TheInfoList

Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss
mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...

,
physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of interest. In classical antiquity, there was no real ancient analog of a modern sci ...

,
astronomer An astronomer is a in the field of who focuses their studies on a specific question or field outside the scope of . They observe s such as s, s, , s and – in either (by analyzing the data) or . Examples of topics or fields astronomers stud ...

,
geographer A geographer is a physical scientist, social scientist or humanist whose area of study is geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, feat ...
,
logician Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to re ...

and
engineer Engineers, as practitioners of engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The dis ...

who founded the studies of
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
and
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

and made pioneering and influential discoveries in many other branches of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
such as
analytic number theory 300px, Riemann zeta function ''ζ''(''s'') in the complex plane. The color of a point ''s'' encodes the value of ''ζ''(''s''): colors close to black denote values close to zero, while hue encodes the value's Argument (complex analysis)">argument ...
,
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
, and
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...
. He introduced much of modern mathematical terminology and
notation In linguistics Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The traditional areas of linguistic analysis include p ...
, including the notion of a
mathematical function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. He is also known for his work in
mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million ...

,
fluid dynamics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...
,
optics Optics is the branch of that studies the behaviour and properties of , including its interactions with and the construction of that use or it. Optics usually describes the behaviour of , , and light. Because light is an , other forms of s ...

,
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses mathematics, phys ...
and
music theory Music theory is the study of the practices and possibilities of music Music is the of arranging s in time through the of melody, harmony, rhythm, and timbre. It is one of the aspects of all human societies. General include common elem ...
. 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 A scholar is a person who pursues academic and intellectual activities, particularly those that develop expertise in an area of Studying, study. A ...

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 This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of ma ...

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 more than 850 publications are collected in 92 ''quarto'' volumes, (including his Opera Omnia) more than anyone else in the field. He spent most of his adult life in

,
Russia Russia ( rus, link=no, Россия, Rossiya, ), or the Russian Federation, is a country spanning Eastern Europe Eastern Europe is the eastern region of Europe. There is no consistent definition of the precise area it covers, partly becau ...
, and in
Berlin Berlin (; ) is the and by both area and population. Its 3,769,495 inhabitants, as of 31 December 2019 makes it the , according to population within city limits. One of 's , Berlin is surrounded by the state of , and contiguous with , Brande ...

, then the capital of
Prussia Prussia, , Old Prussian Distribution of the Baltic tribes, circa 1200 CE (boundaries are approximate). Old Prussian was a Western Baltic language belonging to the Balto-Slavic branch of the Indo-European languages The Indo-Europ ...
. Euler is credited for popularizing the Greek letter (lowercase pi) to denote Archimedes' constant (the ratio of a circle's circumference to its diameter), as well as first employing the term to describe a function's
y-axis A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, the letter to express the imaginary unit , and the Greek letter (capital sigma) to express summations. He gave the current definition of the constant , the base of the
natural logarithm The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natura ...
, now known as
Euler's number The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828, and can be characterized in many ways. It is the base of a logarithm, base of the natural logarithm. It is the Limit of a sequence, limit of ...
. Euler was also the first practitioner of
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
(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 Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographe ...
). He became famous, among others, for solving the Basel Problem, after proving that the sum of the infinite series 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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...
equals 2, a number now commonly known as the
Euler characteristic#REDIRECT Euler characteristic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathema ...
. In the field of physics, Euler reformulated
Newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...

's laws of physics into
new laws The New Laws ( Spanish: ''Leyes Nuevas''), also known as the New Laws of the Indies for the Good Treatment and Preservation of the Indians ( Spanish: ''Leyes y ordenanzas nuevamente hechas por su Majestad para la gobernación de las Indias y buen ...
in his two-volume work ''Mechanica'' to explain the motion of rigid bodies more easily. 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 Basel ( , ) or Basle ( ; french: link=no, Bâle ; it, Basilea ; rm, Basilea ) is a city in northwestern Switzerland on the river High Rhine, Rhine. Basel is Switzerland's List of cities in Switzerland, third-most-populous city (after Zürich and ...

, Switzerland, to Paul III Euler, a pastor of the
Reformed Church Calvinism (also called the Reformed tradition, Reformed Christianity, Reformed Protestantism, or the Reformed faith) is a major branch of Protestantism Protestantism is a form of Christianity Christianity is an Abrahamic religions, A ...

, and Marguerite ( Brucker), another pastor's daughter. 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 Riehen (Swiss German: ''Rieche'') is a municipalities of Switzerland, municipality in the Cantons of Switzerland, canton of Basel-Stadt in Switzerland. Together with the city of Basel and Bettingen, Riehen is one of three municipalities in the cant ...
, Switzerland, where his father became pastor in the local church and Leonhard spent most of his childhood. Paul was a friend of the
Bernoulli family The Bernoulli family () of Basel , french: link=no, Bâlois(e), it, Basilese , neighboring_municipalities= Allschwil (BL), Hégenheim (FR-68), Binningen, Switzerland, Binningen (BL), Birsfelden (BL), Bottmingen (BL), Huningue (FR-68), Münche ...
, interested in mathematics, and took classes 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 Leibn ...
.
Johann Bernoulli Johann Bernoulli (also known as Jean or John; – 1 January 1748) was a Swiss Swiss may refer to: * the adjectival form of Switzerland ,german: Schweizer(in),french: Suisse(sse), it, svizzero/svizzera or , rm, Svizzer/Svizra , government ...

, then regarded as Europe's foremost mathematician, would eventually be an important influence on young Leonhard. Euler's formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, at only thirteen years of age, he enrolled at the
University of Basel The University of Basel (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of ...
. In 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of
René Descartes René Descartes ( or ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, Mathematics, mathematician, and scientist who invented analytic geometry, linking the previously sep ...

and
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

. Afterwards he enrolled in the theological faculty of the University of Basel. He was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered Euler's talent for mathematics. It was during this time that Euler, encouraged by the results of Johann Bernoulli's tutorial, obtained his father's consent to become a mathematician instead of a pastor. 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 prize competition (offered annually and later biennially by the academy beginning in 1720) for the first time. The problem that year was to find the best way to place the masts on a ship.
Pierre Bouguer Pierre Bouguer () (16 February 1698, Le Croisic, Croisic – 15 August 1758, Paris) was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture". Career Bouguer's father, Jean Bougu ...
, who became known as "the father of naval architecture", won and Euler took second place. Euler eventually entered this competition 15 times, winning 12 of them.

# Career

## Saint Petersburg

Johann Bernoulli's two sons,
Daniel Daniel is a masculine Masculinity (also called manhood or manliness) is a set of attributes, behavior Behavior (American English) or behaviour (British English; American and British English spelling differences#-our, -or, see spelling ...
and
NicolausNicolaus 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 system ...
, entered into service at the Imperial Russian Academy of Sciences in

in 1725, leaving Euler with the assurance they'd 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 refers to anything related to Russia, including: *Russians (русские, ''russkiye''), an ethnic group of the East Slavic peoples, primarily living in Russia and neighboring countries *Rossiyane (россияне), Russian language term ...
, settled into life in Saint Petersburg and took on an additional job as a medic in the
Russian Navy )Slow – "''Гвардейский встречный марш Военно-морского флота''" () , mascot = , equipment = 1 aircraft carrier An aircraft carrier is a warship that serve ...
. The Academy at Saint Petersburg, established by
Peter the Great Peter the Great ( rus, Пётр Вели́кий, Pyotr Velíkiy, ˈpʲɵtr vʲɪˈlʲikʲɪj), Peter I ( rus, Пётр Первый, Pyotr Pyervyy, ˈpʲɵtr ˈpʲɛrvɨj) or Pyotr Alekséyevich ( rus, Пётр Алексе́евич, p=ˈp ...

, 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 Emperor of all the Russias, Empress Regn ...

, 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 . 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 anglicization of names, anglicized as Anne, served as regent of the duchy of Courland from 1711 until 1730 and then ruled as Emperor of Russia, ...

assumed. 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 a
lieutenant A lieutenant ( or abbreviated Lt., Lt, LT, Lieut and similar) is a commissioned officer An officer is a person who holds a position of authority as a member of an armed force A military, also known collectively as armed forces, i ...
. 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 has 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 for permission to leave to Berlin, arguing he was in need for 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. He lived for 25 years in
Berlin Berlin (; ) is the and by both area and population. Its 3,769,495 inhabitants, as of 31 December 2019 makes it the , according to population within city limits. One of 's , Berlin is surrounded by the state of , and contiguous with , Brande ...

, where he wrote several hundred articles. In 1748 his text on functions called the ''
Introductio in analysin infinitorum ''Introductio in analysin infinitorum'' (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Throug ...
'' was published and in 1755 a text on
differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 volum ...
'' was published. In 1755, he was elected a foreign member of the
Royal Swedish Academy of Sciences The Royal Swedish Academy of Sciences ( Swedish: ''Kungliga Vetenskapsakademien'') is one of the royal academies of Sweden Sweden ( sv, Sverige ), officially the Kingdom of Sweden ( sv, links=no, Konungariket Sverige ), is a Nordic co ...
and of the
French Academy of Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization that exists to promote an discipli ...
. Notable students of Euler in Berlin included
Stepan Rumovsky Stepan Yakovlevich Rumovsky (russian: Степан Яковлевич Румовский; , Vladimir Governorate A governorate is an administrative division of a country. It is headed by a governor. As English-speaking nations tend to call regio ...
, 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 Locality may refer to: * Locality (association), an association of community regeneration organizations in England * Locality (linguistics) * Locality (settlement) * Suburbs and localities (Australia), in which ...

, in which he lived with his family and widowed mother. Euler became the tutor for Friederike Charlotte of Brandenburg-Schwedt, the Princess of
Anhalt-Dessau Anhalt-Dessau was a principality of the Holy Roman Empire The Holy Roman Empire ( la, Sacrum Imperium Romanum; german: Heiliges Römisches Reich) was a multi-ethnic complex of territories in Western Europe, Western and Central Europe that devel ...

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, mechanics, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Enc ...
,
Frederick IIFrederick II, Frederik II or Friedrich II may refer to: * Frederick II, Holy Roman Emperor (1194–1250), King of Sicily from 1198; Holy Roman Emperor from 1220 * Frederick II of Denmark (1534–1588), king of Denmark and Norway 1559–1588 * Freder ...

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, in many ways the polar opposite of
Voltaire François-Marie Arouet (; 21 November 169430 May 1778), known by his ''nom de plume A pen name, also called a ''nom de plume'' () or a literary double, is a pseudonym A pseudonym () or alias () (originally: ψευδώνυμος in Greek) is a ...

, 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, he maintained a strong connection to the Academy in St. Petersburg and also published 109 papers in Russia. He also assisted students from the Academy in St. Petersburg and at times accommodated Russian students in his house in Berlin. In 1760, with the
Seven Years' War The Seven Years' War (1756–1763) is widely considered to be the first global conflict in history, and was a struggle for world supremacy between Kingdom of Great Britain, Great Britain and Kingdom of France, France. In Europe, the conflict ar ...
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 the Empress of Russia The emperor or empress of all the Russias or All Russia, ''Imperator Vseros ...
of Russia later adding a further payment of 4000 roubles—an exorbitant amount at the time. Euler decided to leave Berlin in 1766 and return to Russia.

The political situation in Russia stabilized after 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 Anders Johan Lexell (24 December 1740 – ) was a Swedish-speaking population of Finland, Finnish-Swedish astronomer, mathematician, and physicist who spent most of his life in Russian Empire, Imperial Russia, where he was known as Andrei Ivano ...
. 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 The Neva (russian: Нева́, ; fi, Neva) is a river A river is a natural flowing watercourse, usually freshwater, flowing towards an ocean, sea, lake or another river. In some cases a river flows into the ground and becomes dry at the ...

. 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 (; ; March 18, 1690 – November 20, 1764) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
. 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 the 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 using light in the visible spectrum Laser beams with visible spectrum The visible spectrum is the portion of the electromagnetic spectrum that is visual perception, ...
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 χάρτης ''chartēs'', "papyrus, sheet of paper, map"; and γράφειν ''graphein'', "write") is the study and practice of making and using s. Combining , , and technique, cartography builds on the premise that rea ...
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 In Greek mythology and later Roman mythology, the Cyclopes ( ; el, Κύκλωπες, ''Kýklōpes'', "Circle-eyes" or "Round-eyes"; singular Cyclops ; , ''Kýklōps'') are giant one-eyed creatures. Three groups of Cyclopes can be distinguished ...

". 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 A lens is a transmissive optical Optics is the branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physic ...
in his left eye was discovered, and a few weeks later a failed surgical restoration rendered him almost totally blind. 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, who, according to Greek mythology Greek mythology is the body of myths originally told by the Ancient Greece, ancient Greeks, and ...

and its
orbit In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or po ...

with Lexell when he collapsed and died from a
brain hemorrhage Intracerebral hemorrhage (ICH), also known as cerebral bleed and intraparenchymal bleed, is a sudden bleeding into intraparenchymal bleed, the tissues of the brain, into its intraventricular hemorrhages, ventricles, or into both.https://my.clevela ...
. wrote a short obituary for the
and Russian mathematician
Nicolas Fuss Nicolas Fuss (29 January 1755 – 4 January 1826), also known as Nikolai Fuss, was a Swiss Swiss may refer to: * the adjectival form of Switzerland *Swiss people Places *Swiss, Missouri *Swiss, North Carolina *Swiss, West Virginia *Swiss, Wisconsi ...
, 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 mathematician and philosopher
Marquis de Condorcet Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet (; 17 September 1743 – 29 March 1794), known as Nicolas de Condorcet, was a French philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is th ...

, wrote: Euler was buried next to Katharina at the
Smolensk Lutheran Cemetery The Smolenskoye Cemetery (in German ''Smolensker Friedhof'') is a Lutheran cemetery on Dekabristov Island in Saint Petersburg Saint Petersburg ( rus, links=no, Санкт-Петербург, a=Ru-Sankt Peterburg Leningrad Petrograd Piter.ogg ...
on
Vasilievsky Island Vasilyevsky Island (russian: Васи́льевский о́стров, Vasilyevsky Ostrov, V.O.) is an island in Saint Petersburg, St. Petersburg, Russia, bordered by the Bolshaya Neva River, Bolshaya Neva and Malaya Neva Rivers (in the del ...
. In 1837, the
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 cemeteries ...
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 supposing that that was the site of the Neva Battle in 1240 when Alexander Ne ...

.

# Contributions to mathematics and physics

Euler worked in almost all areas of mathematics, such as
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

,
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...
,
trigonometry Trigonometry (from ', "triangle" and ', "measure") is a branch of that studies relationships between side lengths and s of s. The field emerged in the during the 3rd century BC from applications of to . The Greeks focused on the , while ...

,
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

, and
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

, as well as continuum physics, lunar theory and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto (text), quarto volumes. Euler's name is associated with a List of topics named after Leonhard Euler, large number of topics.

## Mathematical notation

Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function (mathematics), function 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 The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natura ...
(now also known as Euler's number), the Greek letter Sigma, Σ for summations and the letter to denote the imaginary unit. The use of the Greek letter ''pi (letter), π'' to denote the pi, ratio of a circle's circumference to its diameter was also popularized by Euler, although it originated with Welsh people, Welsh mathematician William Jones (mathematician), William Jones.

## Analysis

The development of
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...
was at the forefront of 18th-century mathematical research, and the Bernoulli family, Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigor, mathematical rigour (in particular his reliance on the principle of the generality of algebra), his ideas led to many great advances. Euler is well known in Mathematical analysis, analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as $e^x = \sum_^\infty = \lim_ \left(\frac + \frac + \frac + \cdots + \frac\right).$ Euler's use of power series enabled him to solve the famous Basel problem in 1735 (he provided a more elaborate argument in 1741): $\sum_^\infty = \lim_\left(\frac + \frac + \frac + \cdots + \frac\right) = \frac.$ He introduced the constant $\gamma = \lim_ \left( 1+ \frac + \frac + \frac + \cdots + \frac - \ln(n) \right) \approx 0.5772,$ now known as Euler's constant or the Euler–Mascheroni constant, and studied its relationship with the harmonic series (mathematics), harmonic series, the gamma function, and values of the Riemann zeta function. Euler introduced the use of the exponential function and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers, 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 functions. For any real number (taken to be radians), Euler's formula states that the Exponential function#On the complex plane, complex exponential function satisfies $e^ = \cos \varphi + i\sin \varphi.$ A special case of the above formula is known as Euler's identity, $e^ +1 = 0$ called "the most remarkable formula in mathematics" by Richard P. Feynman, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, , and . Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. He found a way to calculate integrals with complex limits, foreshadowing the development of modern
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
. He invented the calculus of variations and formulated the Euler–Lagrange equation for reducing optimization problems in this area to the solution of differential equations. 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 300px, Riemann zeta function ''ζ''(''s'') in the complex plane. The color of a point ''s'' encodes the value of ''ζ''(''s''): colors close to black denote values close to zero, while hue encodes the value's Argument (complex analysis)">argument ...
. In breaking ground for this new field, Euler created the theory of Generalized hypergeometric series, hypergeometric series, q-series, hyperbolic functions, hyperbolic trigonometric functions and the analytic theory of generalized continued fraction, continued fractions. For example, he proved the infinitude of primes using the divergence of the harmonic series (mathematics), harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to the development of the prime number theorem.

## Number theory

Euler's interest in number theory can be traced to the influence of
Christian Goldbach Christian Goldbach (; ; March 18, 1690 – November 20, 1764) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...
, 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. Euler developed some of Fermat's ideas and disproved some of his conjectures, such as his conjecture that all numbers of the form $2^+1$ (Fermat numbers) are prime. Euler linked the nature of prime distribution with ideas in analysis. He proved that Proof that the sum of the reciprocals of the primes diverges, the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and the prime numbers; this is known as the Proof of the Euler product formula for the Riemann zeta function, Euler product formula for the Riemann zeta function. Euler invented the totient function φ(''n''), the number of positive integers less than or equal to the integer ''n'' that are coprime to ''n''. Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem. He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. He proved that the relationship shown between even perfect numbers and Mersenne primes earlier proved by Euclid was one-to-one, a result otherwise known as the Euclid–Euler theorem. Euler also conjectured the law of quadratic reciprocity. The concept is regarded as a fundamental theorem of 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 This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of ma ...

, particularly ''Disquisitiones Arithmeticae''. By 1772 Euler had proved that 231 − 1 = 2147483647, 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867. Euler contributed major developments to the theory of Partition (number theory), 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 Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographe ...
. The city of Königsberg,
Prussia Prussia, , Old Prussian Distribution of the Baltic tribes, circa 1200 CE (boundaries are approximate). Old Prussian was a Western Baltic language belonging to the Balto-Slavic branch of the Indo-European languages The Indo-Europ ...
was set on the Pregolya, Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian path, Eulerian circuit. This solution is considered to be the first theorem of
graph theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
. Euler also discovered the Planar graph#Euler's formula, formula $V - E + F = 2$ relating the number of vertices, edges and faces of a Convex polytope, convex polyhedron, and hence of a planar graph. The constant in this formula is now known as the
Euler characteristic#REDIRECT Euler characteristic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathema ...
for the graph (or other mathematical object), and is related to the genus (mathematics), genus of the object. The study and generalization of this formula, specifically by Augustin-Louis Cauchy, Cauchy and Simon Antoine Jean L'Huilier, L'Huilier, is at the origin of
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

.

## 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, Fourier series, Euler numbers, the constants and pi, , continued fractions and integrals. He integrated Gottfried Leibniz, Leibniz's
differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
with Newton's Method of Fluxions, and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. The most notable of these approximations are Euler's method and the Euler–Maclaurin formula. 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, Euler applied these techniques to celestial problems. His work in astronomy was recognized by multiple Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the solar parallax, parallax of the Sun. His calculations contributed to the development of accurate History of longitude, longitude tables. Euler made important contributions in
optics Optics is the branch of that studies the behaviour and properties of , including its interactions with and the construction of that use or it. Optics usually describes the behaviour of , , and light. Because light is an , other forms of s ...

. He disagreed with Newton's corpuscular theory of light, which was then the prevailing theory. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christiaan Huygens would become the dominant mode of thought, at least until the development of the wave-particle duality, quantum theory of light. In
fluid dynamics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...
, Euler was the first to predict the phenomenon of cavitation, in 1754, long before its first observation in the late 19th century, and the Euler number (physics), Euler number used in fluid flow calculations comes from his related work on the efficiency of turbines. In 1757 he published an important set of equations for inviscid flow in
fluid dynamics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...
, that are now known as the Euler equations (fluid dynamics), Euler equations. Euler is well known in structural engineering for his formula giving Euler's critical load, the critical buckling load of an ideal strut, which depends only on its length and flexural stiffness.

## Logic

Euler is credited with using closed curves to illustrate syllogism, syllogistic reasoning (1768). These diagrams have become known as Euler diagrams. An Euler diagram is a diagrammatic means of representing Set (mathematics), sets and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict Set (mathematics), sets. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the element (mathematics), elements of the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important; the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection (set theory), intersection, subset and Disjoint sets, disjointness). Curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection (set theory), intersection of the sets). A curve that is contained completely within the interior zone of another represents a subset of it. Euler diagrams (and their refinement to Venn diagrams) were incorporated as part of instruction in set theory 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 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, including for instance the introduction of binary logarithms as a way of describing numerically 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 did not leave 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 $ds=\sum_i(k_ip_i-k_i)+1,$ 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 stayed 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, =''x'', 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 an Contributions of Leonhard Euler to mathematics#Works, extensive bibliography. His books include: * ''Mechanica'' (1736).
''Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti'' (1744)
Latin translation: ''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 Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Throug ...
'' (1748). English translation: ''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 volum ...
'' (1755). * ''Vollständige Anleitung zur Algebra'' (1765). * ''Institutiones calculi integralis'' (1768–1770). * ''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. After several delays in the 19th century, a definitive collection of Euler's works, entitled ''Opera Omnia Leonhard Euler, Opera Omnia'', has been published since 1911 by the Euler Commission of the Swiss Academies of Arts and Sciences, Swiss Academy of Sciences. A chronological catalog of Euler's works was compiled by Swedish mathematician Gustaf Eneström and published from 1910 to 1913, and Euler's works are often cited by their number in the Eneström index, 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.Euler Archive
, University of the Pacific
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''.

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