrium of a fluid|journal=Académie Royale des Sciences et des Belles-Lettres de Berlin, Mémoires|year=1757|volume=11|pages=217–273|language=fr|url=https://scholarlycommons.pacific.edu/euler-works/225/|access-date=12 June 2021}} Translated into English as {{cite arXiv|title=Translation of Leonhard Euler's: General Principles of the Motion of Fluids|first=Uriel|last=Frisch|author-link=Uriel Frisch|eprint=0802.2383|year=2008|class=nlin.CD}}</ref>

<ref name=Feynman>{{cite book |last1= Feynman|first1= Richard|author-link1=Richard Feynman|title= The Feynman Lectures on Physics |title-link=The Feynman Lectures on Physics |volume= I|page=10 |chapter= Chapter 22: Algebra |date= 1970|chapter-url=https://feynmanlectures.caltech.edu/I_22.html#Ch22-S5}}</ref>

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

<ref name=fussletter>{{cite web |last1=Fuß |first1=Paul Heinrich |last2=Gauß |first2=Carl Friedrich |title=Carl Friedrich Gauß → Paul Heinrich Fuß, Göttingen, 1849 Sept. 11 |url=https://gauss.adw-goe.de/handle/gauss/218 |date=11 September 1849}}</ref>

<ref name=gibbons>{{cite book |first1=Alan |last1=Gibbons |title=Algorithmic Graph Theory |url=https://books.google.com/books?id=Be6t04pgggwC&pg=PA72 |year=1985 |publisher=[[Cambridge University Press]] |isbn=978-0-521-28881-1 |page=72 |access-date=12 November 2015 }}</ref>

<ref name=gindikin>{{cite book|last=Gindikin|first= Simon|author-link=Simon Gindikin|title=Tales of Mathematicians and Physicists|publisher=[[Springer Publishing]]|year=2007|isbn=978-0-387-48811-0|contribution=Leonhard Euler|pages=171–212|doi=10.1007/978-0-387-48811-0_7}} See in particular [https://books.google.com/books?id=Ng1Vn4byYHUC&pg=PA182 p. 182].</ref>

<ref name=Grinstein>{{cite encyclopedia | title = Euler, Leonhard (1707–1783)|encyclopedia=Encyclopedia of Mathematics Education | last1 = Grinstein | first1 = Louise | last2 = Lipsey | first2 = Sally I. | publisher = [[Routledge]] | year = 2001 | page = 235 | isbn = 978-0-415-76368-4}}</ref>

<ref name=hopwil>{{cite book
 | last1 = Hopkins | first1 = Brian
 | last2 = Wilson | first2 = Robin
 | author-link2 = Robin Wilson (mathematician)
 | contribution = Euler's science of combinations
 | mr = 3890500
 | pages = 395–408
 | publisher = Elsevier | location = Amsterdam
 | series = Stud. Hist. Philos. Math.
 | title = Leonhard Euler: Life, Work and Legacy
 | volume = 5
 | year = 2007}}</ref>

<ref name="pluss">{{Cite web|last=Plüss|first=Matthias|title=Der Goethe der Mathematik |url=http://www.snf.ch/de/fokusForschung/newsroom/Seiten/news-160620-horizonte-der-goethe-der-mathematik.aspx |access-date=16 June 2021 |website=[[Swiss National Science Foundation]]}}</ref>

<ref name=kleinert>{{cite journal|last=Kleinert|first=Andreas|doi=10.4467/23921749pkhn_pau.16.002.5258|journal=Prace Komisji Historii Nauki PAU|pages=13–35|publisher=[[Jagiellonian University]]|title=Leonhardi Euleri Opera omnia: Editing the works and correspondence of Leonhard Euler|volume=14|year=2015|doi-access=free}}</ref>

<ref name=Laplace>{{harvnb|Dunham|1999|p=xiii}} "Lisez Euler, lisez Euler, c'est notre maître à tous."</ref>

<ref name=li>{{cite journal
 | last = Li | first = Shengcai
 | date = October 2015
 | doi = 10.1098/rsfs.2015.0020
 | issue = 5
 | journal = [[Interface Focus]]
 |article-number= 20150020
 | publisher = [[Royal Society]]
 | title = Tiny bubbles challenge giant turbines: Three Gorges puzzle
 | volume = 5| pmid = 26442144
 | pmc = 4549846
 }}</ref>

<ref name=lemanski>{{cite journal | last = Lemanski | first = Jens | journal = Logic-Philosophical Studies | pages = 98–122 | title = Means or end? On the valuation of logic diagrams | url = https://philarchive.org/rec/LEMMOE | volume = 14 | year = 2016}}</ref>

<ref name=Lhuillier>{{cite journal|last=L'Huillier|first=S.-A.-J.|author-link=Simon Antoine Jean L'Huilier|title=Mémoire sur la polyèdrométrie|journal=[[Annales de Gergonne|Annales de mathématiques pures et appliquées]]|volume=3|year=1812–1813|pages=169–189|url=http://www.numdam.org/item/AMPA_1812-1813__3__169_0/|access-date=10 June 2021}}</ref>

<ref name=Lynch>{{cite news |last=Lynch |first=Peter |date=September 2017 |title=Euler and the failed fountain of Sanssouci — that's maths: Frederick the Great ignored the advice of a genius in maths and physics |url=https://www.irishtimes.com/news/science/euler-and-the-failed-fountain-of-sanssouci-1.3205969 |work=Irish Times |access-date=26 December 2023}}</ref>

<ref name=logic>{{cite journal |last=Baron |first=Margaret E. |author-link=Margaret Baron|title=A note on the historical development of logic diagrams |journal=[[The Mathematical Gazette]] |volume=53 |issue=383 |pages=113–125 |date=May 1969 |doi=10.2307/3614533 |jstor=3614533|s2cid=125364002 }}</ref>

<ref name="mathg">{{MathGenealogy|id=38586}} Retrieved 2021-07-02</ref>

<ref name=novaacta>{{cite journal|year=1783|title=Éloge de M. Léonhard Euler|trans-title=Eulogy for Leonhard Euler|language=fr|first1=Nicolas|last1=Fuss |author-link1=Nicolas Fuss |journal=Nova Acta Academiae Scientiarum Imperialis Petropolitanae|volume=1|pages=159–212|url=https://www.biodiversitylibrary.org/item/38629#page/177/mode/1up|via=Bioheritage Diversity Library|access-date=19 May 2018}} Translated into English as {{cite web|url=http://www-history.mcs.st-and.ac.uk/~history/Extras/Euler_Fuss_Eulogy.html|title=Eulogy of Leonhard Euler by Nicolas Fuss|access-date=30 August 2006|work=[[MacTutor History of Mathematics archive]]|publisher=[[University of St Andrews]] |translator-first=John S. D.|translator-last=Glaus}}</ref>

<ref name=optics>{{cite journal
| last = Home | first = R. W. | author-link = Roderick Weir Home
| 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 | mr = 0962700
}}</ref>

<ref name=paris-acad>{{cite web|url=http://eulerarchive.maa.org/historica/places/paris.html|title=The Paris Academy|work=Euler Archive|publisher=[[Mathematical Association of America]]|access-date=29 July 2021}}</ref>

<ref name=persee>{{cite journal|last=Marty|first=Jacques|title=Quelques aspects des travaux de Diderot en " mathématiques mixtes "|language=fr|trans-title=Some aspects of Diderot's work in general mathematics|journal=Recherches sur Diderot et sur l'Encyclopédie|volume=4|issue=1|pages=145–147|url=http://www.persee.fr/web/revues/home/prescript/article/rde_0769-0886_1988_num_4_1_954|year=1988|access-date=20 April 2012}}</ref>

<ref name=rodgers>{{cite journal | last = Rodgers | first = Peter | date = June 2014 | doi = 10.1016/j.jvlc.2013.08.006 | issue = 3 | journal = Journal of Visual Languages & Computing | pages = 134–155 | title = A survey of Euler diagrams | volume = 25 | s2cid = 2571971 | url = https://kar.kent.ac.uk/35163/1/JVLC_Euler_Survey.pdf | access-date = 23 July 2021 }}</ref>

<ref name=sten>{{cite book | last = Stén | first = Johan C.-E. | contribution = Academic events in Saint Petersburg | doi = 10.1007/978-3-319-00618-5_7 | pages = 119–135 | publisher = [[Birkhäuser]] | series = Vita Mathematica | title = A Comet of the Enlightenment | volume = 17 | year = 2014| isbn = 978-3-319-00617-8 }} See in particular footnote 37, p. 131.</ref>

<ref name=thiele>{{cite book |last=Thiele |first=Rüdiger | author-link = Rüdiger Thiele|chapter=The mathematics and science of Leonhard Euler |editor1-first=Michael |editor1-last=Kinyon |editor2-first=Glen |editor2-last=van Brummelen |editor-link2=Glen Van Brummelen |title=Mathematics and the Historian's Craft: The Kenneth O. May Lectures |chapter-url=https://books.google.com/books?id=3ZTedZtwYMoC&pg=PA81+ |year=2005 |publisher=[[Springer Publishing]] |isbn=978-0-387-25284-1 |pages=81–140}}</ref>

<ref name=theology>{{cite book|last1=Euler|first1=Leonhard |author-link1=Leonhard Euler |via=Euler Archive |title=Rettung der Göttlichen Offenbahrung gegen die Einwürfe der Freygeister |others=Eneström index 92 |language=de|trans-title=Defense of divine revelation against the objections of the freethinkers|location=Berlin|publisher=Ambrosius Haude and Johann Carl Spener|year=1747|url=https://scholarlycommons.pacific.edu/euler-works/92/|access-date=12 June 2021}}</ref>

<ref name=volumes>{{cite journal |last = Finkel |first = B. F. |year = 1897 |title = Biography – Leonhard Euler |journal = [[The American Mathematical Monthly]] |volume = 4 | issue = 12 |jstor = 2968971|pages = 297–302|doi=10.2307/2968971 | mr=1514436 }}</ref>

<ref name="wolff">{{harvnb|Calinger|1996|pp=153–154}}</ref>

<ref name=yousch>{{cite book |title=Dictionary of Scientific Biography|title-link=Dictionary of Scientific Biography|volume= 4: Richard Dedekind – Firmicus Maternus | editor-first = Charles Coulston | editor-last = Gillispie | editor-link = Charles Coulston Gillispie | publisher = [[Charles Scribner's Sons]] | location = New York | year = 1971 | isbn = 978-0-684-16964-4 |first1=A. P. |last1=Youschkevitch|author-link1=Adolph P. Yushkevich|pages=467–484|contribution=Euler, Leonhard }}</ref>

<ref name=lagarias>{{cite journal |last=Lagarias |first=Jeffrey C. | author-link = Jeffrey Lagarias |date=October 2013 |title=Euler's constant: Euler's work and modern developments |journal=[[Bulletin of the American Mathematical Society]] |volume=50 |issue=4 |page=556 |doi=10.1090/s0273-0979-2013-01423-x |arxiv=1303.1856 |s2cid=119612431|mr=3090422}}</ref>

<ref name=davis>{{cite journal | last = Davis | first = Philip J. | author-link = Philip J. Davis | doi = 10.2307/2309786 | journal = [[The American Mathematical Monthly]] | jstor = 2309786 | mr = 106810 | pages = 849–869 | title = Leonhard Euler's integral: A historical profile of the gamma function | url = https://archive.org/details/sim_american-mathematical-monthly_1959-11_66_9/page/n97 | volume = 66 | year = 1959}}</ref>

<ref name=nickalls>{{cite journal | last = Nickalls | first = R. W. D. | date = March 2009 | issue = 526 | journal = [[The Mathematical Gazette]] | jstor = 40378672 | pages = 66–75 | title = The quartic equation: invariants and Euler's solution revealed | volume = 93| doi = 10.1017/S0025557200184190 | s2cid = 16741834 }}</ref>

<ref name=patterson>{{cite book | last = Patterson | first = S. J. | author-link = Samuel James Patterson | doi = 10.1017/CBO9780511623707 | isbn = 978-0-521-33535-5 | mr = 933558 | page = 1 | publisher = [[Cambridge University Press]] | location = Cambridge | series = Cambridge Studies in Advanced Mathematics | title = An introduction to the theory of the Riemann zeta-function | url = https://books.google.com/books?id=IdHLCgAAQBAJ&pg=PA1 | volume = 14 | year = 1988 | access-date = 6 June 2021 }}</ref>

<ref name="shiu">{{Cite journal |last=Shiu |first=Peter |date=November 2007 |title=Euler's contribution to number theory |journal=[[The Mathematical Gazette]] |volume=91 |issue=522 |pages=453–461 |doi=10.1017/S0025557200182099 |jstor=40378418 |s2cid=125064003}}</ref>

<ref name=stillwell>{{cite book|title=Mathematics and Its History|series=[[Undergraduate Texts in Mathematics]]|first=John|last=Stillwell|author-link=John Stillwell|publisher=Springer|year=2010|isbn=978-1-4419-6052-8|page=40|url=https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA40|access-date=6 June 2021}}.</ref>

<ref name=butcher>{{cite book |last1=Butcher |first1=John C. |author1-link=John C. Butcher |title=Numerical Methods for Ordinary Differential Equations |publisher=[[John Wiley & Sons]] |location=New York |isbn=978-0-471-96758-3 |year=2003 |page=45 |url=https://books.google.com/books?id=okzpIwEX8aEC&pg=PA45 |access-date=8 June 2021 }}</ref>

<ref name=mills>{{cite journal | last = Mills | first = Stella | doi = 10.1007/BF00328047 | issue = 1–3 | journal = [[Archive for History of Exact Sciences]] | mr = 795457 | pages = 1–13 | title = The independent derivations by Leonhard Euler and Colin Maclaurin of the Euler–Maclaurin summation formula | volume = 33 | year = 1985| s2cid = 122119093 }}</ref>

<ref name=ojalvo>{{cite journal | last = Ojalvo | first = Morris | date = December 2007 | doi = 10.1061/(asce)0733-9445(2007)133:12(1686) | issue = 12 | journal = [[Journal of Structural Engineering]] | pages = 1686–1689 | title = Three hundred years of bar theory | volume = 133}}</ref>

<ref name=davidson>{{cite journal | last1 = Davidson | first1 = Michael W. |author-link1=Michael W. Davidson | date = February 2011 | doi = 10.1017/s1551929511000046 | issue = 2 | journal = Microscopy Today | pages = 52–54 | title = Pioneers in Optics: Leonhard Euler and Étienne-Louis Malus | volume = 19| s2cid = 122853454 | doi-access = free }}</ref>

<ref name=pesic>{{cite book|first=Peter|last=Pesic|title=Music and the Making of Modern Science|publisher=[[MIT Press]]|year=2014|contribution-url=https://books.google.com/books?id=HfPvAwAAQBAJ&pg=PA133|contribution=Euler: the mathematics of musical sadness; Euler: from sound to light|pages=133–160|isbn=978-0-262-02727-4|access-date=10 June 2021}}</ref>

<ref name=schmadel>{{cite encyclopedia|title=(2002) Euler|date=2007|encyclopedia=Dictionary of Minor Planet Names|page=162|editor-last=Schmadel|editor-first=Lutz D. |editor-link1=Lutz D. Schmadel |place=Berlin, [[Heidelberg]]|publisher=[[Springer Publishing]]|language=en|doi=10.1007/978-3-540-29925-7_2003|isbn=978-3-540-29925-7|doi-access=free }}</ref>

<ref name=swiss6>{{Cite web|title=Schweizerische Nationalbank (SNB) – Sechste Banknotenserie (1976)|url=https://www.snb.ch/de/iabout/cash/history/id/cash_history_serie6#t7|access-date=15 June 2021|website=[[Swiss National Bank]]}}</ref>

<ref name=swiss7>{{Cite web|title=Schweizerische Nationalbank (SNB) – Siebte Banknotenserie (1984)|url=https://www.snb.ch/de/iabout/cash/history/id/cash_history_serie7#t7|access-date=15 June 2021|website=[[Swiss National Bank]]}}</ref>

<ref name=tegg>{{cite encyclopedia|encyclopedia=London encyclopaedia; or, Universal dictionary of science, art, literature and practical mechanics: comprising a popular view of the present state of knowledge, Volume 4|first=Thomas|last=Tegg|author-link=Thomas Tegg|year=1829|title=Binary logarithms|pages=142–143|url=https://books.google.com/books?id=E-ZTAAAAYAAJ&pg=PA142|access-date=13 June 2021}}</ref>

<ref name=emery>{{cite book|first=Eric|last=Emery|title=Temps et musique|location=Lausanne|publisher=L'Âge d'homme|year=2000|pages=344–345}}</ref>

<ref name=mattheson>{{cite book|first=Johannes|last=Mattheson|author-link=Johann Mattheson|title=Grosse General-Baß-Schule|location=Hamburg|year=1731|volume=I|pages=104–106|oclc=30006387 |url=https://archive.org/details/bub_gb_yRxDAAAAcAAJ}} Mentioned by Euler. Also: {{cite book|first=Johannes|last=Mattheson|title=Exemplarische Organisten-Probe|location=Hamburg|year=1719|pages=57–59 |oclc= |url=https://archive.org/details/exemplarischeor00mattgoog }}</ref>

<ref name=perret>See:
* {{cite book|first=Wilfrid|last=Perret|title=Some Questions of Musical Theory|location=Cambridge|publisher= W. Heffer & Sons|year=1926|pages=60–62 |oclc=3212114}}
* {{cite web|title=What is an Euler-Fokker genus?|url=http://www.huygens-fokker.org/microtonality/efg.html|access-date=12 June 2015|publisher=[[Huygens-Fokker Foundation]] |work=Microtonality}}</ref>

<ref name="de harmoniae">{{cite journal |last1=Euler |first1=Leonhard |author-link1=Leonhard Euler |title=De harmoniae veris principiis per speculum musicum repraesentatis |journal=Novi Commentarii Academiae Scientiarum Petropolitanae |date=1774 |volume=18 |pages=330–353 |url=https://scholarlycommons.pacific.edu/euler-works/457/ |access-date=12 September 2022 |others=Eneström index 457}}</ref>

<ref name=gollin>{{cite conference|first=Edward|last=Gollin|contribution=Combinatorial and transformational aspects of Euler's ''Speculum Musicum''|title=Mathematics and Computation in Music: First International Conference, MCM 2007 Berlin, Germany, May 18–20, 2007, Revised Selected Papers|series=Communications in Computer and Information Science |volume=37|editor1-first=T.|editor1-last=Klouche|editor2-first=Th.|editor2-last=Noll|publisher=Springer|year=2009|pages=406–411|doi=10.1007/978-3-642-04579-0_40|isbn=978-3-642-04578-3 }}</ref>

<ref name="lindley">{{Cite book |last1=Lindley |first1=Mark |title=Mathematical Models of Musical Scales: A New Approach |last2=Turner-Smith |first2=Ronald |date=1993 |publisher=Verlag für Systematische Musikwissenschaft  |isbn=978-3-922626-66-4 |location=Bonn |pages=234–239 |oclc=27789639 |author-link=Mark Lindley}} See also {{cite book|first=Catherine|last=Nolan|contribution=Music Theory and Mathematics|title=The Cambridge History of Western Music Theory|editor-first=Th.|editor-last=Christensen|location=New York|publisher=[[Cambridge University Press]] |year=2002|pages=278–279 |isbn=978-1-139-05347-1 |oclc=828741887}}</ref>

<ref name=bailhache>{{cite web|first=Patrice|last=Bailhache|title=La Musique traduite en Mathématiques: Leonhard Euler|url=http://patrice.bailhache.free.fr/thmusique/euler.html|access-date=12 June 2015|language=fr|work=Communication au colloque du Centre François Viète, "Problèmes de traduction au XVIIIe siècle", Nantes|date=17 January 1997}}</ref>

<ref name=fraser>{{cite book |last=Fraser |first=Craig G. |title=Leonhard Euler's 1744 book on the calculus of variations |url=https://books.google.com/books?id=UdGBy8iLpocC&pg=PA168|isbn=978-0-08-045744-4 |date=11 February 2005 |publisher=Elsevier }} In {{harvnb|Grattan-Guinness|2005|pp=168–180}}</ref>

<ref name=reich>{{cite book |last1=Reich |first1=Karin | author-link = Karin Reich |title='Introduction' to analysis |url=https://books.google.com/books?id=UdGBy8iLpocC&pg=PA181 |isbn=978-0-08-045744-4 |date=11 February 2005 |publisher=Elsevier }} In {{harvnb|Grattan-Guinness|2005|pp=181–190}}</ref>

<ref name=ferraro>{{cite book | last1 = Ferraro | first1 = Giovanni | editor-last = Baker | editor-first = Roger | contribution = Euler's treatises on infinitesimal analysis: ''Introductio in analysin infinitorum, institutiones calculi differentialis, institutionum calculi integralis'' | mr = 2384378 | pages = 39–101 | publisher = Kendrick Press | location = Heber City, UT | title = Euler Reconsidered: Tercentenary Essays | year = 2007 |url=http://oak.ucc.nau.edu/cgg/Ferraro%20entire.pdf |archive-url=https://web.archive.org/web/20220912015017/http://oak.ucc.nau.edu/cgg/Ferraro%20entire.pdf |archive-date=12 September 2022}}</ref>

<ref name=demidov>{{cite book |first=S. S. |last=Demidov |title=Treatise on the differential calculus |url=https://books.google.com/books?id=UdGBy8iLpocC&pg=PA191 |date=2005 |publisher=Elsevier |isbn=978-0-08-045744-4 |access-date=12 November 2015 }} In {{harvnb|Grattan-Guinness|2005|pp=191–198}}.</ref>

<ref name="revaninf">Reviews of ''Introduction to Analysis of the Infinite'':
* {{Cite journal |last=Aiton |first=E. J. |title=Introduction to analysis of the infinite. Book I. Transl. by John D. Blanton. (English) |journal=[[zbMATH]] |zbl=0657.01013}}
* {{cite journal | last = Shiu | first = P. | date = December 1990 | doi = 10.2307/3618156 | issue = 470 | journal = [[The Mathematical Gazette]] | jstor = 3618156 | pages = 392–393 | title = Introduction to analysis of the infinite (Book II), by Leonard Euler (translated by John D. Blanton). | volume = 74}}
* {{cite journal|first=Doru|last=Ştefănescu|title=Euler, Leonhard Introduction to analysis of the infinite. Book I. Translated from the Latin and with an introduction by John D. Blanton.|journal=[[Mathematical Reviews]]|mr=1025504}}</ref>

<ref name="fuss">{{Cite book |last1=Euler |first1=Leonhard |author-link1=Leonhard Euler |title=Opera postuma mathematica et physica anno 1844 detecta quae Academiae scientiarum petropolitanae obtulerunt ejusque auspicus ediderunt auctoris pronepotes Paulus Henricus Fuss et Nicolaus Fuss. |last2=Fuss |first2=Nikola Ivanovich |last3=Fuss |first3=Paul |publisher=[[Russian Academy of Sciences|Imperatorskaia akademīia nauk (Russia)]] |year=1862 |oclc=9094558695 |author-link2=Nicolas Fuss}}</ref>

<ref name=righteye>{{cite journal | last1 = Asensi | first1 = Victor | last2 = Asensi | first2 = Jose M. | date = March 2013 | doi = 10.1093/cid/cit170 | issue = 1 | journal = [[Clinical Infectious Diseases]] | pages = 158–159 | title = Euler's right eye: the dark side of a bright scientist | volume = 57| pmid = 23487386 }}</ref>
<ref name="series ii done">{{cite web |title=The works |url=https://bernoulli-euler-gesellschaft.ch/en/opera_omnia |website=Bernoulli-Euler Society |access-date=11 September 2022 |archive-url=https://web.archive.org/web/20220911182123/https://bernoulli-euler-gesellschaft.ch/en/opera_omnia |archive-date=11 September 2022}}</ref>
<ref name="ivb">{{cite web |title=Leonhardi Euleri Opera Omnia (LEOO) |url=https://bez.unibas.ch/de/projekte/opera-omnia-leonhard-euler/ |website=Bernoulli Euler Center |access-date=11 September 2022 |archive-url=https://web.archive.org/web/20220911205554/https://bez.unibas.ch/de/projekte/opera-omnia-leonhard-euler/ |archive-date=11 September 2022}}</ref>
<ref name="assad">{{Cite journal |last=Assad |first=Arjang A. |date=2007 |title=Leonhard Euler: A brief appreciation |url=https://onlinelibrary.wiley.com/doi/10.1002/net.20158 |journal=Networks |language=en |volume=49 |issue=3 |pages=190–198 |doi=10.1002/net.20158|s2cid=11298706 }}</ref>
<ref name="Ho2">{{Cite book |author=Marquis de Condorcet  |author-link=Marquis de Condorcet |url=http://www.math.dartmouth.edu/~euler/docs/translations/E092trans.pdf |title=Comparison to the Last Edition of Euler's Letters Published by de Condorcet, with the Original Edition: A Defense of the Revelation Against the Objections of Freethinkers, by Mr. Euler Followed by Thoughts by the Author on Religion, Omitted From the Last Edition of his Letters to a Princess of Germany |year=1805 |translator-last=Ho |translator-first=Andie |access-date=26 July 2021 }}</ref>
<ref name="new look">{{Cite book |last1=Varadarajan |first1= V. S. |author-link1=Veeravalli S. Varadarajan |title=Euler through time: a new look at old themes |date=2006 |publisher=[[American Mathematical Society]] |isbn=978-0-8218-3580-7 |oclc=803144928}}</ref>
<ref name="zum werk leonhard">{{Cite book |url=https://link.springer.com/content/pdf/bfm%3A978-3-0348-7121-1%2F1.pdf |title=Zum Werk Leonhard Eulers: Vorträge des Euler-Kolloquiums im Mai 1983 in Berlin |date=May 1983 |publisher=[[Birkhäuser Verlag]] |isbn=978-3-0348-7122-8 |editor-last=Knobloch |editor-first=Eberhard |editor-link=Eberhard Knobloch |doi=10.1007/978-3-0348-7121-1 |editor-last2=Louhivaara |editor-first2=I. S. |editor-last3=Winkler |editor-first3=J.}}</ref>
<ref name="euler archive">{{Cite web |title=The Euler Archive |url=https://scholarlycommons.pacific.edu/euler/ |website=[[University of the Pacific (United States)|University of the Pacific]]}}</ref>
<ref name="BEA">{{Cite encyclopedia |title=Rumovsky, Stepan Yakovlevich |encyclopedia=[[Biographical Encyclopedia of Astronomers]] |publisher=Springer |place=New York |year=2007 |isbn=978-0-387-30400-7 |last=Balashov |first=Yuri |editor1-last=Hockey |editor1-first=Thomas |display-editors=1 |editor2-last=Trimble |editor2-first=Virginia |editor2-link=Virginia Louise Trimble  |pages=991–992 |editor3-last=Williams |editor3-first=Thomas |editor4-last=Bracher |editor4-first=Katherine |editor5-last=Jarrell |editor5-first=Richard |editor6-last=Marché |editor6-first=Jordan D. |editor7-last=Ragep |editor7-first=F. Jamil |doi=10.1007/978-0-387-30400-7_1196 }}</ref>
<ref name="ClarkGolinski1999">{{cite book|last1=Clark|first1=William|last2=Golinski|first2=Jan|last3=Schaffer|first3=Simon|title=The Sciences in Enlightened Europe|url=https://books.google.com/books?id=ttGgd6mec1MC&pg=PA395|year=1999|publisher=[[University of Chicago Press]]|isbn=978-0-226-10940-4|page=395|access-date=15 June 2021}}</ref>

<ref name="zum 300 knobloch">{{Cite journal |last=Knobloch |first=Eberhard |author-link=Eberhard Knobloch |date=2007 |title=Leonhard Euler 1707–1783. Zum 300. Geburtstag eines langjährigen Wahlberliners |journal=Mitteilungen der Deutschen Mathematiker-Vereinigung |volume=15 |issue=4 |pages=276–288 |doi=10.1515/dmvm-2007-0092 |s2cid=122271644|doi-access=free }}</ref>
<ref name="vucinich">{{Cite journal|last1=Vucinich|first1=Alexander |author-link1=Alexander Vucinich |date=1960|title=Mathematics in Russian Culture|journal=[[Journal of the History of Ideas]]|volume=21|issue=2|pages=164–165|doi=10.2307/2708192|jstor=2708192|issn=0022-5037}}</ref>
<ref name="math at prussian">{{cite book |last1=Knobloch |first1=Eberhard |author-link1=Eberhard Knobloch |contribution=Mathematics at the Prussian Academy of Sciences 1700–1810 |date=1998 |doi=10.1007/978-3-0348-8787-8_1 |title=Mathematics in Berlin|editor1-first= Heinrich|editor1-last=Begehr|editor2-first=Helmut|editor2-last= Koch |editor-link2=Helmut Koch |editor3-first=Jürg|editor3-last=Kramer|editor4-first= Norbert|editor4-last=Schappacher |editor-link4=Norbert Schappacher |editor5-first= Ernst-Jochen|editor5-last= Thiele |pages=1–8 |place=Basel |publisher=[[Birkhäuser|Birkhäuser Basel]]  |isbn=978-3-7643-5943-0 }}</ref>
<ref name="historian's craft">{{cite book |last1=Thiele |first1=Rüdiger |author-link1=Rüdiger Thiele |title=Mathematics and the Historian's Craft |chapter=The Mathematics and Science of Leonhard Euler (1707–1783) |date=2005  |series=CMS Books in Mathematics |pages=81–140 |place=New York |publisher=[[Springer Publishing]] |doi=10.1007/0-387-28272-6_6 |isbn=978-0-387-25284-1 }}</ref>
<ref name="fountains">{{Cite journal |last=Eckert |first=Michael |date=2002 |title=Euler and the Fountains of Sanssouci |journal=[[Archive for History of Exact Sciences]] |volume=56 |issue=6 |pages=451–468 |doi=10.1007/s004070200054 |s2cid=121790508 |issn=0003-9519}}</ref>
<ref name="lexell's theorem">{{Cite journal|last1=Maehara|first1=Hiroshi|last2=Martini|first2=Horst|date=2017|title=On Lexell's Theorem|journal=[[The American Mathematical Monthly]]|volume=124|issue=4|pages=337–344|doi=10.4169/amer.math.monthly.124.4.337|issn=0002-9890|jstor=10.4169/amer.math.monthly.124.4.337|s2cid=125175471}}</ref>
}}

===Sources===
{{Refbegin|30em}}
* {{cite journal |last= Calinger |first=Ronald | year = 1996| title = Leonhard Euler: The First St. Petersburg Years (1727–1741)| journal = [[Historia Mathematica]] | volume = 23| issue = 2| pages= 121–166 | doi = 10.1006/hmat.1996.0015| doi-access = free}}
* {{cite book |last=Calinger |first=Ronald |date=2016 |title=Leonhard Euler: Mathematical Genius in the Enlightenment |url=http://press.princeton.edu/titles/10531.html |publisher=[[Princeton University Press]] |isbn=978-0-691-11927-4 |access-date=4 January 2017 }}
* {{cite book |author-link=William Dunham (mathematician) |first=William |last=Dunham |title=Euler: The Master of Us All |url=https://books.google.com/books?id=uKOVNvGOkhQC |year=1999 |publisher=[[Mathematical Association of America]] |isbn=978-0-88385-328-3 |series=Dolciani Mathematical Expositions |volume=22 |access-date=12 November 2015 }}
* {{cite book|first1=Leonhard|last1=Euler |author-link1=Leonhard Euler |via=Euler archive|url=https://scholarlycommons.pacific.edu/euler-works/33/|title=Tentamen novae theoriae musicae|trans-title=An attempt at a new theory of music, exposed in all clearness, according to the most well-founded principles of harmony|language=la|location=St. Petersburg|publisher=[[Russian Academy of Sciences|Imperial Academy of Sciences]]|year=1739|access-date=12 June 2021 }}
* {{cite book | first = Giovanni | last = Ferraro | title = The Rise and Development of the Theory of Series up to the Early 1820s |publisher=[[Springer Science+Business Media]] | url = https://books.google.com/books?id=vLBJSmA9zgAC | date = 2008 | isbn = 978-0-387-73467-5 | access-date = 27 May 2021 }}
* {{Cite book |first1=I. R. |last1=Gekker |first2=A. A. |last2=Euler |chapter=Leonhard Euler's family and descendants |editor1-first=Nikolaĭ Nikolaevich |editor1-last=Bogolyubov |editor1-link=Nikolay Bogolyubov |editor2-first=G. K. |editor2-last=Mikhaĭlov |editor3-first=Adolph Pavlovich |editor3-last=Yushkevich |editor3-link=Adolph P. Yushkevich |title=Euler and Modern Science |url=https://books.google.com/books?id=Ta9bz1wv79AC |year=2007 |publisher=[[Mathematical Association of America]] |isbn=978-0-88385-564-5 |translator=Robert Burns |access-date=12 November 2015 }}
* {{Cite journal|last=Gautschi|first=Walter|author-link=Walter Gautschi|date=2008|title=Leonhard Euler: His Life, the Man, and His Works|journal=[[SIAM Review]]|volume=50|issue=1|pages=3–33|doi=10.1137/070702710|jstor=20454060|issn=0036-1445 |bibcode = 2008SIAMR..50....3G |citeseerx=10.1.1.177.8766 }}
* {{cite book |editor1-first=Ivor |editor1-last=Grattan-Guinness |editor1-link=Ivor Grattan-Guinness |title=Landmark Writings in Western Mathematics 1640–1940 |url=https://books.google.com/books?id=UdGBy8iLpocC |year=2005 |publisher=Elsevier |isbn=978-0-08-045744-4 }}
* {{cite book|first=David S.|last=Richeson|author-link= David Richeson |title=Euler's Gem: The Polyhedron Formula and the Birth of Topology|at=[https://books.google.com/books?id=KUYLhOVkaV4C&pg=PA17 p. 17]|year=2012|publisher=[[Princeton University Press]]|isbn=978-1-4008-3856-1|title-link= Euler's Gem}}
{{Refend}}

==Further reading==
{{refbegin|60em}}
* {{cite book |first1=Robert E. |last1=Bradley |first2=Lawrence A. |last2=D'Antonio |first3=Charles Edward |last3=Sandifer |title=Euler at 300: An Appreciation |url=https://books.google.com/books?id=tK_KRmTf9nUC |year=2007 |publisher=[[Mathematical Association of America]] |isbn=978-0-88385-565-2 }}
* {{cite book |editor1-first=Robert E. |editor1-last=Bradley |editor2-first=Charles Edward |editor2-last=Sandifer |title=Leonhard Euler: Life, Work and Legacy |url=https://books.google.com/books?id=75vJL_Y-PvsC |year=2007 |publisher=Elsevier |isbn=978-0-444-52728-8 |series=Studies in the History and Philosophy of Mathematics |volume=5 |access-date=8 June 2021 }}
* {{cite book |first1=William |last1=Dunham |author-link1=William Dunham (mathematician) |title=The Genius of Euler: Reflections on his Life and Work |url=https://books.google.com/books?id=A6by_UpQikIC |year=2007 |publisher=[[Mathematical Association of America]] |isbn=978-0-88385-558-4 }}
* {{cite book|editor1-last=Hascher|editor1-first=Xavier|editor2-last=Papadopoulos|editor2-first=Athanase|year=2015|title=Leonhard Euler: Mathématicien, physicien et théoricien de la musique|language=fr|location=Paris|publisher=CNRS Editions|isbn=978-2-271-08331-9|url=https://books.google.com/books?id=KVcGrgEACAAJ|access-date=8 June 2021 }}
* {{cite book |first=C. Edward |last=Sandifer |title=The Early Mathematics of Leonhard Euler |url=https://books.google.com/books?id=CvBxLr_0uBQC |year=2007 |publisher=[[Mathematical Association of America]] |isbn=978-0-88385-559-1 }}
* {{cite book |first=C. Edward |last=Sandifer |title=How Euler Did It |url=https://books.google.com/books?id=sohHs7ExOsYC |year=2007 |publisher=[[Mathematical Association of America]] |isbn=978-0-88385-563-8 }}
* {{cite book |first=C. Edward |last=Sandifer |title=How Euler Did Even More |url=https://books.google.com/books?id=3c6iBQAAQBAJ |year=2015 |publisher=[[Mathematical Association of America]] |isbn=978-0-88385-584-3 |access-date=8 June 2021 }}
* {{cite journal|editor-first=Doris|editor-last=Schattschneider|editor-link=Doris Schattschneider |date=November 1983 | title = A Tribute to Leonhard Euler 1707–1783 (special issue) | journal = [[Mathematics Magazine]] | volume = 56 | issue = 5 | jstor = i326726}}
{{refend}}

==External links==
{{Sister project links|wikt=no|b=no|n=no|q=Leonhard Euler|s=Author:Leonhard Euler|v=no|voy=no|species=no|d=q7604}}
* {{MathGenealogy|id=38586}}
* [http://eulerarchive.maa.org/ The Euler Archive]: Composition of Euler works with translations into English
* [http://lettre.digital/euler/ Opera-Bernoulli-Euler] (compiled works of Euler, Bernoulli family, and contemporary peers)
* [http://www.euler-2007.ch/en/index.htm Euler Tercentenary 2007]
* [http://www.eulersociety.org/ The Euler Society]
* [https://euler.bbaw.de/ Euleriana] at the [[Berlin-Brandenburg Academy of Sciences and Humanities]]
* [http://www.math.dartmouth.edu/~euler/historica/family-tree.html Euler Family Tree]
* [https://web.archive.org/web/20110518054936/http://friedrich.uni-trier.de/oeuvres/20/219/ Euler's Correspondence with Frederick the Great, King of Prussia]
* {{Librivox author |id=12395}}
* {{MacTutor Biography|id=Euler|mode=cs1}}
* {{cite web|title=An Evening with Leonhard Euler|date=24 September 2009 |publication-date=9 November 2009|last1=Dunham |first1=William |author-link1=William Dunham (mathematician)|publisher=philoctetesctr|website=YouTube|url=https://www.youtube.com/watch?v=h-DV26x6n_Q |location=[[Muhlenberg College]]}} (talk given by [[William Dunham (mathematician)|William Dunham]] at )
* {{cite web |last1=Dunham |first1=William |title=A Tribute to Euler – William Dunham|date=14 October 2008 |location=Muhlenberg College|publication-date=23 November 2011 |publisher=PoincareDuality |website=YouTube |url=https://www.youtube.com/watch?v=fEWj93XjON0}}

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