Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
,
physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate cau ...
,
astronomer
An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either ...
,
geographer
A geographer is a physical scientist, social scientist or humanist whose area of study is geography, the study of Earth's natural environment and human society, including how society and nature interacts. The Greek prefix "geo" means "earth" a ...
,
logician
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
and
engineer
Engineers, as practitioners of engineering, are professionals who invent, design, analyze, build and test machines, complex systems, structures, gadgets and materials to fulfill functional objectives and requirements while considering the limit ...
who founded the studies of
graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
and
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and made pioneering and influential discoveries in many other branches of
mathematics such as
analytic number theory,
complex analysis, and
infinitesimal calculus. He introduced much of modern mathematical terminology and
notation
In linguistics and semiotics, a notation is a system of graphics or symbols, characters and abbreviated expressions, used (for example) in artistic and scientific disciplines to represent technical facts and quantities by convention. Therefore, ...
, including the notion of a
mathematical function
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
. He is also known for his work in
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to object ...
,
fluid dynamics,
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
,
astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
and
music theory.
Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it."
Euler is also widely considered to be the most prolific; his 866 publications as well as his correspondences are collected in the ''
Opera Omnia Leonhard Euler
''Opera Omnia Leonhard Euler (Leonhardi Euleri Opera omnia)'' is the compilation of Leonhard Euler's scientific writings. The project of this compilation was undertaken by the Euler Committee of the Swiss Academy of Sciences, established in 1908 ...
'' which, when completed, will consist of 81 ''
quarto
Quarto (abbreviated Qto, 4to or 4º) is the format of a book or pamphlet produced from full sheets printed with eight pages of text, four to a side, then folded twice to produce four leaves. The leaves are then trimmed along the folds to produc ...
'' volumes.
He spent most of his adult life in
Saint Petersburg
Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...
,
Russia
Russia (, , ), or the Russian Federation, is a transcontinental country spanning Eastern Europe and Northern Asia. It is the largest country in the world, with its internationally recognised territory covering , and encompassing one-eig ...
, and in
Berlin
Berlin ( , ) is the capital and List of cities in Germany by population, largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's List of cities in the European Union by population within ci ...
, then the capital of
Prussia
Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an ...
.
Euler is credited for popularizing the Greek letter
(lowercase
pi) to denote
the ratio of a circle's circumference to its diameter, as well as first using the notation
for the value of a function, the letter
to express the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, the Greek letter
(capital
sigma) to express
summations, the Greek letter
(uppercase
delta
Delta commonly refers to:
* Delta (letter) (Δ or δ), a letter of the Greek alphabet
* River delta, at a river mouth
* D ( NATO phonetic alphabet: "Delta")
* Delta Air Lines, US
* Delta variant of SARS-CoV-2 that causes COVID-19
Delta may also ...
) for
finite differences
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the ...
, and lowercase letters to represent the sides of a triangle while representing the angles as capital letters.
He gave the current definition of the constant
, the base of the
natural logarithm, now known as
Euler's number
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expressi ...
.
Euler was also the first practitioner of
graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
(partly as a solution for the problem of the
Seven Bridges of Königsberg
The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology.
The city of Königsberg in Prussia (n ...
). He became famous for, among many other accomplishments, solving the
Basel problem
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
, after proving that the sum of the
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
of squared integer
reciprocals equaled exactly , and for discovering that the sum of the numbers of vertices and faces minus edges of a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on ...
equals 2, a number now commonly known as the
Euler characteristic. In the field of physics, Euler reformulated
Newton's laws of physics into
new laws in his two-volume work ''
Mechanica
''Mechanica'' ( la, Mechanica sive motus scientia analytice exposita; 1736) is a two-volume work published by mathematician Leonhard Euler which describes analytically the mathematics governing movement.
Euler both developed the techniques of ...
'' to better explain the motion of rigid bodies. He also made substantial contributions to the study of
elastic deformations of solid objects.
Early life
Leonhard Euler was born on 15 April 1707, in
Basel
, french: link=no, Bâlois(e), it, Basilese
, neighboring_municipalities= Allschwil (BL), Hégenheim (FR-68), Binningen (BL), Birsfelden (BL), Bottmingen (BL), Huningue (FR-68), Münchenstein (BL), Muttenz (BL), Reinach (BL), Riehen (BS ...
, Switzerland, to Paul III Euler, a pastor of the
Reformed Church, and Marguerite ( Brucker), whose ancestors include a number of well-known scholars in the classics. He was the oldest of four children, having two younger sisters, Anna Maria and Maria Magdalena, and a younger brother, Johann Heinrich. Soon after the birth of Leonhard, the Euler family moved from Basel to the town of
Riehen, Switzerland, where his father became pastor in the local church and Leonhard spent most of his childhood.
From a young age, Euler received schooling in mathematics from his father, who had taken courses from
Jacob Bernoulli
Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
some years earlier at the
University of Basel
The University of Basel (Latin: ''Universitas Basiliensis'', German: ''Universität Basel'') is a university in Basel, Switzerland. Founded on 4 April 1460, it is Switzerland's oldest university and among the world's oldest surviving universit ...
. Around the age of eight, Euler was sent to live at his maternal grandmother's house and enrolled in the Latin school in Basel. In addition, he received private tutoring from Johannes Burckhardt, a young theologian with a keen interest in mathematics.
In 1720, at thirteen years of age, Euler enrolled at the
University of Basel
The University of Basel (Latin: ''Universitas Basiliensis'', German: ''Universität Basel'') is a university in Basel, Switzerland. Founded on 4 April 1460, it is Switzerland's oldest university and among the world's oldest surviving universit ...
. Attending university at such a young age was not unusual at the time. The course on elementary mathematics was given by
Johann Bernoulli
Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is known for his contributions to infinitesimal calculus and educating Le ...
, the younger brother of the deceased Jacob Bernoulli (who had taught Euler's father). Johann Bernoulli and Euler soon got to know each other better. Euler described Bernoulli in his autobiography:
:"the famous professor Johann Bernoulli
..made it a special pleasure for himself to help me along in the mathematical sciences. Private lessons, however, he refused because of his busy schedule. However, he gave me a far more salutary advice, which consisted in myself getting a hold of some of the more difficult mathematical books and working through them with great diligence, and should I encounter some objections or difficulties, he offered me free access to him every Saturday afternoon, and he was gracious enough to comment on the collected difficulties, which was done with such a desired advantage that, when he resolved one of my objections, ten others at once disappeared, which certainly is the best method of making happy progress in the mathematical sciences."
It was during this time that Euler, backed by Bernoulli, obtained his father's consent to become a mathematician instead of a pastor.
In 1723, Euler received a
Master of Philosophy
The Master of Philosophy (MPhil; Latin ' or ') is a postgraduate degree. In the United States, an MPhil typically includes a taught portion and a significant research portion, during which a thesis project is conducted under supervision. An MPhil ...
with a dissertation that compared the philosophies of
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
and
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the grea ...
. Afterwards he enrolled in the theological faculty of the University of Basel.
In 1726, Euler completed a dissertation on the
propagation of sound with the title ''De Sono''
with which he unsuccessfully attempted to obtain a position at the University of Basel. In 1727, he entered the
Paris Academy
The French Academy of Sciences (French: ''Académie des 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. It was at th ...
prize competition (offered annually and later biennially by the academy beginning in 1720)
[ for the first time. The problem posed that year was to find the best way to place the masts on a ship. Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place. Over the years, Euler entered this competition 15 times,][ winning 12 of them.
]
Career
Saint Petersburg
Johann Bernoulli's two sons, Daniel and Nicolaus Nicolaus is a masculine given name. It is a Latin, Greek and German form of Nicholas. Nicolaus may refer to:
In science:
* Nicolaus Copernicus, Polish astronomer who provided the first modern formulation of a heliocentric theory of the solar syst ...
, entered into service at the Imperial Russian Academy of Sciences in Saint Petersburg
Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg, r=Sankt-Peterburg, p=ˈsankt pʲɪtʲɪrˈburk), formerly known as Petrograd (1914–1924) and later Leningrad (1924–1991), i ...
in 1725, leaving Euler with the assurance they would recommend him to a post when one was available. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia.[ Retrieved 2 July 2021.] 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 Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.
Euler arrived in Saint Petersburg in 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 worked in close collaboration. Euler mastered Russian
Russian(s) refers to anything related to Russia, including:
*Russians (, ''russkiye''), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries
*Rossiyane (), Russian language term for all citizens and peo ...
, settled into life in Saint Petersburg and took on an additional job as a medic in the Russian Navy.
The academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy's benefactress, Catherine I
Catherine I ( rus, Екатери́на I Алексе́евна Миха́йлова, Yekaterína I Alekséyevna Mikháylova; born , ; – ) was the second wife and empress consort of Peter the Great, and Empress Regnant of Russia from 1725 un ...
, who had continued the progressive policies of her late husband, died before Euler's arrival to Saint Petersburg. The Russian conservative nobility then gained power upon the ascension of the twelve-year-old Peter II. The nobility, suspicious of the academy's foreign scientists, cut funding for Euler and his colleagues and prevented the entrance of foreign and non-aristocratic students into the Gymnasium and Universities.
Conditions improved slightly after the death of Peter II in 1730 and the German-influenced Anna of Russia
Anna Ioannovna (russian: Анна Иоанновна; ), also russified as Anna Ivanovna and sometimes anglicized as Anne, served as regent of the duchy of Courland from 1711 until 1730 and then ruled as Empress of Russia from 1730 to 1740. Much ...
assumed power. Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731. He also left the Russian Navy, refusing a promotion to lieutenant
A lieutenant ( , ; abbreviated Lt., Lt, LT, Lieut and similar) is a commissioned officer rank in the armed forces of many nations.
The meaning of lieutenant differs in different militaries (see comparative military ranks), but it is often ...
. Two years later, Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. In January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell. Frederick II had made an attempt to recruit the services of Euler for his newly established Berlin Academy in 1740, but Euler initially preferred to stay in St Petersburg. But after Emperor Anna died and Frederick II agreed to pay 1600 ecus (the same as Euler earned in Russia) he agreed to move to Berlin. In 1741, he requested permission to leave to Berlin, arguing he was in need of a milder climate for his eyesight. The Russian academy gave its consent and would pay him 200 rubles per year as one of its active members.
Berlin
Concerned about the continuing turmoil in Russia, Euler left St. Petersburg in June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia
Frederick II (german: Friedrich II.; 24 January 171217 August 1786) was King in Prussia from 1740 until 1772, and King of Prussia from 1772 until his death in 1786. His most significant accomplishments include his military successes in the S ...
. He lived for 25 years in Berlin
Berlin ( , ) is the capital and List of cities in Germany by population, largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's List of cities in the European Union by population within ci ...
, where he wrote several hundred articles. In 1748 his text on functions called the ''Introductio in analysin infinitorum
''Introductio in analysin infinitorum'' (Latin: ''Introduction to the Analysis of the Infinite'') is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the ''Introducti ...
'' was published and in 1755 a text on differential calculus called the ''Institutiones calculi differentialis
''Institutiones calculi differentialis'' (''Foundations of differential calculus'') is a mathematical work written in 1748 by Leonhard Euler and published in 1755 that lays the groundwork for the differential calculus. It consists of a single volu ...
'' was published. In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences[ and of the French Academy of Sciences.] Notable students of Euler in Berlin included Stepan Rumovsky, later considered as the first Russian astronomer. In 1748 he declined an offer from the University of Basel to succeed the recently deceased Johann Bernoulli. In 1753 he bought a house in Charlottenburg
Charlottenburg () is a locality of Berlin within the borough of Charlottenburg-Wilmersdorf. Established as a town in 1705 and named after Sophia Charlotte of Hanover, Queen consort of Prussia, it is best known for Charlottenburg Palace, the ...
, in which he lived with his family and widowed mother.
Euler became the tutor for Friederike Charlotte of Brandenburg-Schwedt
Friederike Charlotte Leopoldine Louise of Brandenburg-Schwedt (also often referred to as the Princess of Prussia; 18 August 1745 in Schwedt – 23 January 1808 in Altona) was a German aristocrat who lived as a secular canoness and ruled a ...
, the Princess of Anhalt-Dessau
Anhalt-Dessau was a principality of the Holy Roman Empire and later a duchy of the German Confederation. Ruled by the House of Ascania, it was created in 1396 following the partition of the Principality of Anhalt-Zerbst, and finally merged into t ...
and Frederick's niece. He wrote over 200 letters to her in the early 1760s, which were later compiled into a 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 and offered valuable insights into Euler's personality and religious beliefs. It was translated into multiple languages, published across Europe and in the United States, and became more widely read than any of his mathematical works. The popularity of the ''Letters'' testifies to Euler's ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist.
Despite Euler's immense contribution to the academy's prestige and having been put forward as a candidate for its presidency by Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclopéd ...
, Frederick II named himself as its president. The Prussian king had a large circle of intellectuals in his court, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs. He was, in many ways, the polar opposite of Voltaire
François-Marie Arouet (; 21 November 169430 May 1778) was a French Enlightenment writer, historian, and philosopher. Known by his ''nom de plume'' M. de Voltaire (; also ; ), he was famous for his wit, and his criticism of Christianity—es ...
, who enjoyed a high place of prestige at Frederick's court. Euler was not a skilled debater and often made it a point to argue subjects that he knew little about, making him the frequent target of Voltaire's wit. Frederick also expressed disappointment with Euler's practical engineering abilities, stating:
Throughout his stay in Berlin, Euler maintained a strong connection to the academy in St. Petersburg and also published 109 papers in Russia. He also assisted students from the St. Petersburg academy and at times accommodated Russian students in his house in Berlin. In 1760, with the Seven Years' War
The Seven Years' War (1756–1763) was a global conflict that involved most of the European Great Powers, and was fought primarily in Europe, the Americas, and Asia-Pacific. Other concurrent conflicts include the French and Indian War (175 ...
raging, Euler's farm in Charlottenburg was sacked by advancing Russian troops. Upon learning of this event, General Ivan Petrovich Saltykov paid compensation for the damage caused to Euler's estate, with Empress Elizabeth
Elizabeth Petrovna (russian: Елизаве́та (Елисаве́та) Петро́вна) (), also known as Yelisaveta or Elizaveta, reigned as Empress of Russia from 1741 until her death in 1762. She remains one of the most popular List of ...
of Russia later adding a further payment of 4000 rubles—an exorbitant amount at the time. Euler decided to leave Berlin in 1766 and return to Russia.
During his Berlin years (1741–1766), Euler was at the peak of his productivity. He wrote 380 works, 275 of which were published. This included 125 memoirs in the Berlin Academy and over 100 memoirs sent to the St. Petersburg Academy
The Russian Academy of Sciences (RAS; russian: Росси́йская акаде́мия нау́к (РАН) ''Rossíyskaya akadémiya naúk'') consists of the national academy of Russia; a network of scientific research institutes from across t ...
, which had retained him as a member and paid him an annual stipend. Euler’s ''Introductio in Analysin Infinitorum'' was published in two parts in 1748. In addition to his own research, Euler supervised the library, the observatory, the botanical garden, and the publication of calendars and maps from which the academy derived income. He was even involved in the design of the water fountains at Sanssouci
Sanssouci () is a historical building in Potsdam, near Berlin. Built by Prussian King Frederick the Great as his summer palace, it is often counted among the German rivals of Versailles. While Sanssouci is in the more intimate Rococo style and ...
, the King’s summer palace.
Return to Russia
The political situation in Russia stabilized after Catherine the Great's accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Petersburg Academy. His conditions were quite exorbitant—a 3000 ruble annual salary, a pension for his wife, and the promise of high-ranking appointments for his sons. At the university he was assisted by his student Anders Johan Lexell. While living in St. Petersburg, a fire in 1771 destroyed his home.
Personal life
On 7 January 1734, he married Katharina Gsell (1707–1773), daughter of Georg Gsell, a painter from the Academy Gymnasium in Saint Petersburg. The young couple bought a house by the Neva River.
Of their thirteen children, only five survived childhood, three sons and two daughters. Their first son was Johann Albrecht Euler, whose godfather was Christian Goldbach
Christian Goldbach (; ; 18 March 1690 – 20 November 1764) was a German mathematician connected with some important research mainly in number theory; he also studied law and took an interest in and a role in the Russian court. After traveling ...
.
Three years after his wife's death in 1773, Euler married her half-sister, Salome Abigail Gsell (1723–1794). This marriage lasted until his death in 1783.
His brother Johann Heinrich settled in St. Petersburg in 1735 and was employed as a painter at the academy.
Eyesight deterioration
Euler's eyesight
Visual perception is the ability to interpret the surrounding environment through photopic vision (daytime vision), color vision, scotopic vision (night vision), and mesopic vision (twilight vision), using light in the visible spectrum reflect ...
worsened throughout his mathematical career. In 1738, three years after nearly expiring from fever, he became almost blind in his right eye. Euler blamed the cartography
Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an i ...
he performed for the St. Petersburg Academy for his condition, but the cause of his blindness remains the subject of speculation. Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as " Cyclops". Euler remarked on his loss of vision, stating "Now I will have fewer distractions." In 1766 a cataract
A cataract is a cloudy area in the lens of the eye that leads to a decrease in vision. Cataracts often develop slowly and can affect one or both eyes. Symptoms may include faded colors, blurry or double vision, halos around light, trouble ...
in his left eye was discovered. Though couching of the cataract temporarily improved his vision, complications ultimately rendered him almost totally blind in the left eye as well. However, his condition appeared to have little effect on his productivity. With the aid of his scribes, Euler's productivity in many areas of study increased; and, in 1775, he produced, on average, one mathematical paper every week.
Death
In St. Petersburg on 18 September 1783, after a lunch with his family, Euler was discussing the newly discovered planet Uranus
Uranus is the seventh planet from the Sun. Its name is a reference to the Greek god of the sky, Uranus ( Caelus), who, according to Greek mythology, was the great-grandfather of Ares (Mars), grandfather of Zeus (Jupiter) and father of ...
and its orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
with Lexell when he collapsed and died from a brain hemorrhage. wrote a short obituary for the Russian Academy of Sciences
The Russian Academy of Sciences (RAS; russian: Росси́йская акаде́мия нау́к (РАН) ''Rossíyskaya akadémiya naúk'') consists of the national academy of Russia; a network of scientific research institutes from across ...
and Russian mathematician Nicolas Fuss, one of Euler's disciples, wrote a more detailed eulogy, which he delivered at a memorial meeting. In his eulogy for the French Academy
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
** French language, which originated in France, and its various dialects and accents
** French people, a nation and ethnic group identified with France ...
, French mathematician and philosopher Marquis de Condorcet, wrote:
Euler was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island
Vasilyevsky Island (russian: Васи́льевский о́стров, Vasilyevsky Ostrov, V.O.) is an island in St. Petersburg, Russia, bordered by the Bolshaya Neva and Malaya Neva Rivers (in the delta of the Neva River) in the south ...
. In 1837, the Russian Academy of Sciences
The Russian Academy of Sciences (RAS; russian: Росси́йская акаде́мия нау́к (РАН) ''Rossíyskaya akadémiya naúk'') consists of the national academy of Russia; a network of scientific research institutes from across ...
installed a new monument, replacing his overgrown grave plaque. To commemorate the 250th anniversary of Euler's birth in 1957, his tomb was moved to the Lazarevskoe Cemetery
Lazarevskoe Cemetery (russian: Лазаревское кладбище) is a historic cemetery in the centre of Saint Petersburg, and the oldest surviving cemetery in the city. It is part of the Alexander Nevsky Lavra, and is one of four cemeterie ...
at the Alexander Nevsky Monastery
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 Saint Petersburg, in the belief that this was the site of the Neva Battle in 1240 when Alex ...
.
Contributions to mathematics and physics
Euler worked in almost all areas of mathematics, including geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, infinitesimal calculus, trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies ...
, algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, and number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, as well as continuum physics, lunar theory, and other areas of physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. 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 (abbreviated Qto, 4to or 4º) is the format of a book or pamphlet produced from full sheets printed with eight pages of text, four to a side, then folded twice to produce four leaves. The leaves are then trimmed along the folds to produc ...
volumes. It has been proposed that Euler was responsible for a third of all the scientific and mathematical output of the 18th century. Euler's name is associated with a large number of topics. Euler's work averages 800 pages a year from 1725 to 1783. He also wrote over 4500 letters and hundreds of manuscripts. It has been estimated that Leonard Euler was the author of a quarter of the combined output in mathematics, physics, mechanics, astronomy, and navigation in the 18th century.
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 or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
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
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in al ...
, the letter for the base of the natural logarithm (now also known as Euler's number
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expressi ...
), the Greek letter Σ for summations and the letter to denote the imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
.[ The use of the Greek letter '' π'' to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it originated with ]Welsh
Welsh may refer to:
Related to Wales
* Welsh, referring or related to Wales
* Welsh language, a Brittonic Celtic language spoken in Wales
* Welsh people
People
* Welsh (surname)
* Sometimes used as a synonym for the ancient Britons (Celtic peop ...
mathematician William Jones.[
]
Analysis
The development of infinitesimal calculus was at the forefront of 18th-century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus 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 In the history of mathematics, the generality of algebra was a 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-Louis Lagrange,. particularly ...
), his ideas led to many great advances.
Euler is well known in analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
for his frequent use and development of power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
, the expression of functions as sums of infinitely many terms, such as
Euler's use of power series enabled him to solve the famous Basel problem
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 ...
in 1735 (he provided a more elaborate argument in 1741):
He introduced the constant
now known as Euler's constant
Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma ().
It is defined as the limiting difference between the harmonic series and the natural ...
or the Euler–Mascheroni constant, and studied its relationship with the harmonic series, the gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
, and values of the Riemann zeta function.[
Euler introduced the use of the ]exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
and logarithms
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
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
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s. For any real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
(taken to be radians), Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...
states that the complex exponential
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, al ...
function satisfies
which was called "the most remarkable formula in mathematics" by Richard P. Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...
A special case of the above formula is known as Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality
e^ + 1 = 0
where
: is Euler's number, the base of natural logarithms,
: is the imaginary unit, which by definition satisfies , and
: is pi, the ratio of the circ ...
,
Euler elaborated the theory of higher transcendental function
In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.
In other words, a transcendental function "transcends" algebra in that it cannot be expressed alge ...
s by introducing the gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
[ and introduced a new method for solving ]quartic equation
In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is
:ax^4+bx^3+cx^2+dx+e=0 \,
where ''a'' ≠ 0.
The quartic is the highest order polynomi ...
s.[ He found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis. He invented the calculus of variations and formulated the ]Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
for reducing optimization problems in this area to the solution of differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s.
Euler pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product
(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^),
with (a;q)_0 = 1.
It is a ''q''-analog of the Pochhammer sym ...
, hyperbolic trigonometric functions, and the analytic theory of continued fractions
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
. For example, he proved the infinitude of primes using the divergence of the harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to the development of the prime number theorem.
Number theory
Euler's interest in number theory can be traced to the influence of Christian Goldbach
Christian Goldbach (; ; 18 March 1690 – 20 November 1764) was a German mathematician connected with some important research mainly in number theory; he also studied law and took an interest in and a role in the Russian court. After traveling ...
, his friend in the St. Petersburg Academy. Much of Euler's early work on number theory was based on the work of Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
. Euler developed some of Fermat's ideas and disproved some of his conjectures, such as his conjecture that all numbers of the form (Fermat numbers
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form
:F_ = 2^ + 1,
where ''n'' is a non-negative integer. The first few Fermat numbers are:
: 3, 5, 17, 257, 65537, 42949672 ...
) are prime.
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 prime numbers; this is known as the Euler product formula for the Riemann zeta function.[
Euler invented the ]totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ...
φ(''n''), the number of positive integers less than or equal to the integer ''n'' that are coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
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
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
...
s, which had fascinated mathematicians since Euclid
Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
. He proved that the relationship shown between even perfect numbers and Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
s (which he had earlier proved) was one-to-one, a result otherwise known as the Euclid–Euler theorem
The Euclid–Euler theorem is a theorem in number theory that relates perfect numbers to Mersenne primes. It states that an even number is perfect if and only if it has the form , where is a prime number. The theorem is named after mathematician ...
.[ Euler also conjectured the law of ]quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...
. The concept is regarded as a fundamental theorem within number theory, and his ideas paved the way for the work of Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, particularly ''Disquisitiones Arithmeticae
The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...
''. By 1772 Euler had proved that 231 − 1 = 2,147,483,647
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.
The primality of this number was proven by Leonhard Euler, who reported the proof in a letter to Daniel ...
is a Mersenne prime. It may have remained the largest known prime
The largest known prime number () is , a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018.
A prime number is a posi ...
until 1867.[
Euler also contributed major developments to the theory of partitions of an integer.][
]
Graph theory
In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg
The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler in 1736 laid the foundations of graph theory and prefigured the idea of topology.
The city of Königsberg in Prussia (n ...
. The city of Königsberg
Königsberg (, ) was the historic Prussian city that is now Kaliningrad, Russia. Königsberg was founded in 1255 on the site of the ancient Old Prussian settlement ''Twangste'' by the Teutonic Knights during the Northern Crusades, and was name ...
, Prussia
Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an ...
was set on the Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian circuit
In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends ...
. This solution is considered to be the first theorem of graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
.
Euler also discovered the formula relating the number of vertices, edges, and faces of a convex polyhedron
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
, and hence of a planar graph. The constant in this formula is now known as the Euler characteristic for the graph (or other mathematical object), and is related to the genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
of the object.[ The study and generalization of this formula, specifically by ]Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
and L'Huilier, is at the origin of topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
.
Physics, astronomy, and engineering
Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the Bernoulli numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
, Fourier series, Euler number
In mathematics, the Euler numbers are a sequence ''En'' of integers defined by the Taylor series expansion
:\frac = \frac = \sum_^\infty \frac \cdot t^n,
where \cosh (t) is the hyperbolic cosine function. The Euler numbers are related to a ...
s, the constants and , continued fractions, and integrals. He integrated Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
's differential calculus with Newton's Method of Fluxions
''Method of Fluxions'' ( la, De Methodis Serierum et Fluxionum) is a mathematical treatise by Sir Isaac Newton which served as the earliest written formulation of modern calculus. The book was completed in 1671, and published in 1736. Fluxion ...
, and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
of integrals, inventing what are now known as the Euler approximations. The most notable of these approximations are Euler's method
In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit met ...
[ and the ]Euler–Maclaurin formula
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using ...
.[
Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering.][ Besides successfully applying his analytic tools to problems in ]classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, Euler applied these techniques to celestial problems. His work in astronomy was recognized by multiple Paris Academy
The French Academy of Sciences (French: ''Académie des 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. It was at th ...
Prizes over the course of his career. His accomplishments include determining with great accuracy the orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
s of comet
A comet is an icy, small Solar System body that, when passing close to the Sun, warms and begins to release gases, a process that is called outgassing. This produces a visible atmosphere or coma, and sometimes also a tail. These phenomena ...
s and other celestial bodies, understanding the nature of comets, and calculating the parallax of the Sun. His calculations contributed to the development of accurate longitude tables.[
Euler made important contributions in ]optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
.[ He disagreed with Newton's ]corpuscular theory of light
In optics, the corpuscular theory of light states that light is made up of small discrete particles called " corpuscles" (little particles) which travel in a straight line with a finite velocity and possess impetus. This was based on an alternate ...
, which was the prevailing theory of the time. His 1740s papers on optics helped ensure that the wave theory of light
In physics, physical optics, or wave optics, is the branch of optics that studies Interference (wave propagation), interference, diffraction, Polarization (waves), polarization, and other phenomena for which the ray approximation of geometric opti ...
proposed by Christiaan Huygens would become the dominant mode of thought, at least until the development of the quantum theory of light.
In fluid dynamics, Euler was the first to predict the phenomenon of cavitation, in 1754, long before its first observation in the late 19th century, and the Euler number
In mathematics, the Euler numbers are a sequence ''En'' of integers defined by the Taylor series expansion
:\frac = \frac = \sum_^\infty \frac \cdot t^n,
where \cosh (t) is the hyperbolic cosine function. The Euler numbers are related to a ...
used in fluid flow calculations comes from his related work on the efficiency of turbine
A turbine ( or ) (from the Greek , ''tyrbē'', or Latin ''turbo'', meaning vortex) is a rotary mechanical device that extracts energy from a fluid flow and converts it into useful work. The work produced by a turbine can be used for generating ...
s. In 1757 he published an important set of equations for inviscid flow
In fluid dynamics, inviscid flow is the flow of an inviscid (zero-viscosity) fluid, also known as a superfluid. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, suc ...
in fluid dynamics, that are now known as the Euler equations
200px, Leonhard Euler (1707–1783)
In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
.[
Euler is well known in ]structural engineering
Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and muscles' that create the form and shape of man-made structures. Structural engineers also must understand and cal ...
for his formula giving Euler's critical load
Euler's critical load is the compressive load at which a slender column will suddenly bend or buckle. It is given by the formula:
P_ = \frac
where
*P_, Euler's critical load (longitudinal compression load on column),
*E, Young's modulus of the ...
, the critical buckling
In structural engineering, buckling is the sudden change in shape ( deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a ...
load of an ideal strut, which depends only on its length and flexural stiffness.
Logic
Euler is credited with using closed curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s to illustrate syllogistic reasoning (1768). These diagrams have become known as Euler diagram
An Euler diagram (, ) is a diagrammatic means of representing sets and their relationships. They are particularly useful for explaining complex hierarchies and overlapping definitions. They are similar to another set diagramming technique, Ven ...
s.[
An Euler diagram is a diagrammatic means of representing sets and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict sets. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important; the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships ( intersection, subset, and ]disjointness
In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, and are ''disjoint sets,'' while and are not disjoint. A c ...
). Curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection of the sets). A curve that is contained completely within the interior zone of another represents a subset of it.
Euler diagrams (and their refinement to Venn diagram
A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships ...
s) were incorporated as part of instruction in set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
as part of the new math movement in the 1960s.[ Since then, they have come into wide use as a way of visualizing combinations of characteristics.][
]
Music
One of Euler's more unusual interests was the application of Music and mathematics, mathematical ideas in music. In 1739 he wrote the ''Tentamen novae theoriae musicae'' (''Attempt at a New Theory of Music''), hoping to eventually incorporate musical theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians. Even when dealing with music, Euler's approach is mainly mathematical,[ for instance, his introduction of binary logarithms as a way of numerically describing the subdivision of octaves into fractional parts.][ His writings on music are not particularly numerous (a few hundred pages, in his total production of about thirty thousand pages), but they reflect an early preoccupation and one that remained with him throughout his life.][
A first point of Euler's musical theory is the definition of "genres", i.e. of possible divisions of the octave using the prime numbers 3 and 5. Euler describes 18 such genres, with the general definition 2mA, where A is the "exponent" of the genre (i.e. the sum of the exponents of 3 and 5) and 2m (where "m is an indefinite number, small or large, so long as the sounds are perceptible"), expresses that the relation holds independently of the number of octaves concerned. The first genre, with A = 1, is the octave itself (or its duplicates); the second genre, 2m.3, is the octave divided by the fifth (fifth + fourth, C–G–C); the third genre is 2m.5, major third + minor sixth (C–E–C); the fourth is 2m.32, two-fourths and a tone (C–F–B–C); the fifth is 2m.3.5 (C–E–G–B–C); etc. Genres 12 (2m.33.5), 13 (2m.32.52) and 14 (2m.3.53) are corrected versions of the Genus (music), diatonic, chromatic and enharmonic, respectively, of the Ancients. Genre 18 (2m.33.52) is the "diatonico-chromatic", "used generally in all compositions",][ and which turns out to be identical with the system described by Johann Mattheson.][ Euler later envisaged the possibility of describing genres including the prime number 7.][
Euler devised a specific graph, the ''Speculum musicum'',] to illustrate the diatonico-chromatic genre, and discussed paths in this graph for specific intervals, recalling his interest in the Seven Bridges of Königsberg (see #Graph theory, above). The device drew renewed interest as the Tonnetz in neo-Riemannian theory (see also Lattice (music)).[
Euler further used the principle of the "exponent" to propose a derivation of the ''gradus suavitatis'' (degree of suavity, of agreeableness) of intervals and chords from their prime factors – one must keep in mind that he considered just intonation, i.e. 1 and the prime numbers 3 and 5 only.][ Formulas have been proposed extending this system to any number of prime numbers, e.g. in the form
where ''p''''i'' are prime numbers and ''k''''i'' their exponents.][
]
Personal philosophy and religious beliefs
Euler opposed the concepts of Gottfried Leibniz, Leibniz's monadism and the philosophy of Christian Wolff (philosopher), Christian Wolff. Euler insisted that knowledge is founded in part on the basis of precise quantitative laws, something that monadism and Wolffian science were unable to provide. Euler's religious leanings might also have had a bearing on his dislike of the doctrine; he went so far as to label Wolff's ideas as "heathen and atheistic".
Euler was a religious person throughout his life. Much of what is known of Euler's religious beliefs can be deduced from his ''Letters to a German Princess'' and an earlier work, ''Rettung der Göttlichen Offenbahrung gegen die Einwürfe der Freygeister'' (''Defense of the Divine Revelation against the Objections of the Freethinkers''). These works show that Euler was a devout Christian who believed the Bible to be inspired; the ''Rettung'' was primarily an argument for the Biblical inspiration, divine inspiration of scripture.
There is a famous legend inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg Academy. The French philosopher Denis Diderot was visiting Russia on Catherine the Great's invitation. However, the Empress was alarmed that the philosopher's arguments for atheism were influencing members of her court, and so Euler was asked to confront the Frenchman. Diderot was informed that a learned mathematician had produced a proof of the existence of God: he agreed to view the proof as it was presented in court. Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced this Non sequitur (literary device), non-sequitur: "Sir, , hence God exists—reply!"
Diderot, to whom (says the story) all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. Embarrassed, he asked to leave Russia, a request that was graciously granted by the Empress. However amusing the anecdote may be, it is wikt:apocryphal, apocryphal, given that Diderot himself did research in mathematics.[
The legend was apparently first told by Dieudonné Thiébault with embellishment by Augustus De Morgan.]
Commemorations
Euler was featured on both the sixth[ and seventh][ series of the Swiss 10-Swiss franc, franc banknote and on numerous Swiss, German, and Russian postage stamps. In 1782 he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences.] The asteroid 2002 Euler was named in his honour.[
]
Selected bibliography
Euler has Contributions of Leonhard Euler to mathematics#Works, an extensive bibliography. His books include:
* ''Mechanica
''Mechanica'' ( la, Mechanica sive motus scientia analytice exposita; 1736) is a two-volume work published by mathematician Leonhard Euler which describes analytically the mathematics governing movement.
Euler both developed the techniques of ...
'' (1736)
''Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti'' (1744)
ref name=fraser/> (''A method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense'')[
* '']Introductio in analysin infinitorum
''Introductio in analysin infinitorum'' (Latin: ''Introduction to the Analysis of the Infinite'') is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the ''Introducti ...
'' (1748)[ (''Introduction to Analysis of the Infinite'')][
* '']Institutiones calculi differentialis
''Institutiones calculi differentialis'' (''Foundations of differential calculus'') is a mathematical work written in 1748 by Leonhard Euler and published in 1755 that lays the groundwork for the differential calculus. It consists of a single volu ...
'' (1755)[ (''Foundations of differential calculus'')
* ''Elements of Algebra, Vollständige Anleitung zur Algebra'' (1765)][ (''Elements of Algebra'')
* ''Institutiones calculi integralis'' (1768–1770)][ (''Foundations of integral calculus'')
* ''Letters to a German Princess'' (1768–1772)
* ''Dioptrica'', published in three volumes beginning in 1769][
It took until 1830 for the bulk of Euler's posthumous works to be individually published,][ with an additional batch of 61 unpublished works discovered by Paul Heinrich von Fuss (Euler's great-grandson and Nicolas Fuss's son) and published as a collection in 1862.][ A chronological catalog of Euler's works was compiled by Swedish mathematician Gustaf Eneström and published from 1910 to 1913. The catalog, known as the Eneström index, numbers Euler's works from E1 to E866.][ The Euler Archive was started at Dartmouth College][ before moving to the Mathematical Association of America][ and, most recently, to University of the Pacific (United States), University of the Pacific in 2017.]
In 1907, the Swiss Academies of Arts and Sciences, Swiss Academy of Sciences created the Euler Commission and charged it with the publication of Euler’s complete works. After several delays in the 19th century,[ the first volume of the ''Opera Omnia Leonhard Euler, Opera Omnia'', was published in 1911.] However, the discovery of new manuscripts continued to increase the magnitude of this project. Fortunately, the publication of Euler’s Opera Omnia has made steady progress, with over 70 volumes (averaging 426 pages each) published by 2006 and 80 volumes published by 2022. These volumes are organized into four series. The first series compiles the works on analysis, algebra, and number theory; it consists of 29 volumes and numbers over 14,000 pages. The 31 volumes of Series II, amounting to 10,660 pages, contain the works on mechanics, astronomy, and engineering. Series III contains 12 volumes on physics. Series IV, which contains the massive amount of Euler’s correspondences, unpublished manuscripts, and notes only began compilation in 1967. The series is projected to span 16 volumes, eight volumes of which have been released .
File:Acta Eruditorum - II geometria, 1744 – BEIC 13411238.jpg, Illustration from ''Solutio problematis... a. 1743 propositi'' published in Acta Eruditorum, 1744
File:Methodus inveniendi - Leonhard Euler - 1744.jpg, The title page of Euler's ''Methodus inveniendi lineas curvas''.
File:Leonhard Euler World Map AD1760.jpg, Euler's 1760 world map.
File:Euler Tab. Geogr. Africae 1753 UTA.jpg, Euler's 1753 map of Africa.
Notes
References
Sources
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Further reading
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External links
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The Euler Archive
Composition of Euler works with translations into English
Opera-Bernoulli-Euler
(compiled works of Euler, Bernoulli family, and contemporary peers)
The Euler Society
Euleriana
at the Berlin-Brandenburg Academy of Sciences and Humanities
Euler Family Tree
Euler's Correspondence with Frederick the Great, King of Prussia
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* (talk given by William Dunham (mathematician), William Dunham at )
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{{DEFAULTSORT:Euler, Leonhard
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