Leonhard Euler

Leonhard Euler

Overview
Leonhard Euler (ˈɔʏlɐ, English approximation, "Oiler"; 15 April 1707 18 September 1783) was a pioneering Swiss
Swiss (people)
The Swiss are citizens or natives of Switzerland. The demonym derives from the toponym of Schwyz and has been in widespread use to refer to the Old Swiss Confederacy since the 16th century....

 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 and physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

. He made important discoveries in fields as diverse as infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

 and graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, such as the notion of a mathematical function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. He is also renowned for his work in mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

, fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

, and astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

.

Euler spent most of his adult life in St.
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Unanswered Questions
Quotations

Madam, I have come from a country where people are hanged if they talk.

In Berlin, to the Queen Mother of Prussia, on his lack of conversation in his meeting with her, on his return from Russia; as quoted in Science in Russian Culture : A History to 1860 (1963) Alexander Vucinich

Now I will have less distraction.

Upon losing the use of his right eye; as quoted in In Mathematical Circles (1969) by H. Eves

Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.

As quoted in Calculus Gems (1992) by G. Simmons

To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.

As quoted in Fundamentals of Teaching Mathematics at University Level (2000) by Benjamin Baumslag, p. 214 :Translated as Introduction to Analysis of the Infinite (1988-89) by John Blanton (Book I ISBN 0387968245; Book II ISBN 0387971327).

Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.

For the sake of brevity, we will always represent e (mathematical constant)|this number 2.718281828459... by the letter e.

Encyclopedia
Leonhard Euler (ˈɔʏlɐ, English approximation, "Oiler"; 15 April 1707 18 September 1783) was a pioneering Swiss
Swiss (people)
The Swiss are citizens or natives of Switzerland. The demonym derives from the toponym of Schwyz and has been in widespread use to refer to the Old Swiss Confederacy since the 16th century....

 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 and physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

. He made important discoveries in fields as diverse as infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

 and graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, such as the notion of a mathematical function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. He is also renowned for his work in mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

, fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

, and astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

.

Euler spent most of his adult life in St. Petersburg, Russia
Russian Empire
The Russian Empire was a state that existed from 1721 until the Russian Revolution of 1917. It was the successor to the Tsardom of Russia and the predecessor of the Soviet Union...

, and in Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...

, Prussia
Kingdom of Prussia
The Kingdom of Prussia was a German kingdom from 1701 to 1918. Until the defeat of Germany in World War I, it comprised almost two-thirds of the area of the German Empire...

. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto
Quarto (text)
Quarto is a book or pamphlet produced from full 'blanksheets', each of which is printed with eight pages of text, four to a side, then folded two times to produce four leaves...

 volumes. A statement attributed to Pierre-Simon Laplace
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...

 expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Euler, read Euler, he is the master of us all."

Early years



Euler was born on April 15, 1707, in Basel
Basel
Basel or Basle In the national languages of Switzerland the city is also known as Bâle , Basilea and Basilea is Switzerland's third most populous city with about 166,000 inhabitants. Located where the Swiss, French and German borders meet, Basel also has suburbs in France and Germany...

 to Paul Euler, a pastor
Pastor
The word pastor usually refers to an ordained leader of a Christian congregation. When used as an ecclesiastical styling or title, this role may be abbreviated to "Pr." or often "Ps"....

 of the Reformed Church. His mother was Marguerite Brucker, a pastor's daughter. He had two younger sisters named Anna Maria and Maria Magdalena. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen
Riehen
Riehen is a municipality in the canton of Basel-Stadt in Switzerland. Together with the city of Basel and Bettingen, Riehen is one of three municipalities in the canton....

, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli family—Johann Bernoulli
Johann Bernoulli
Johann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...

, who was then regarded as Europe's foremost mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, would eventually be the most important influence on young Leonhard. Euler's early formal education started in Basel, where he was sent to live with his maternal grandmother. At the age of thirteen he enrolled at the University of Basel
University of Basel
The University of Basel is located in Basel, Switzerland, and is considered to be one of leading universities in the country...

, and in 1723, received his Master of Philosophy with a dissertation that compared the philosophies of Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...

 and Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. Euler was at this point studying theology
Theology
Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...

, Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

, and Hebrew
Hebrew language
Hebrew is a Semitic language of the Afroasiatic language family. Culturally, is it considered by Jews and other religious groups as the language of the Jewish people, though other Jewish languages had originated among diaspora Jews, and the Hebrew language is also used by non-Jewish groups, such...

 at his father's urging, in order to become a pastor, but Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

 with the title De Sono. At that time, he was pursuing an (ultimately unsuccessful) attempt to obtain a position at the University of Basel. In 1727, he entered the Paris Academy
French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...

 Prize Problem
competition, where the problem that year was to find the best way to place the mast
Mast (sailing)
The mast of a sailing vessel is a tall, vertical, or near vertical, spar, or arrangement of spars, which supports the sails. Large ships have several masts, with the size and configuration depending on the style of ship...

s on a ship. He won second place, losing only to Pierre Bouguer
Pierre Bouguer
Pierre Bouguer was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture"....

—a man now known as "the father of naval architecture". Euler subsequently won this coveted annual prize twelve times in his career.

St. Petersburg


Around this time Johann Bernoulli's two sons, Daniel
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

 and Nicolas
Nicolaus II Bernoulli
Nicolaus II Bernoulli, a.k.a. Niklaus Bernoulli, Nikolaus Bernoulli, was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family....

, were working at the Imperial Russian Academy of Sciences
Russian Academy of Sciences
The Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

 in St Petersburg. On July 10, 1726, Nicolas died of appendicitis
Appendicitis
Appendicitis is a condition characterized by inflammation of the appendix. It is classified as a medical emergency and many cases require removal of the inflamed appendix, either by laparotomy or laparoscopy. Untreated, mortality is high, mainly because of the risk of rupture leading to...

 after spending a year in Russia, and when Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.


Euler arrived in the Russian capital on 17 May 1727. He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he often worked in close collaboration. Euler mastered Russian
Russian language
Russian is a Slavic language used primarily in Russia, Belarus, Uzbekistan, Kazakhstan, Tajikistan and Kyrgyzstan. It is an unofficial but widely spoken language in Ukraine, Moldova, Latvia, Turkmenistan and Estonia and, to a lesser extent, the other countries that were once constituent republics...

 and settled into life in St Petersburg. He also took on an additional job as a medic in the Russian Navy.

The Academy at St. Petersburg, established by Peter the Great
Peter I of Russia
Peter the Great, Peter I or Pyotr Alexeyevich Romanov Dates indicated by the letters "O.S." are Old Style. All other dates in this article are New Style. ruled the Tsardom of Russia and later the Russian Empire from until his death, jointly ruling before 1696 with his half-brother, Ivan V...

, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy in order to lessen the faculty's teaching burden, and the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions.

The Academy's benefactress, Catherine I
Catherine I of Russia
Catherine I , the second wife of Peter the Great, reigned as Empress of Russia from 1725 until her death.-Life as a peasant woman:The life of Catherine I was said by Voltaire to be nearly as extraordinary as that of Peter the Great himself. There are no documents that confirm her origins. Born on...

, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility then gained power upon the ascension of the twelve-year-old Peter II
Peter II of Russia
Pyotr II Alekseyevich was Emperor of Russia from 1727 until his death. He was the only son of Tsarevich Alexei Petrovich, son of Peter I of Russia by his first wife Eudoxia Lopukhina, and Princess Charlotte, daughter of Duke Louis Rudolph of Brunswick-Lüneburg and sister-in-law of Charles VI,...

. The nobility were suspicious of the academy's foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues.

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

On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell
Georg Gsell
Georg Gsell was a Swiss Baroque painter, art consultant and art dealer. Was recruited by Peter the Great in 1716 and went to Russia...

, a painter from the Academy Gymnasium. The young couple bought a house by the Neva River
Neva River
The Neva is a river in northwestern Russia flowing from Lake Ladoga through the western part of Leningrad Oblast to the Neva Bay of the Gulf of Finland. Despite its modest length , it is the third largest river in Europe in terms of average discharge .The Neva is the only river flowing from Lake...

. Of their thirteen children, only five survived childhood.

Berlin


Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy
Prussian Academy of Sciences
The Prussian Academy of Sciences was an academy established in Berlin on 11 July 1700, four years after the Akademie der Künste or "Arts Academy", to which "Berlin Academy" may also refer.-Origins:...

, which he had been offered by Frederick the Great of Prussia. He lived for twenty-five years in Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...

, where he wrote over 380 articles. In Berlin, he published the two works which he would be most renowned for: the Introductio in analysin infinitorum
Introductio in analysin infinitorum
Introductio in analysin infinitorum is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis...

, a text on functions published in 1748, and the Institutiones calculi differentialis
Institutiones calculi differentialis
Institutiones calculi differentialis is a mathematical work written in 1748 by Leonhard Euler and published in 1755 that lays the groundwork for the differential calculus...

, published in 1755 on differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

. In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2...

.

In addition, Euler was asked to tutor the Princess of Anhalt-Dessau, Frederick's niece. Euler wrote over 200 letters to her, which were later compiled into a best-selling volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess. This work contained Euler's exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler's personality and religious beliefs. This book became more widely read than any of his mathematical works, and it was published across Europe and in the United States. The popularity of the 'Letters' testifies to Euler's ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist.

Despite Euler's immense contribution to the Academy's prestige, he was eventually forced to leave Berlin. This was partly because of a conflict of personality with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy. Voltaire
Voltaire
François-Marie Arouet , better known by the pen name Voltaire , was a French Enlightenment writer, historian and philosopher famous for his wit and for his advocacy of civil liberties, including freedom of religion, free trade and separation of church and state...

 was among those in Frederick's employ, and the Frenchman enjoyed a prominent position in the king's social circle. Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes. He was in many ways the direct opposite of Voltaire. Euler had limited training in rhetoric
Rhetoric
Rhetoric is the art of discourse, an art that aims to improve the facility of speakers or writers who attempt to inform, persuade, or motivate particular audiences in specific situations. As a subject of formal study and a productive civic practice, rhetoric has played a central role in the Western...

, and tended to debate matters that he knew little about, making him a frequent target of Voltaire's wit. Frederick
Frederick II of Prussia
Frederick II was a King in Prussia and a King of Prussia from the Hohenzollern dynasty. In his role as a prince-elector of the Holy Roman Empire, he was also Elector of Brandenburg. He was in personal union the sovereign prince of the Principality of Neuchâtel...

 also expressed disappointment with Euler's practical engineering abilities:

Eyesight deterioration


Euler's eyesight worsened throughout his mathematical career. Three years after suffering a near-fatal fever
Fever
Fever is a common medical sign characterized by an elevation of temperature above the normal range of due to an increase in the body temperature regulatory set-point. This increase in set-point triggers increased muscle tone and shivering.As a person's temperature increases, there is, in...

 in 1735 he became nearly blind in his right eye, but Euler rather blamed his condition on the painstaking work on cartography
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

 he performed for the St. Petersburg Academy. Euler's sight in that eye worsened throughout his stay in Germany, so much so that Frederick referred to him as "Cyclops
Cyclops
A cyclops , in Greek mythology and later Roman mythology, was a member of a primordial race of giants, each with a single eye in the middle of his forehead...

". Euler later suffered a cataract
Cataract
A cataract is a clouding that develops in the crystalline lens of the eye or in its envelope, varying in degree from slight to complete opacity and obstructing the passage of light...

 in his good left eye, rendering him almost totally blind a few weeks after its discovery in 1766. Even so, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and photographic memory
Eidetic memory
Eidetic , commonly referred to as photographic memory, is a medical term, popularly defined as the ability to recall images, sounds, or objects in memory with extreme precision and in abundant volume. The word eidetic, referring to extraordinarily detailed and vivid recall not limited to, but...

. For example, Euler could repeat the Aeneid
Aeneid
The Aeneid is a Latin epic poem, written by Virgil between 29 and 19 BC, that tells the legendary story of Aeneas, a Trojan who travelled to Italy, where he became the ancestor of the Romans. It is composed of roughly 10,000 lines in dactylic hexameter...

 of Virgil
Virgil
Publius Vergilius Maro, usually called Virgil or Vergil in English , was an ancient Roman poet of the Augustan period. He is known for three major works of Latin literature, the Eclogues , the Georgics, and the epic Aeneid...

 from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last. With the aid of his scribes, Euler's productivity on many areas of study actually increased. He produced on average one mathematical paper every week in the year 1775.

Return to Russia


The situation in Russia had improved greatly since the accession to the throne of Catherine the Great
Catherine II of Russia
Catherine II, also known as Catherine the Great , Empress of Russia, was born in Stettin, Pomerania, Prussia on as Sophie Friederike Auguste von Anhalt-Zerbst-Dornburg...

, and in 1766 Euler accepted an invitation to return to the St. Petersburg Academy and spent the rest of his life in Russia. His second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife Katharina after 40 years of marriage. Three years after his wife's death Euler married her half sister, Salome Abigail Gsell (1723–1794). This marriage would last until his death.

In St Petersburg on 18 September 1783, after a lunch with his family, during a conversation with a fellow academician
Academician
The title Academician denotes a Full Member of an art, literary, or scientific academy.In many countries, it is an honorary title. There also exists a lower-rank title, variously translated Corresponding Member or Associate Member, .-Eastern Europe and China:"Academician" may also be a functional...

 Anders Johan Lexell
Anders Johan Lexell
Anders Johan Lexell was a Swedish-born Russian astronomer, mathematician, and physicist who spent most of his life in Russia where he is known as Andrei Ivanovich Leksel .Lexell made important discoveries in polygonometry and celestial mechanics; the latter led to a comet named in...

 about the newly-discovered Uranus
Uranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

 and its orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

, Euler suffered a brain hemorrhage and died a few hours later. A short obituary for the Russian Academy of Sciences
Russian Academy of Sciences
The Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

 was written by Jacob von Shtelin and a more detailed eulogy was written and delivered at a memorial meeting by Russian mathematician Nicolas Fuss
Nicolas Fuss
Nicolas Fuss , also known as Nikolai Fuss, was a Swiss mathematician.Fuss was born in Basel, Switzerland. He moved to Saint Petersburg to serve as a mathematical assistant to Leonhard Euler from 1773–1783, and remained there until his death...

, one of the Euler's disciples. In the eulogy written for the French Academy by the French mathematician and philosopher Marquis de Condorcet
Marquis de Condorcet
Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet , known as Nicolas de Condorcet, was a French philosopher, mathematician, and early political scientist whose Condorcet method in voting tally selects the candidate who would beat each of the other candidates in a run-off election...

, he commented,
He was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island
Vasilievsky Island
Vasilyevsky Island is an island in Saint Petersburg, Russia, bordered by the rivers Bolshaya Neva and Malaya Neva in the south and northeast, and by the Gulf of Finland in the west. Vasilyevsky Island is separated from Dekabristov Island by the Smolenka River...

. In 1785, the Russian Academy of Sciences
Russian Academy of Sciences
The Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

 put a marble bust of Leonhard Euler on a pedestal next to the Director's seat and, in 1837, placed a headstone on Euler's grave. To commemorate the 250th anniversary of Euler's birth, the headstone was moved in 1956, together with his remains, to the 18th-century necropolis at the Alexander Nevsky Monastery
Alexander Nevsky Lavra
Saint Alexander Nevsky Lavra or Saint Alexander Nevsky Monastery was founded by Peter I of Russia in 1710 at the eastern end of the Nevsky Prospekt in St. Petersburg supposing that that was the site of the Neva Battle in 1240 when Alexander Nevsky, a prince, defeated the Swedes; however, the battle...

.


Contributions to mathematics and physics


Euler worked in almost all areas of mathematics: geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

, trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

, algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, as well as continuum physics, lunar theory
Lunar theory
Lunar theory attempts to account for the motions of the Moon. There are many irregularities in the Moon's motion, and many attempts have been made over a long history to account for them. After centuries of being heavily problematic, the lunar motions are nowadays modelled to a very high degree...

 and other areas of physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

. 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
Quarto (text)
Quarto is a book or pamphlet produced from full 'blanksheets', each of which is printed with eight pages of text, four to a side, then folded two times to produce four leaves...

 volumes. Euler's name is associated with a large number of topics.

Mathematical notation


Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter for the base of the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

 (now also known as Euler's number), the Greek letter Σ
Sigma
Sigma is the eighteenth letter of the Greek alphabet, and carries the 'S' sound. In the system of Greek numerals it has a value of 200. When used at the end of a word, and the word is not all upper case, the final form is used, e.g...

 for summations and the letter to denote the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

. The use of the Greek letter
Pi (letter)
Pi is the sixteenth letter of the Greek alphabet, representing . In the system of Greek numerals it has a value of 80. Letters that arose from pi include Cyrillic Pe , Coptic pi , and Gothic pairthra .The upper-case letter Π is used as a symbol for:...

to denote the ratio of a circle's circumference to its diameter
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

 was also popularized by Euler, although it did not originate with him.

Analysis


The development of infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

 was at the forefront of 18th century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour (in particular his reliance on the principle of the generality of algebra
Generality of algebra
In the history of mathematics, the generality of algebra is phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph Lagrange...

), his ideas led to many great advances.
Euler is well known in analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

 for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as


Notably, Euler directly proved the power series expansions for and the inverse tangent function. (Indirect proof via the inverse power series technique was given by Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 and Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the famous Basel problem
Basel problem
The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...

 in 1735 (he provided a more elaborate argument in 1741):


Euler introduced the use of the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

 and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s, thus greatly expanding the scope of mathematical applications of logarithms. He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s. For any real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 
{{Redirect|Euler}}
{{Portal:Mathematics/Featured article template}}
Leonhard Euler (ˈɔʏlɐ, ({{Audio|LeonhardEulerByDrsDotChRadio.ogg|Swiss German pronunciation}}) ({{Audio|De-Leonard_Euler.ogg|Standard German pronunciation}}) English approximation, "Oiler"; 15 April 1707{{ndash}} 18 September 1783) was a pioneering Swiss
Swiss (people)
The Swiss are citizens or natives of Switzerland. The demonym derives from the toponym of Schwyz and has been in widespread use to refer to the Old Swiss Confederacy since the 16th century....

 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 and physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

. He made important discoveries in fields as diverse as infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

 and graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, such as the notion of a mathematical function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. He is also renowned for his work in mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

, fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

, and astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

.

Euler spent most of his adult life in St. Petersburg, Russia
Russian Empire
The Russian Empire was a state that existed from 1721 until the Russian Revolution of 1917. It was the successor to the Tsardom of Russia and the predecessor of the Soviet Union...

, and in Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...

, Prussia
Kingdom of Prussia
The Kingdom of Prussia was a German kingdom from 1701 to 1918. Until the defeat of Germany in World War I, it comprised almost two-thirds of the area of the German Empire...

. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto
Quarto (text)
Quarto is a book or pamphlet produced from full 'blanksheets', each of which is printed with eight pages of text, four to a side, then folded two times to produce four leaves...

 volumes. A statement attributed to Pierre-Simon Laplace
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...

 expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Euler, read Euler, he is the master of us all."

Early years



Euler was born on April 15, 1707, in Basel
Basel
Basel or Basle In the national languages of Switzerland the city is also known as Bâle , Basilea and Basilea is Switzerland's third most populous city with about 166,000 inhabitants. Located where the Swiss, French and German borders meet, Basel also has suburbs in France and Germany...

 to Paul Euler, a pastor
Pastor
The word pastor usually refers to an ordained leader of a Christian congregation. When used as an ecclesiastical styling or title, this role may be abbreviated to "Pr." or often "Ps"....

 of the Reformed Church. His mother was Marguerite Brucker, a pastor's daughter. He had two younger sisters named Anna Maria and Maria Magdalena. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen
Riehen
Riehen is a municipality in the canton of Basel-Stadt in Switzerland. Together with the city of Basel and Bettingen, Riehen is one of three municipalities in the canton....

, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli family—Johann Bernoulli
Johann Bernoulli
Johann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...

, who was then regarded as Europe's foremost mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, would eventually be the most important influence on young Leonhard. Euler's early formal education started in Basel, where he was sent to live with his maternal grandmother. At the age of thirteen he enrolled at the University of Basel
University of Basel
The University of Basel is located in Basel, Switzerland, and is considered to be one of leading universities in the country...

, and in 1723, received his Master of Philosophy with a dissertation that compared the philosophies of Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...

 and Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. Euler was at this point studying theology
Theology
Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...

, Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

, and Hebrew
Hebrew language
Hebrew is a Semitic language of the Afroasiatic language family. Culturally, is it considered by Jews and other religious groups as the language of the Jewish people, though other Jewish languages had originated among diaspora Jews, and the Hebrew language is also used by non-Jewish groups, such...

 at his father's urging, in order to become a pastor, but Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

 with the title De Sono. At that time, he was pursuing an (ultimately unsuccessful) attempt to obtain a position at the University of Basel. In 1727, he entered the Paris Academy
French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...

 Prize Problem
competition, where the problem that year was to find the best way to place the mast
Mast (sailing)
The mast of a sailing vessel is a tall, vertical, or near vertical, spar, or arrangement of spars, which supports the sails. Large ships have several masts, with the size and configuration depending on the style of ship...

s on a ship. He won second place, losing only to Pierre Bouguer
Pierre Bouguer
Pierre Bouguer was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture"....

—a man now known as "the father of naval architecture". Euler subsequently won this coveted annual prize twelve times in his career.

St. Petersburg


Around this time Johann Bernoulli's two sons, Daniel
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

 and Nicolas
Nicolaus II Bernoulli
Nicolaus II Bernoulli, a.k.a. Niklaus Bernoulli, Nikolaus Bernoulli, was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family....

, were working at the Imperial Russian Academy of Sciences
Russian Academy of Sciences
The Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

 in St Petersburg. On July 10, 1726, Nicolas died of appendicitis
Appendicitis
Appendicitis is a condition characterized by inflammation of the appendix. It is classified as a medical emergency and many cases require removal of the inflamed appendix, either by laparotomy or laparoscopy. Untreated, mortality is high, mainly because of the risk of rupture leading to...

 after spending a year in Russia, and when Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.


Euler arrived in the Russian capital on 17 May 1727. He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he often worked in close collaboration. Euler mastered Russian
Russian language
Russian is a Slavic language used primarily in Russia, Belarus, Uzbekistan, Kazakhstan, Tajikistan and Kyrgyzstan. It is an unofficial but widely spoken language in Ukraine, Moldova, Latvia, Turkmenistan and Estonia and, to a lesser extent, the other countries that were once constituent republics...

 and settled into life in St Petersburg. He also took on an additional job as a medic in the Russian Navy.

The Academy at St. Petersburg, established by Peter the Great
Peter I of Russia
Peter the Great, Peter I or Pyotr Alexeyevich Romanov Dates indicated by the letters "O.S." are Old Style. All other dates in this article are New Style. ruled the Tsardom of Russia and later the Russian Empire from until his death, jointly ruling before 1696 with his half-brother, Ivan V...

, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy in order to lessen the faculty's teaching burden, and the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions.

The Academy's benefactress, Catherine I
Catherine I of Russia
Catherine I , the second wife of Peter the Great, reigned as Empress of Russia from 1725 until her death.-Life as a peasant woman:The life of Catherine I was said by Voltaire to be nearly as extraordinary as that of Peter the Great himself. There are no documents that confirm her origins. Born on...

, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility then gained power upon the ascension of the twelve-year-old Peter II
Peter II of Russia
Pyotr II Alekseyevich was Emperor of Russia from 1727 until his death. He was the only son of Tsarevich Alexei Petrovich, son of Peter I of Russia by his first wife Eudoxia Lopukhina, and Princess Charlotte, daughter of Duke Louis Rudolph of Brunswick-Lüneburg and sister-in-law of Charles VI,...

. The nobility were suspicious of the academy's foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues.

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

On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell
Georg Gsell
Georg Gsell was a Swiss Baroque painter, art consultant and art dealer. Was recruited by Peter the Great in 1716 and went to Russia...

, a painter from the Academy Gymnasium. The young couple bought a house by the Neva River
Neva River
The Neva is a river in northwestern Russia flowing from Lake Ladoga through the western part of Leningrad Oblast to the Neva Bay of the Gulf of Finland. Despite its modest length , it is the third largest river in Europe in terms of average discharge .The Neva is the only river flowing from Lake...

. Of their thirteen children, only five survived childhood.

Berlin


Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy
Prussian Academy of Sciences
The Prussian Academy of Sciences was an academy established in Berlin on 11 July 1700, four years after the Akademie der Künste or "Arts Academy", to which "Berlin Academy" may also refer.-Origins:...

, which he had been offered by Frederick the Great of Prussia. He lived for twenty-five years in Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...

, where he wrote over 380 articles. In Berlin, he published the two works which he would be most renowned for: the Introductio in analysin infinitorum
Introductio in analysin infinitorum
Introductio in analysin infinitorum is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis...

, a text on functions published in 1748, and the Institutiones calculi differentialis
Institutiones calculi differentialis
Institutiones calculi differentialis is a mathematical work written in 1748 by Leonhard Euler and published in 1755 that lays the groundwork for the differential calculus...

, published in 1755 on differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

. In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2...

.

In addition, Euler was asked to tutor the Princess of Anhalt-Dessau, Frederick's niece. Euler wrote over 200 letters to her{{When|date=June 2010}}, which were later compiled into a best-selling volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess. This work contained Euler's exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler's personality and religious beliefs. This book became more widely read than any of his mathematical works, and it was published across Europe and in the United States. The popularity of the 'Letters' testifies to Euler's ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist.

Despite Euler's immense contribution to the Academy's prestige, he was eventually forced to leave Berlin. This was partly because of a conflict of personality with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy. Voltaire
Voltaire
François-Marie Arouet , better known by the pen name Voltaire , was a French Enlightenment writer, historian and philosopher famous for his wit and for his advocacy of civil liberties, including freedom of religion, free trade and separation of church and state...

 was among those in Frederick's employ, and the Frenchman enjoyed a prominent position in the king's social circle. Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes. He was in many ways the direct opposite of Voltaire. Euler had limited training in rhetoric
Rhetoric
Rhetoric is the art of discourse, an art that aims to improve the facility of speakers or writers who attempt to inform, persuade, or motivate particular audiences in specific situations. As a subject of formal study and a productive civic practice, rhetoric has played a central role in the Western...

, and tended to debate matters that he knew little about, making him a frequent target of Voltaire's wit. Frederick
Frederick II of Prussia
Frederick II was a King in Prussia and a King of Prussia from the Hohenzollern dynasty. In his role as a prince-elector of the Holy Roman Empire, he was also Elector of Brandenburg. He was in personal union the sovereign prince of the Principality of Neuchâtel...

 also expressed disappointment with Euler's practical engineering abilities:

{{quote|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
Sanssouci
Sanssouci is the name of the former summer palace of Frederick the Great, King of Prussia, in Potsdam, near Berlin. It is often counted among the German rivals of Versailles. While Sanssouci is in the more intimate Rococo style and is far smaller than its French Baroque counterpart, it too is...

. My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry!}}


Eyesight deterioration


Euler's eyesight worsened throughout his mathematical career. Three years after suffering a near-fatal fever
Fever
Fever is a common medical sign characterized by an elevation of temperature above the normal range of due to an increase in the body temperature regulatory set-point. This increase in set-point triggers increased muscle tone and shivering.As a person's temperature increases, there is, in...

 in 1735 he became nearly blind in his right eye, but Euler rather blamed his condition on the painstaking work on cartography
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

 he performed for the St. Petersburg Academy. Euler's sight in that eye worsened throughout his stay in Germany, so much so that Frederick referred to him as "Cyclops
Cyclops
A cyclops , in Greek mythology and later Roman mythology, was a member of a primordial race of giants, each with a single eye in the middle of his forehead...

". Euler later suffered a cataract
Cataract
A cataract is a clouding that develops in the crystalline lens of the eye or in its envelope, varying in degree from slight to complete opacity and obstructing the passage of light...

 in his good left eye, rendering him almost totally blind a few weeks after its discovery in 1766. Even so, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and photographic memory
Eidetic memory
Eidetic , commonly referred to as photographic memory, is a medical term, popularly defined as the ability to recall images, sounds, or objects in memory with extreme precision and in abundant volume. The word eidetic, referring to extraordinarily detailed and vivid recall not limited to, but...

. For example, Euler could repeat the Aeneid
Aeneid
The Aeneid is a Latin epic poem, written by Virgil between 29 and 19 BC, that tells the legendary story of Aeneas, a Trojan who travelled to Italy, where he became the ancestor of the Romans. It is composed of roughly 10,000 lines in dactylic hexameter...

 of Virgil
Virgil
Publius Vergilius Maro, usually called Virgil or Vergil in English , was an ancient Roman poet of the Augustan period. He is known for three major works of Latin literature, the Eclogues , the Georgics, and the epic Aeneid...

 from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last. With the aid of his scribes, Euler's productivity on many areas of study actually increased. He produced on average one mathematical paper every week in the year 1775.

Return to Russia


The situation in Russia had improved greatly since the accession to the throne of Catherine the Great
Catherine II of Russia
Catherine II, also known as Catherine the Great , Empress of Russia, was born in Stettin, Pomerania, Prussia on as Sophie Friederike Auguste von Anhalt-Zerbst-Dornburg...

, and in 1766 Euler accepted an invitation to return to the St. Petersburg Academy and spent the rest of his life in Russia. His second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife Katharina after 40 years of marriage. Three years after his wife's death Euler married her half sister, Salome Abigail Gsell (1723–1794). This marriage would last until his death.

In St Petersburg on 18 September 1783, after a lunch with his family, during a conversation with a fellow academician
Academician
The title Academician denotes a Full Member of an art, literary, or scientific academy.In many countries, it is an honorary title. There also exists a lower-rank title, variously translated Corresponding Member or Associate Member, .-Eastern Europe and China:"Academician" may also be a functional...

 Anders Johan Lexell
Anders Johan Lexell
Anders Johan Lexell was a Swedish-born Russian astronomer, mathematician, and physicist who spent most of his life in Russia where he is known as Andrei Ivanovich Leksel .Lexell made important discoveries in polygonometry and celestial mechanics; the latter led to a comet named in...

 about the newly-discovered Uranus
Uranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

 and its orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

, Euler suffered a brain hemorrhage and died a few hours later. A short obituary for the Russian Academy of Sciences
Russian Academy of Sciences
The Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

 was written by Jacob von Shtelin and a more detailed eulogy was written and delivered at a memorial meeting by Russian mathematician Nicolas Fuss
Nicolas Fuss
Nicolas Fuss , also known as Nikolai Fuss, was a Swiss mathematician.Fuss was born in Basel, Switzerland. He moved to Saint Petersburg to serve as a mathematical assistant to Leonhard Euler from 1773–1783, and remained there until his death...

, one of the Euler's disciples. In the eulogy written for the French Academy by the French mathematician and philosopher Marquis de Condorcet
Marquis de Condorcet
Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet , known as Nicolas de Condorcet, was a French philosopher, mathematician, and early political scientist whose Condorcet method in voting tally selects the candidate who would beat each of the other candidates in a run-off election...

, he commented,

{{quote|...il cessa de calculer et de vivre—... he ceased to calculate and to live.}}

He was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island
Vasilievsky Island
Vasilyevsky Island is an island in Saint Petersburg, Russia, bordered by the rivers Bolshaya Neva and Malaya Neva in the south and northeast, and by the Gulf of Finland in the west. Vasilyevsky Island is separated from Dekabristov Island by the Smolenka River...

. In 1785, the Russian Academy of Sciences
Russian Academy of Sciences
The Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

 put a marble bust of Leonhard Euler on a pedestal next to the Director's seat and, in 1837, placed a headstone on Euler's grave. To commemorate the 250th anniversary of Euler's birth, the headstone was moved in 1956, together with his remains, to the 18th-century necropolis at the Alexander Nevsky Monastery
Alexander Nevsky Lavra
Saint Alexander Nevsky Lavra or Saint Alexander Nevsky Monastery was founded by Peter I of Russia in 1710 at the eastern end of the Nevsky Prospekt in St. Petersburg supposing that that was the site of the Neva Battle in 1240 when Alexander Nevsky, a prince, defeated the Swedes; however, the battle...

.


Contributions to mathematics and physics


{{E (mathematical constant)}}

Euler worked in almost all areas of mathematics: geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

, trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

, algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, as well as continuum physics, lunar theory
Lunar theory
Lunar theory attempts to account for the motions of the Moon. There are many irregularities in the Moon's motion, and many attempts have been made over a long history to account for them. After centuries of being heavily problematic, the lunar motions are nowadays modelled to a very high degree...

 and other areas of physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

. 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
Quarto (text)
Quarto is a book or pamphlet produced from full 'blanksheets', each of which is printed with eight pages of text, four to a side, then folded two times to produce four leaves...

 volumes. Euler's name is associated with a large number of topics.

Mathematical notation


Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter {{math|e}} for the base of the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

 (now also known as Euler's number), the Greek letter Σ
Sigma
Sigma is the eighteenth letter of the Greek alphabet, and carries the 'S' sound. In the system of Greek numerals it has a value of 200. When used at the end of a word, and the word is not all upper case, the final form is used, e.g...

 for summations and the letter {{math|i}} to denote the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

. The use of the Greek letter {{pi}}
Pi (letter)
Pi is the sixteenth letter of the Greek alphabet, representing . In the system of Greek numerals it has a value of 80. Letters that arose from pi include Cyrillic Pe , Coptic pi , and Gothic pairthra .The upper-case letter Π is used as a symbol for:...

to denote the ratio of a circle's circumference to its diameter
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

 was also popularized by Euler, although it did not originate with him.

Analysis


The development of infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

 was at the forefront of 18th century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour (in particular his reliance on the principle of the generality of algebra
Generality of algebra
In the history of mathematics, the generality of algebra is phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph Lagrange...

), his ideas led to many great advances.
Euler is well known in analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

 for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as


Notably, Euler directly proved the power series expansions for {{math|e}} and the inverse tangent function. (Indirect proof via the inverse power series technique was given by Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 and Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the famous Basel problem
Basel problem
The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...

 in 1735 (he provided a more elaborate argument in 1741):


Euler introduced the use of the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

 and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s, thus greatly expanding the scope of mathematical applications of logarithms. He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s. For any real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 
{{Redirect|Euler}}
{{Portal:Mathematics/Featured article template}}
Leonhard Euler (ˈɔʏlɐ, ({{Audio|LeonhardEulerByDrsDotChRadio.ogg|Swiss German pronunciation}}) ({{Audio|De-Leonard_Euler.ogg|Standard German pronunciation}}) English approximation, "Oiler"; 15 April 1707{{ndash}} 18 September 1783) was a pioneering Swiss
Swiss (people)
The Swiss are citizens or natives of Switzerland. The demonym derives from the toponym of Schwyz and has been in widespread use to refer to the Old Swiss Confederacy since the 16th century....

 mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

 and physicist
Physicist
A physicist is a scientist who studies or practices physics. Physicists study a wide range of physical phenomena in many branches of physics spanning all length scales: from sub-atomic particles of which all ordinary matter is made to the behavior of the material Universe as a whole...

. He made important discoveries in fields as diverse as infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

 and graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

, such as the notion of a mathematical function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

. He is also renowned for his work in mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

, fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

, and astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

.

Euler spent most of his adult life in St. Petersburg, Russia
Russian Empire
The Russian Empire was a state that existed from 1721 until the Russian Revolution of 1917. It was the successor to the Tsardom of Russia and the predecessor of the Soviet Union...

, and in Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...

, Prussia
Kingdom of Prussia
The Kingdom of Prussia was a German kingdom from 1701 to 1918. Until the defeat of Germany in World War I, it comprised almost two-thirds of the area of the German Empire...

. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto
Quarto (text)
Quarto is a book or pamphlet produced from full 'blanksheets', each of which is printed with eight pages of text, four to a side, then folded two times to produce four leaves...

 volumes. A statement attributed to Pierre-Simon Laplace
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...

 expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Euler, read Euler, he is the master of us all."

Early years



Euler was born on April 15, 1707, in Basel
Basel
Basel or Basle In the national languages of Switzerland the city is also known as Bâle , Basilea and Basilea is Switzerland's third most populous city with about 166,000 inhabitants. Located where the Swiss, French and German borders meet, Basel also has suburbs in France and Germany...

 to Paul Euler, a pastor
Pastor
The word pastor usually refers to an ordained leader of a Christian congregation. When used as an ecclesiastical styling or title, this role may be abbreviated to "Pr." or often "Ps"....

 of the Reformed Church. His mother was Marguerite Brucker, a pastor's daughter. He had two younger sisters named Anna Maria and Maria Magdalena. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen
Riehen
Riehen is a municipality in the canton of Basel-Stadt in Switzerland. Together with the city of Basel and Bettingen, Riehen is one of three municipalities in the canton....

, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli family—Johann Bernoulli
Johann Bernoulli
Johann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...

, who was then regarded as Europe's foremost mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, would eventually be the most important influence on young Leonhard. Euler's early formal education started in Basel, where he was sent to live with his maternal grandmother. At the age of thirteen he enrolled at the University of Basel
University of Basel
The University of Basel is located in Basel, Switzerland, and is considered to be one of leading universities in the country...

, and in 1723, received his Master of Philosophy with a dissertation that compared the philosophies of Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...

 and Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. Euler was at this point studying theology
Theology
Theology is the systematic and rational study of religion and its influences and of the nature of religious truths, or the learned profession acquired by completing specialized training in religious studies, usually at a university or school of divinity or seminary.-Definition:Augustine of Hippo...

, Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

, and Hebrew
Hebrew language
Hebrew is a Semitic language of the Afroasiatic language family. Culturally, is it considered by Jews and other religious groups as the language of the Jewish people, though other Jewish languages had originated among diaspora Jews, and the Hebrew language is also used by non-Jewish groups, such...

 at his father's urging, in order to become a pastor, but Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

 with the title De Sono. At that time, he was pursuing an (ultimately unsuccessful) attempt to obtain a position at the University of Basel. In 1727, he entered the Paris Academy
French Academy of Sciences
The French Academy of Sciences is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research...

 Prize Problem
competition, where the problem that year was to find the best way to place the mast
Mast (sailing)
The mast of a sailing vessel is a tall, vertical, or near vertical, spar, or arrangement of spars, which supports the sails. Large ships have several masts, with the size and configuration depending on the style of ship...

s on a ship. He won second place, losing only to Pierre Bouguer
Pierre Bouguer
Pierre Bouguer was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture"....

—a man now known as "the father of naval architecture". Euler subsequently won this coveted annual prize twelve times in his career.

St. Petersburg


Around this time Johann Bernoulli's two sons, Daniel
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

 and Nicolas
Nicolaus II Bernoulli
Nicolaus II Bernoulli, a.k.a. Niklaus Bernoulli, Nikolaus Bernoulli, was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family....

, were working at the Imperial Russian Academy of Sciences
Russian Academy of Sciences
The Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

 in St Petersburg. On July 10, 1726, Nicolas died of appendicitis
Appendicitis
Appendicitis is a condition characterized by inflammation of the appendix. It is classified as a medical emergency and many cases require removal of the inflamed appendix, either by laparotomy or laparoscopy. Untreated, mortality is high, mainly because of the risk of rupture leading to...

 after spending a year in Russia, and when Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.


Euler arrived in the Russian capital on 17 May 1727. He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he often worked in close collaboration. Euler mastered Russian
Russian language
Russian is a Slavic language used primarily in Russia, Belarus, Uzbekistan, Kazakhstan, Tajikistan and Kyrgyzstan. It is an unofficial but widely spoken language in Ukraine, Moldova, Latvia, Turkmenistan and Estonia and, to a lesser extent, the other countries that were once constituent republics...

 and settled into life in St Petersburg. He also took on an additional job as a medic in the Russian Navy.

The Academy at St. Petersburg, established by Peter the Great
Peter I of Russia
Peter the Great, Peter I or Pyotr Alexeyevich Romanov Dates indicated by the letters "O.S." are Old Style. All other dates in this article are New Style. ruled the Tsardom of Russia and later the Russian Empire from until his death, jointly ruling before 1696 with his half-brother, Ivan V...

, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy in order to lessen the faculty's teaching burden, and the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions.

The Academy's benefactress, Catherine I
Catherine I of Russia
Catherine I , the second wife of Peter the Great, reigned as Empress of Russia from 1725 until her death.-Life as a peasant woman:The life of Catherine I was said by Voltaire to be nearly as extraordinary as that of Peter the Great himself. There are no documents that confirm her origins. Born on...

, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility then gained power upon the ascension of the twelve-year-old Peter II
Peter II of Russia
Pyotr II Alekseyevich was Emperor of Russia from 1727 until his death. He was the only son of Tsarevich Alexei Petrovich, son of Peter I of Russia by his first wife Eudoxia Lopukhina, and Princess Charlotte, daughter of Duke Louis Rudolph of Brunswick-Lüneburg and sister-in-law of Charles VI,...

. The nobility were suspicious of the academy's foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues.

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

On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell
Georg Gsell
Georg Gsell was a Swiss Baroque painter, art consultant and art dealer. Was recruited by Peter the Great in 1716 and went to Russia...

, a painter from the Academy Gymnasium. The young couple bought a house by the Neva River
Neva River
The Neva is a river in northwestern Russia flowing from Lake Ladoga through the western part of Leningrad Oblast to the Neva Bay of the Gulf of Finland. Despite its modest length , it is the third largest river in Europe in terms of average discharge .The Neva is the only river flowing from Lake...

. Of their thirteen children, only five survived childhood.

Berlin


Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy
Prussian Academy of Sciences
The Prussian Academy of Sciences was an academy established in Berlin on 11 July 1700, four years after the Akademie der Künste or "Arts Academy", to which "Berlin Academy" may also refer.-Origins:...

, which he had been offered by Frederick the Great of Prussia. He lived for twenty-five years in Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...

, where he wrote over 380 articles. In Berlin, he published the two works which he would be most renowned for: the Introductio in analysin infinitorum
Introductio in analysin infinitorum
Introductio in analysin infinitorum is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis...

, a text on functions published in 1748, and the Institutiones calculi differentialis
Institutiones calculi differentialis
Institutiones calculi differentialis is a mathematical work written in 1748 by Leonhard Euler and published in 1755 that lays the groundwork for the differential calculus...

, published in 1755 on differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

. In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2...

.

In addition, Euler was asked to tutor the Princess of Anhalt-Dessau, Frederick's niece. Euler wrote over 200 letters to her{{When|date=June 2010}}, which were later compiled into a best-selling volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess. This work contained Euler's exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler's personality and religious beliefs. This book became more widely read than any of his mathematical works, and it was published across Europe and in the United States. The popularity of the 'Letters' testifies to Euler's ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist.

Despite Euler's immense contribution to the Academy's prestige, he was eventually forced to leave Berlin. This was partly because of a conflict of personality with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy. Voltaire
Voltaire
François-Marie Arouet , better known by the pen name Voltaire , was a French Enlightenment writer, historian and philosopher famous for his wit and for his advocacy of civil liberties, including freedom of religion, free trade and separation of church and state...

 was among those in Frederick's employ, and the Frenchman enjoyed a prominent position in the king's social circle. Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes. He was in many ways the direct opposite of Voltaire. Euler had limited training in rhetoric
Rhetoric
Rhetoric is the art of discourse, an art that aims to improve the facility of speakers or writers who attempt to inform, persuade, or motivate particular audiences in specific situations. As a subject of formal study and a productive civic practice, rhetoric has played a central role in the Western...

, and tended to debate matters that he knew little about, making him a frequent target of Voltaire's wit. Frederick
Frederick II of Prussia
Frederick II was a King in Prussia and a King of Prussia from the Hohenzollern dynasty. In his role as a prince-elector of the Holy Roman Empire, he was also Elector of Brandenburg. He was in personal union the sovereign prince of the Principality of Neuchâtel...

 also expressed disappointment with Euler's practical engineering abilities:

{{quote|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
Sanssouci
Sanssouci is the name of the former summer palace of Frederick the Great, King of Prussia, in Potsdam, near Berlin. It is often counted among the German rivals of Versailles. While Sanssouci is in the more intimate Rococo style and is far smaller than its French Baroque counterpart, it too is...

. My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry!}}


Eyesight deterioration


Euler's eyesight worsened throughout his mathematical career. Three years after suffering a near-fatal fever
Fever
Fever is a common medical sign characterized by an elevation of temperature above the normal range of due to an increase in the body temperature regulatory set-point. This increase in set-point triggers increased muscle tone and shivering.As a person's temperature increases, there is, in...

 in 1735 he became nearly blind in his right eye, but Euler rather blamed his condition on the painstaking work on cartography
Cartography
Cartography is the study and practice of making maps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.The fundamental problems of traditional cartography are to:*Set the map's...

 he performed for the St. Petersburg Academy. Euler's sight in that eye worsened throughout his stay in Germany, so much so that Frederick referred to him as "Cyclops
Cyclops
A cyclops , in Greek mythology and later Roman mythology, was a member of a primordial race of giants, each with a single eye in the middle of his forehead...

". Euler later suffered a cataract
Cataract
A cataract is a clouding that develops in the crystalline lens of the eye or in its envelope, varying in degree from slight to complete opacity and obstructing the passage of light...

 in his good left eye, rendering him almost totally blind a few weeks after its discovery in 1766. Even so, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and photographic memory
Eidetic memory
Eidetic , commonly referred to as photographic memory, is a medical term, popularly defined as the ability to recall images, sounds, or objects in memory with extreme precision and in abundant volume. The word eidetic, referring to extraordinarily detailed and vivid recall not limited to, but...

. For example, Euler could repeat the Aeneid
Aeneid
The Aeneid is a Latin epic poem, written by Virgil between 29 and 19 BC, that tells the legendary story of Aeneas, a Trojan who travelled to Italy, where he became the ancestor of the Romans. It is composed of roughly 10,000 lines in dactylic hexameter...

 of Virgil
Virgil
Publius Vergilius Maro, usually called Virgil or Vergil in English , was an ancient Roman poet of the Augustan period. He is known for three major works of Latin literature, the Eclogues , the Georgics, and the epic Aeneid...

 from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last. With the aid of his scribes, Euler's productivity on many areas of study actually increased. He produced on average one mathematical paper every week in the year 1775.

Return to Russia


The situation in Russia had improved greatly since the accession to the throne of Catherine the Great
Catherine II of Russia
Catherine II, also known as Catherine the Great , Empress of Russia, was born in Stettin, Pomerania, Prussia on as Sophie Friederike Auguste von Anhalt-Zerbst-Dornburg...

, and in 1766 Euler accepted an invitation to return to the St. Petersburg Academy and spent the rest of his life in Russia. His second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife Katharina after 40 years of marriage. Three years after his wife's death Euler married her half sister, Salome Abigail Gsell (1723–1794). This marriage would last until his death.

In St Petersburg on 18 September 1783, after a lunch with his family, during a conversation with a fellow academician
Academician
The title Academician denotes a Full Member of an art, literary, or scientific academy.In many countries, it is an honorary title. There also exists a lower-rank title, variously translated Corresponding Member or Associate Member, .-Eastern Europe and China:"Academician" may also be a functional...

 Anders Johan Lexell
Anders Johan Lexell
Anders Johan Lexell was a Swedish-born Russian astronomer, mathematician, and physicist who spent most of his life in Russia where he is known as Andrei Ivanovich Leksel .Lexell made important discoveries in polygonometry and celestial mechanics; the latter led to a comet named in...

 about the newly-discovered Uranus
Uranus
Uranus is the seventh planet from the Sun. It has the third-largest planetary radius and fourth-largest planetary mass in the Solar System. It is named after the ancient Greek deity of the sky Uranus , the father of Cronus and grandfather of Zeus...

 and its orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

, Euler suffered a brain hemorrhage and died a few hours later. A short obituary for the Russian Academy of Sciences
Russian Academy of Sciences
The Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

 was written by Jacob von Shtelin and a more detailed eulogy was written and delivered at a memorial meeting by Russian mathematician Nicolas Fuss
Nicolas Fuss
Nicolas Fuss , also known as Nikolai Fuss, was a Swiss mathematician.Fuss was born in Basel, Switzerland. He moved to Saint Petersburg to serve as a mathematical assistant to Leonhard Euler from 1773–1783, and remained there until his death...

, one of the Euler's disciples. In the eulogy written for the French Academy by the French mathematician and philosopher Marquis de Condorcet
Marquis de Condorcet
Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet , known as Nicolas de Condorcet, was a French philosopher, mathematician, and early political scientist whose Condorcet method in voting tally selects the candidate who would beat each of the other candidates in a run-off election...

, he commented,

{{quote|...il cessa de calculer et de vivre—... he ceased to calculate and to live.}}

He was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island
Vasilievsky Island
Vasilyevsky Island is an island in Saint Petersburg, Russia, bordered by the rivers Bolshaya Neva and Malaya Neva in the south and northeast, and by the Gulf of Finland in the west. Vasilyevsky Island is separated from Dekabristov Island by the Smolenka River...

. In 1785, the Russian Academy of Sciences
Russian Academy of Sciences
The Russian Academy of Sciences consists of the national academy of Russia and a network of scientific research institutes from across the Russian Federation as well as auxiliary scientific and social units like libraries, publishers and hospitals....

 put a marble bust of Leonhard Euler on a pedestal next to the Director's seat and, in 1837, placed a headstone on Euler's grave. To commemorate the 250th anniversary of Euler's birth, the headstone was moved in 1956, together with his remains, to the 18th-century necropolis at the Alexander Nevsky Monastery
Alexander Nevsky Lavra
Saint Alexander Nevsky Lavra or Saint Alexander Nevsky Monastery was founded by Peter I of Russia in 1710 at the eastern end of the Nevsky Prospekt in St. Petersburg supposing that that was the site of the Neva Battle in 1240 when Alexander Nevsky, a prince, defeated the Swedes; however, the battle...

.


Contributions to mathematics and physics


{{E (mathematical constant)}}

Euler worked in almost all areas of mathematics: geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

, trigonometry
Trigonometry
Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

, algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, as well as continuum physics, lunar theory
Lunar theory
Lunar theory attempts to account for the motions of the Moon. There are many irregularities in the Moon's motion, and many attempts have been made over a long history to account for them. After centuries of being heavily problematic, the lunar motions are nowadays modelled to a very high degree...

 and other areas of physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

. 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
Quarto (text)
Quarto is a book or pamphlet produced from full 'blanksheets', each of which is printed with eight pages of text, four to a side, then folded two times to produce four leaves...

 volumes. Euler's name is associated with a large number of topics.

Mathematical notation


Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter {{math|e}} for the base of the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...

 (now also known as Euler's number), the Greek letter Σ
Sigma
Sigma is the eighteenth letter of the Greek alphabet, and carries the 'S' sound. In the system of Greek numerals it has a value of 200. When used at the end of a word, and the word is not all upper case, the final form is used, e.g...

 for summations and the letter {{math|i}} to denote the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

. The use of the Greek letter {{pi}}
Pi (letter)
Pi is the sixteenth letter of the Greek alphabet, representing . In the system of Greek numerals it has a value of 80. Letters that arose from pi include Cyrillic Pe , Coptic pi , and Gothic pairthra .The upper-case letter Π is used as a symbol for:...

to denote the ratio of a circle's circumference to its diameter
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

 was also popularized by Euler, although it did not originate with him.

Analysis


The development of infinitesimal calculus
Infinitesimal calculus
Infinitesimal calculus is the part of mathematics concerned with finding slope of curves, areas under curves, minima and maxima, and other geometric and analytic problems. It was independently developed by Gottfried Leibniz and Isaac Newton starting in the 1660s...

 was at the forefront of 18th century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour (in particular his reliance on the principle of the generality of algebra
Generality of algebra
In the history of mathematics, the generality of algebra is phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph Lagrange...

), his ideas led to many great advances.
Euler is well known in analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

 for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as


Notably, Euler directly proved the power series expansions for {{math|e}} and the inverse tangent function. (Indirect proof via the inverse power series technique was given by Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

 and Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the famous Basel problem
Basel problem
The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...

 in 1735 (he provided a more elaborate argument in 1741):


Euler introduced the use of the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

 and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s, thus greatly expanding the scope of mathematical applications of logarithms. He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric function
Trigonometric function
In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...

s. For any real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

 {{math, Euler's formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...

 states that the complex exponential function satisfies


A special case of the above formula is known as Euler's identity,
called "the most remarkable formula in mathematics" by Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, {{math|e}}, {{math|i}} and {{pi}}. In 1988, readers of the Mathematical Intelligencer
Mathematical Intelligencer
The Mathematical Intelligencer is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common amongst such journals.-Mathematical Conversations:...

voted it "the Most Beautiful Mathematical Formula Ever". In total, Euler was responsible for three of the top five formulae in that poll.{{cite journal | last= Wells | first= David | year= 1988 | title = Which is the most beautiful? | journal = Mathematical Intelligencer | volume = 10 | issue = 4 | pages= 30–31 | doi= 10.1007/BF03023741 }}
See also: {{cite web | url = http://www.maa.org/mathtourist/mathtourist_03_12_07.html | title = The Mathematical Tourist | accessdate=March 2008 | last = Peterson | first = Ivars }}

De Moivre's formula
De Moivre's formula
In mathematics, de Moivre's formula , named after Abraham de Moivre, states that for any complex number x and integer n it holds that...

 is a direct consequence of Euler's formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...

.

In addition, Euler elaborated the theory of higher transcendental function
Transcendental function
A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...

s by introducing the gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

 and introduced a new method for solving quartic equations. He also found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, and invented the calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

 including its best-known result, the Euler–Lagrange equation.

Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory
Analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...

. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions
Generalized continued fraction
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values....

. For example, he proved the infinitude of primes using the divergence of the harmonic series
Harmonic series (mathematics)
In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...

, 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
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....

.

Number theory


Euler's interest in number theory can be traced to the influence of Christian Goldbach
Christian Goldbach
Christian Goldbach was a German mathematician who also studied law. He is remembered today for Goldbach's conjecture.-Biography:...

, his friend in the St. Petersburg Academy. A lot of Euler's early work on number theory was based on the works of Pierre de Fermat
Pierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...

. Euler developed some of Fermat's ideas, and disproved some of his conjectures.

Euler linked the nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and the prime numbers; this is known as the Euler product formula for the Riemann zeta function
Proof of the Euler product formula for the Riemann zeta function
Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas' ', published by St Petersburg Academy in 1737.-The Euler product formula:...

.

Euler proved Newton's identities
Newton's identities
In mathematics, Newton's identities, also known as the Newton–Girard formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials...

, Fermat's little theorem
Fermat's little theorem
Fermat's little theorem states that if p is a prime number, then for any integer a, a p − a will be evenly divisible by p...

, Fermat's theorem on sums of two squares, and he made distinct contributions to Lagrange's four-square theorem
Lagrange's four-square theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that any natural number can be represented as the sum of four integer squaresp = a_0^2 + a_1^2 + a_2^2 + a_3^2\ where the four numbers are integers...

. He also invented the totient function φ(n) which is the number of positive integers less than or equal to the integer n that are coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

 to n. Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem. He contributed significantly to the theory of perfect number
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...

s, which had fascinated mathematicians since Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

. Euler also made progress toward the prime number theorem
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....

, and he conjectured the law of quadratic reciprocity
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...

. The two concepts are regarded as fundamental theorems of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

.

By 1772 Euler had proved that 231 − 1 = 2,147,483,647
2147483647
The number 2,147,483,647 is the eighth Mersenne prime, equal to 231 − 1. It is one of only four known double Mersenne primes....

 is a Mersenne prime
Mersenne prime
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...

. It may have remained the largest known prime
Largest known prime
The largest known prime number is the largest integer that is currently known to be a prime number.It was proven by Euclid that there are infinitely many prime numbers; thus, there is always a prime greater than the largest known prime. Many mathematicians and hobbyists search for large prime numbers...

 until 1867.

Graph theory



In 1736, Euler solved the problem known as the Seven Bridges of Königsberg
Seven Bridges of Königsberg
The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1735 laid the foundations of graph theory and prefigured the idea of topology....

. The city of Königsberg
Königsberg
Königsberg was the capital of East Prussia from the Late Middle Ages until 1945 as well as the northernmost and easternmost German city with 286,666 inhabitants . Due to the multicultural society in and around the city, there are several local names for it...

, Prussia
Kingdom of Prussia
The Kingdom of Prussia was a German kingdom from 1701 to 1918. Until the defeat of Germany in World War I, it comprised almost two-thirds of the area of the German Empire...

 was set on the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian circuit
Eulerian path
In graph theory, an Eulerian trail is a trail in a graph which visits every edge exactly once. Similarly, an Eulerian circuit or Eulerian cycle is a Eulerian trail which starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven Bridges of...

. This solution is considered to be the first theorem of graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...

, specifically of planar graph
Planar graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints...

 theory.

Euler also discovered the formula {{math|V}} − {{math|E}} + {{math|F}} = 2 relating the number of vertices, edges, and faces of a convex polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

, and hence of a planar graph
Planar graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints...

. The constant in this formula is now known as the Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

 for the graph (or other mathematical object), and is related to the genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...

 of the object. The study and generalization of this formula, specifically by Cauchy
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy was a French mathematician who was an early pioneer of analysis. He started the project of formulating and proving the theorems of infinitesimal calculus in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors...

 and L'Huillier, is at the origin of topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

.

Applied mathematics


Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series
Fourier series
In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines...

, Venn diagrams, Euler numbers, the constants {{math
E (mathematical constant)
The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

 and {{pi}}
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

, continued fractions and integrals. He integrated Leibniz
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

's differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....

 with Newton's Method of Fluxions
Method of Fluxions
Method of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736. Fluxions is Newton's term for differential calculus...

, and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. The most notable of these approximations are Euler's method and the Euler–Maclaurin formula. He also facilitated the use of differential equations, in particular introducing the Euler–Mascheroni constant
Euler–Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....

:


One of Euler's more unusual interests was the application of mathematical ideas in music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually incorporate musical theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.

Physics and astronomy


{{Classical mechanics|cTopic=Scientists}}
Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering. Aside from successfully applying his analytic tools to problems in classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

, Euler also applied these techniques to celestial problems. His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the parallax of the sun. His calculations also contributed to the development of accurate longitude tables
History of longitude
The history of longitude is a record of the effort, by navigators and scientists over several centuries, to discover a means of determining longitude....

.

In addition, Euler made important contributions in optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

. He disagreed with Newton's corpuscular theory of light in the Opticks
Opticks
Opticks is a book written by English physicist Isaac Newton that was released to the public in 1704. It is about optics and the refraction of light, and is considered one of the great works of science in history...

, which was then the prevailing theory. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christian Huygens would become the dominant mode of thought, at least until the development of the quantum theory of light.

In 1757 he published an important set of equations for inviscid flow
Inviscid flow
In fluid dynamics there are problems that are easily solved by using the simplifying assumption of an ideal fluid that has no viscosity. The flow of a fluid that is assumed to have no viscosity is called inviscid flow....

, that are now known as the Euler equations
Euler equations
In fluid dynamics, the Euler equations are a set of equations governing inviscid flow. They are named after Leonhard Euler. The equations represent conservation of mass , momentum, and energy, corresponding to the Navier–Stokes equations with zero viscosity and heat conduction terms. Historically,...

.

Logic


He is also credited with using closed curves to illustrate syllogistic
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...

 reasoning (1768). These diagrams have become known as Euler diagram
Euler diagram
An Euler diagram is a diagrammatic means of representing sets and their relationships. The first use of "Eulerian circles" is commonly attributed to Swiss mathematician Leonhard Euler . They are closely related to Venn diagrams....

s.

Personal philosophy and religious beliefs


Euler and his friend Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

 were opponents of Leibniz's
Gottfried Leibniz
Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

 monadism and the philosophy of Christian Wolff
Christian Wolff (philosopher)
Christian Wolff was a German philosopher.He was the most eminent German philosopher between Leibniz and Kant...

. Euler insisted that knowledge is founded in part on the basis of precise quantitative laws, something that monadism and Wolffian science were unable to provide. Euler's religious leanings might also have had a bearing on his dislike of the doctrine; he went so far as to label Wolff's ideas as "heathen and atheistic".

Much of what is known of Euler's religious beliefs can be deduced from his Letters to a German Princess and an earlier work, Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister (Defense of the Divine Revelation against the Objections of the Freethinkers). These works show that Euler was a devout Christian
Christian
A Christian is a person who adheres to Christianity, an Abrahamic, monotheistic religion based on the life and teachings of Jesus of Nazareth as recorded in the Canonical gospels and the letters of the New Testament...

 who believed the Bible to be inspired; the Rettung was primarily an argument for the divine inspiration of scripture
Biblical inspiration
Biblical inspiration is the doctrine in Christian theology that the authors and editors of the Bible were led or influenced by God with the result that their writings many be designated in some sense the word of God.- Etymology :...

.

There is a famous anecdote inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg academy. The French philosopher Denis Diderot
Denis Diderot
Denis Diderot was a French philosopher, art critic, and writer. He was a prominent person during the Enlightenment and is best known for serving as co-founder and chief editor of and contributor to the Encyclopédie....

 was visiting Russia on Catherine the Great's invitation. However, the Empress was alarmed that the philosopher's arguments for atheism
Atheism
Atheism is, in a broad sense, the rejection of belief in the existence of deities. In a narrower sense, atheism is specifically the position that there are no deities...

 were influencing members of her court, and so Euler was asked to confront the Frenchman. Diderot was later informed that a learned mathematician had produced a proof of the existence of God
Existence of God
Arguments for and against the existence of God have been proposed by philosophers, theologians, scientists, and others. In philosophical terms, arguments for and against the existence of God involve primarily the sub-disciplines of epistemology and ontology , but also of the theory of value, since...

: he agreed to view the proof as it was presented in court. Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced, "Sir, , hence God exists—reply!". Diderot, to whom (says the story) all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. Embarrassed, he asked to leave Russia, a request that was graciously granted by the Empress. This anecdote is apocryphal, however, given that Diderot was a capable mathematician who had published several mathematical treatises of his own.

Commemorations


Euler was featured on the sixth series of the Swiss 10-franc
Swiss franc
The franc is the currency and legal tender of Switzerland and Liechtenstein; it is also legal tender in the Italian exclave Campione d'Italia. Although not formally legal tender in the German exclave Büsingen , it is in wide daily use there...

 banknote and on numerous Swiss, German, and Russian postage stamp
Postage stamp
A postage stamp is a small piece of paper that is purchased and displayed on an item of mail as evidence of payment of postage. Typically, stamps are made from special paper, with a national designation and denomination on the face, and a gum adhesive on the reverse side...

s. The asteroid
Asteroid
Asteroids are a class of small Solar System bodies in orbit around the Sun. They have also been called planetoids, especially the larger ones...

 2002 Euler
2002 Euler
2002 Euler is an asteroid named after the Swiss mathematician and physicist Leonhard Euler. The asteroid was discovered on August 29, 1973, by Tamara Mikhailovna Smirnova....

 was named in his honor. He is also commemorated by the Lutheran Church on their Calendar of Saints
Calendar of Saints (Lutheran)
The Lutheran Calendar of Saints is a listing which details the primary annual festivals and events that are celebrated liturgically by some Lutheran Churches in the United States. The calendars of the Evangelical Lutheran Church in America and the Lutheran Church - Missouri Synod are from the...

 on 24 May—he was a devout Christian (and believer in biblical inerrancy
Biblical inerrancy
Biblical inerrancy is the doctrinal position that the Bible is accurate and totally free of error, that "Scripture in the original manuscripts does not affirm anything that is contrary to fact." Some equate inerrancy with infallibility; others do not.Conservative Christians generally believe that...

) who wrote apologetics
Apologetics
Apologetics is the discipline of defending a position through the systematic use of reason. Early Christian writers Apologetics (from Greek ἀπολογία, "speaking in defense") is the discipline of defending a position (often religious) through the systematic use of reason. Early Christian writers...

 and argued forcefully against the prominent atheists of his time.

Selected bibliography



Euler has an extensive bibliography. His best known books include:
  • Elements of Algebra
    Elements of Algebra
    Elements of Algebra is a mathematics textbook by mathematician Leonhard Euler, originally published circa 1765. His Elements of Algebra is one of the first books to set out algebra in the modern form we would recognize today. However, it is sufficiently different from most modern approaches to the...

    . This elementary algebra text starts with a discussion of the nature of numbers and gives a comprehensive introduction to algebra, including formulae for solutions of polynomial equations.
  • Introductio in analysin infinitorum
    Introductio in analysin infinitorum
    Introductio in analysin infinitorum is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis...

    (1748). English translation Introduction to Analysis of the Infinite by John Blanton (Book I, ISBN 0-387-96824-5, Springer-Verlag 1988; Book II, ISBN 0-387-97132-7, Springer-Verlag 1989).
  • Two influential textbooks on calculus: Institutiones calculi differentialis
    Institutiones calculi differentialis
    Institutiones calculi differentialis is a mathematical work written in 1748 by Leonhard Euler and published in 1755 that lays the groundwork for the differential calculus...

    (1755) and Institutionum calculi integralis (1768–1770).
  • Lettres à une Princesse d'Allemagne (Letters to a German Princess) (1768–1772). Available online (in French). English translation, with notes, and a life of Euler, available online from Google Books: Volume 1, Volume 2
  • Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti (1744). The Latin title translates as a method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense.


A definitive collection of Euler's works, entitled Opera Omnia, has been published since 1911 by the Euler Commission of the Swiss Academy of Sciences
Swiss Academy of Sciences
The Swiss Academies of Arts and Sciences is a Swiss organization that supports and networks the sciences at a regional, national and international level...

. A complete chronological list of Euler's works is available at the following page: The Eneström Index (PDF).

Further reading

  • Lexikon der Naturwissenschaftler, (2000), Heidelberg: Spektrum Akademischer Verlag.
  • Bogolyubov, Mikhailov, and Yushkevich, (2007), Euler and Modern Science, Mathematical Association of America. ISBN 0-88385-564-X. Translated by Robert Burns.
  • Bradley, Robert E., D'Antonio, Lawrence A., and C. Edward Sandifer (2007), Euler at 300: An Appreciation, Mathematical Association of America. ISBN 0-88385-565-8
  • Demidov, S.S., (2005), "Treatise on the differential calculus" in Grattan-Guinness, I.
    Ivor Grattan-Guinness
    Ivor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...

    , ed., Landmark Writings in Western Mathematics. Elsevier: 191–98.
  • Dunham, William
    William Dunham (mathematician)
    William Dunham is an American writer who was originally trained in topology but became interested in the history of mathematics. He has received several awards for writing and teaching on this subject.-Education:...

     (1999) Euler: The Master of Us All, Washington: Mathematical Association of America. ISBN 0-88385-328-0
  • Dunham, William (2007), The Genius of Euler: Reflections on his Life and Work, Mathematical Association of America. ISBN 0-88385-558-5
  • Fraser, Craig G., (2005), "Leonhard Euler's 1744 book on the calculus of variations" in Grattan-Guinness, I.
    Ivor Grattan-Guinness
    Ivor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...

    , ed., Landmark Writings in Western Mathematics. Elsevier: 168–80.
  • Gladyshev, Georgi, P. (2007), “ Leonhard Euler’s methods and ideas live on in the thermodynamic hierarchical theory of biological evolution,International Journal of Applied Mathematics & Statistics (IJAMAS) 11 (N07), Special Issue on Leonhard Paul Euler’s: Mathematical Topics and Applications (M. T. A.).
  • Heimpell, Hermann, Theodor Heuss, Benno Reifenberg (editors). 1956. Die großen Deutschen, volume 2, Berlin: Ullstein Verlag.
  • Nahin, Paul (2006), Dr. Euler's Fabulous Formula, New Jersey: Princeton, ISBN 978-0-691-11822-2
  • du Pasquier, Louis-Gustave, (2008) Leonhard Euler And His Friends, CreateSpace, ISBN 1-4348-3327-5. Translated by John S.D. Glaus.
  • Reich, Karin, (2005), " 'Introduction' to analysis" in Grattan-Guinness, I.
    Ivor Grattan-Guinness
    Ivor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...

    , ed., Landmark Writings in Western Mathematics. Elsevier: 181–90.
  • Richeson, David S. (2008), Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
  • Sandifer, Edward C. (2007), The Early Mathematics of Leonhard Euler, Mathematical Association of America. ISBN 0-88385-559-3
  • Sandifer, Edward C. (2007), How Euler Did It, Mathematical Association of America. ISBN 0-88385-563-1
  • Simmons, J. (1996) The giant book of scientists: The 100 greatest minds of all time, Sydney: The Book Company.
  • Singh, Simon. (1997). Fermat's last theorem, Fourth Estate: New York, ISBN 1-85702-669-1
  • Thiele, Rüdiger. (2005). The mathematics and science of Leonhard Euler, in Mathematics and the Historian's Craft: The Kenneth O. May Lectures, G. Van Brummelen and M. Kinyon (eds.), CMS Books in Mathematics, Springer Verlag. ISBN 0-387-25284-3.

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External links


{{Sister project links|s=Author:Leonhard Euler}}
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{{Persondata
|NAME= Euler, Leonhard
|SHORT DESCRIPTION=Mathematician
|DATE OF BIRTH=15 April 1707
|PLACE OF BIRTH=Basel
Basel
Basel or Basle In the national languages of Switzerland the city is also known as Bâle , Basilea and Basilea is Switzerland's third most populous city with about 166,000 inhabitants. Located where the Swiss, French and German borders meet, Basel also has suburbs in France and Germany...

, Switzerland
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....


|DATE OF DEATH={{death date|df=yes|1783|9|18}}
|PLACE OF DEATH=St Petersburg, Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...


}}
{{DEFAULTSORT:Euler, Leonhard}}