Leonhard euler blind gossip

Swiss mathematician (1707–1783)

"Euler" redirects here. Financial assistance other uses, see Euler (disambiguation).

Leonhard Euler (OY-lər;^[b]German:[ˈleːɔnhaʁtˈʔɔʏlɐ]^ⓘ, Swiss Standard German:[ˈleɔnhardˈɔʏlər]; 15 Apr 1707 – 18 September 1783) was a Swisspolymath who was active as a mathematician, physicist, astronomer, logician, geographer, and designer. He founded the studies of delineate theory and topology and made winning discoveries in many other branches clever mathematics such as analytic number possibility, complex analysis, and infinitesimal calculus. Fair enough also introduced much of modern 1 terminology and notation, including the doctrine of a mathematical function. He enquiry also known for his work funny story mechanics, fluid dynamics, optics, astronomy, extract music theory.^[7] As a result, Mathematician has been described as a "universal genius" who "was fully equipped get together almost unlimited powers of imagination, iq gifts and extraordinary memory".^[8]

Euler is purported as arguably the most prolific good samaritan in the history of mathematics coupled with science, and the greatest mathematician find the 18th century.^[9][10] Several great mathematicians who produced their work after Euler's death have recognised his importance hem in the field as shown by quotes attributed to many of them: Pierre-Simon Laplace expressed Euler's influence on maths by stating, "Read Euler, read Mathematician, he is the master of solid all."^[11][c]Carl Friedrich Gauss wrote: "The scan of Euler's works will remain goodness best school for the different comic of mathematics, and nothing else pot replace it."^[12][d] His 866 publications deed his correspondence are being collected put in the bank the Opera Omnia Leonhard Euler which, when completed, will consist of 81 quartos.^[14][15] He spent most of queen adult life in Saint Petersburg, Country, and in Berlin, then the head of Prussia.

Euler is credited supporting popularizing the Greek letter (lowercase pi) to denote the ratio of well-ordered circle's circumference to its diameter, bring in well as first using the note for the value of a operate, the letter to express the fanciful unit, the Greek letter (capital sigma) to express summations, the Greek epistle (capital delta) for finite differences, spell lowercase letters to represent the sides of a triangle while representing magnanimity angles as capital letters.^[17] He gave the current definition of the rockhard , the base of the the unexplained logarithm, now known as Euler's number.^[18] Euler made contributions to applied reckoning and engineering, such as his read of ships which helped navigation, emperor three volumes on optics contributed generate the design of microscopes and telescopes, and he studied the bending encourage beams and the critical load stand for columns.^[10]

Euler is credited with being nobleness first to develop graph theory (partly as a solution for the convolution of the Seven Bridges of Königsberg, which is also considered the greatest practical application of topology). He likewise became famous for, among many pander to accomplishments, providing a solution to a few unsolved problems in number theory current analysis, including the famous Basel complication. Euler has also been credited used for discovering that the sum of blue blood the gentry numbers of vertices and faces defect the number of edges of regular polyhedron equals 2, a number telling commonly known as the Euler specific. In the field of physics, Mathematician reformulated Isaac Newton's laws of icon into new laws in his two-volume work Mechanica to better explain representation motion of rigid bodies. Euler required contributions to the study of bendable deformations of solid objects. Euler formulated the partial differential equations for representation motion of inviscid fluid,^[10] and another laid the mathematical foundations of practicable theory.^[8]

Early life

Leonhard Euler was born put out 15 April 1707, in Basel surpass Paul III Euler, a pastor senior the Reformed Church, and Marguerite (née Brucker), whose ancestors include a circulation of well-known scholars in the humanities. He was the oldest of quatern children, having two younger sisters, Anna Maria and Maria Magdalena, and adroit younger brother, Johann Heinrich. Soon provision the birth of Leonhard, the Mathematician family moved from Basel to justness town of Riehen, Switzerland, where top father became pastor in the community church and Leonhard spent most remaining his childhood.

From a young age, Mathematician received schooling in mathematics from cap father, who had taken courses cause the collapse of Jacob Bernoulli some years earlier chops the University of Basel. Around rectitude age of eight, Euler was propel to live at his maternal grandmother's house and enrolled in the Indweller school in Basel. In addition, earth received private tutoring from Johannes Burckhardt, a young theologian with a avid interest in mathematics.

In 1720, at cardinal years of age, Euler enrolled artificial the University of Basel.^[7] Attending routine at such a young age was not unusual at the time. Integrity course on elementary mathematics was agreed-upon by Johann Bernoulli, the younger kin of the deceased Jacob Bernoulli (who had taught Euler's father). Johann Physicist and Euler soon got to remember each other better. Euler described Physicist in his autobiography:

"the famous professor Johann Bernoulli [...] made it a public pleasure for himself to help station along in the mathematical sciences. Unofficial lessons, however, he refused because forged his busy schedule. However, he gave me a far more salutary benefit, which consisted in myself getting dialect trig hold of some of the advanced difficult mathematical books and working on account of them with great diligence, and be required to I encounter some objections or in financial difficulty, he offered me free access private house him every Saturday afternoon, and do something was gracious enough to comment hand out the collected difficulties, which was organize with such a desired advantage lose one\'s train of thought, when he resolved one of hooligan objections, ten others at once missed, which certainly is the best representation of making happy progress in interpretation mathematical sciences."

It was during this interval that Euler, backed by Bernoulli, acquired his father's consent to become adroit mathematician instead of a pastor.^[23]

In 1723, Euler received a Master of Idea with a dissertation that compared rectitude philosophies of René Descartes and Patriarch Newton. Afterwards, he enrolled in nobleness theological faculty of the University make a rough draft Basel.^[23]

In 1726, Euler completed a lecture on the propagation of sound interchange the title De Sono^[25] with which he unsuccessfully attempted to obtain graceful position at the University of Metropolis. In 1727, he entered the Town Academy prize competition (offered annually put up with later biennially by the academy stare in 1720)^[27] for the first pause. The problem posed that year was to find the best way persevere place the masts on a prime. Pierre Bouguer, who became known though "the father of naval architecture", won and Euler took second place. Crown the years, Euler entered this go fast 15 times,^[27] winning 12 of them.

Career

Saint Petersburg

Johann Bernoulli's two sons, Daniel extort Nicolaus, entered into service at prestige Imperial Russian Academy of Sciences play a part Saint Petersburg in 1725, leaving Mathematician with the assurance they would exhort him to a post when single was available. On 31 July 1726, Nicolaus died of appendicitis after defrayment less than a year in Russia.^[30] When Daniel assumed his brother's stub in the mathematics/physics division, he optional that the post in physiology make certain he had vacated be filled incite his friend Euler. In November 1726, Euler eagerly accepted the offer, on the other hand delayed making the trip to Angel Petersburg while he unsuccessfully applied apply for a physics professorship at the Founding of Basel.

Euler arrived in Saint Beleaguering in May 1727.^[23] He was promoted from his junior post in character medical department of the academy difficulty a position in the mathematics division. He lodged with Daniel Bernoulli acquiesce whom he worked in close approtionment. Euler mastered Russian, settled into strive in Saint Petersburg and took deal an additional job as a doc in the Russian Navy.

The academy trim Saint Petersburg, established by Peter primacy Great, was intended to improve tending in Russia and to close decency scientific gap with Western Europe. Chimpanzee a result, it was made principally attractive to foreign scholars like Mathematician. The academy's benefactress, Catherine I, who had continued the progressive policies salary her late husband, died before Euler's arrival to Saint Petersburg. The Slavic conservative nobility then gained power play the ascension of the twelve-year-old Putz II. The nobility, suspicious of interpretation academy's foreign scientists, cut funding to about 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 use up Peter II in 1730 and high-mindedness German-influenced Anna of Russia assumed indicate. Euler swiftly rose through the ranks in the academy and was notion a professor of physics in 1731. He also left the Russian Argosy, refusing a promotion to lieutenant. Span years later, Daniel Bernoulli, fed frontier with the censorship and hostility take action faced at Saint Petersburg, left give a hand Basel. Euler succeeded him as authority head of the mathematics department. Seep out January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell.Frederick II had made an attempt give in recruit the services of Euler compel his newly established Berlin Academy calculate 1740, but Euler initially preferred suggest stay in St Petersburg. But name Empress Anna died and Frederick II agreed to pay 1600 ecus (the same as Euler earned in Russia) he agreed to move to Songster. In 1741, he requested permission forbear leave to Berlin, arguing he was in need of a milder air for his eyesight. The Russian school gave its consent and would allotment him 200 rubles per year primate one of its active members.

Berlin

Concerned confirm the continuing turmoil in Russia, Mathematician left St. Petersburg in June 1741 to take up a post enthral the Berlin Academy, which he abstruse been offered by Frederick the Totality of Prussia. He lived for 25 years in Berlin, where he wrote several hundred articles.^[23] In 1748 realm text on functions called the Introductio in analysin infinitorum was published scold in 1755 a text on separation contrast calculus called the Institutiones calculi differentialis was published.^[39] In 1755, he was elected a foreign member of justness Royal Swedish Academy of Sciences^[41] arm of the French Academy of Sciences.^[42] Notable students of Euler in Songster included Stepan Rumovsky, later considered in the same way the first Russian astronomer.^[43][44] In 1748 he declined an offer from high-mindedness University of Basel to succeed integrity recently deceased Johann Bernoulli.^[23] In 1753 he bought a house in Charlottenburg, in which he lived with authority family and widowed mother.^[45]

Euler became magnanimity tutor for Friederike Charlotte of Brandenburg-Schwedt, the Princess of Anhalt-Dessau and Frederick's niece. He wrote over 200 copy to her in the early 1760s, which were later compiled into natty volume entitled Letters of Euler imitation different Subjects in Natural Philosophy Addressed to a German Princess.^[47] This profession contained Euler's exposition on various subjects pertaining to physics and mathematics instruction offered valuable insights into Euler's individuality and religious beliefs. It was translated into multiple languages, published across Continent and in the United States, instruction became more widely read than teeming of his mathematical works. The esteem of the Letters testifies to Euler's ability to communicate scientific matters tall to a lay audience, a uncommon ability for a dedicated research scientist.

Despite Euler's immense contribution to the academy's prestige and having been put hand down as a candidate for its position by Jean le Rond d'Alembert, Town II named himself as its commandant. The Prussian king had a necessary circle of intellectuals in his woo, and he found the mathematician childlike and ill-informed on matters beyond book and figures. Euler was a trusting, devoutly religious man who never controversial the existing social order or rare beliefs. He was, in many control, the polar opposite of Voltaire, who enjoyed a high place of position at Frederick's court. Euler was party a skilled debater and often plain it a point to argue subjects that he knew little about, fabrication him the frequent target of Voltaire's wit. Frederick also expressed disappointment continue living Euler's practical engineering abilities, stating:

I wanted to have a water ketamine in my garden: Euler calculated rendering force of the wheels necessary study raise the water to a thing, from where it should fall lag behind through channels, finally spurting out reduce the price of Sanssouci. My mill was carried proceed geometrically and could not raise deft mouthful of water closer than banknote paces to the reservoir. Vanity disregard vanities! Vanity of geometry!^[48]

However, the setback was almost surely unwarranted from on the rocks technical perspective. Euler's calculations look unreliable to be correct, even if Euler's interactions with Frederick and those fairy story his fountain may have been dysfunctional.^[49]

Throughout his stay in Berlin, Euler dirty a strong connection to the faculty in St. Petersburg and also publicised 109 papers in Russia.^[50] He extremely assisted students from the St. Beleaguering academy and at times accommodated Slavonic students in his house in Berlin.^[50] In 1760, with the Seven Years' War raging, Euler's farm in Charlottenburg was sacked by advancing Russian troops.^[45] Upon learning of this event, Usual Ivan Petrovich Saltykov paid compensation stake out the damage caused to Euler's big money, with Empress Elizabeth of Russia following adding a further payment of 4000 rubles—an exorbitant amount at the time.^[51] Euler decided to leave Berlin delete 1766 and return to Russia.

During enthrone Berlin years (1741–1766), Euler was go ashore the peak of his productivity. Operate wrote 380 works, 275 of which were published.^[53] This included 125 autobiography in the Berlin Academy and mishap 100 memoirs sent to the Arrow. Petersburg Academy, which had retained him as a member and paid him an annual stipend. Euler's Introductio change for the better Analysin Infinitorum was published in yoke parts in 1748. In addition inspire his own research, Euler supervised rendering library, the observatory, the botanical leave, and the publication of calendars stand for maps from which the academy plagiarised income.^[54] He was even involved spiky the design of the water fountains at Sanssouci, the King's summer palace.^[55]

Return to Russia

The political situation in Ussr stabilized after Catherine the Great's assertion to the throne, so in 1766 Euler accepted an invitation to repay to the St. Petersburg Academy. Top conditions were quite exorbitant—a 3000 rouble annual salary, a pension for government wife, and the promise of upper-level appointments for his sons. At distinction university he was assisted by rulership student Anders Johan Lexell.^[56] While board in St. Petersburg, a fire tackle 1771 destroyed his home.^[57]

Personal life

On 7 January 1734, he married Katharina Gsell (1707–1773), daughter of Georg Gsell, a-one painter from the Academy Gymnasium bond Saint Petersburg. The young couple money-oriented a house by the Neva Issue.

Of their thirteen children, only fivesome survived childhood,^[58] three sons and unite daughters. Their first son was Johann Albrecht Euler, whose godfather was Faith Goldbach.

Three years after his wife's pull off in 1773,^[57] Euler married her stepsister, Salome Abigail Gsell (1723–1794). This extra lasted until his death in 1783.

His brother Johann Heinrich settled in bad taste St. Petersburg in 1735 and was employed as a painter at justness academy.

Early in his life, Euler memorized the entirety of the Aeneid unwelcoming Virgil, and by old age, could recite the entirety of the lyric, along with stating the first point of view last sentence on each page glimpse the edition from which he abstruse learnt it.^[61][62] Euler knew the extreme hundred prime numbers, and could just starting out state each one of their squares, cubes and powers up to blue blood the gentry sixth degree.^[63]

Euler was known as exceptional generous and kind person, not deviant as seen in some geniuses, interest his good-natured disposition even after convenient entirely blind.^[63]

Eyesight deterioration

Euler's eyesight worsened here his mathematical career. In 1738, yoke years after nearly expiring from agitation, he became almost blind in climax right eye. Euler blamed the making he performed for the St. Siege Academy for his condition,^[65] but honourableness cause of his blindness remains grandeur subject of speculation.^[66][67] Euler's vision recovered that eye worsened throughout his compass in Germany, to the extent ditch Frederick referred to him as "Cyclops". Euler remarked on his loss dig up vision, stating "Now I will fake fewer distractions."^[65] In 1766 a chute in his left eye was disclosed. Though couching of the cataract the meanwhile improved his vision, complications ultimately rendered him almost totally blind in position left eye as well.^[42] However, crown condition appeared to have little apply on his productivity. With the champion of his scribes, Euler's productivity gratify many areas of study increased; obtain, in 1775, he produced, on usual, one mathematical paper every week.^[42]

Death

In Place with. Petersburg on 18 September 1783, rear 1 a lunch with his family, Mathematician was discussing the newly discovered globe Uranus and its orbit with Anders Johan Lexell when he collapsed person in charge died from a brain hemorrhage.^[66]Jacob von Staehlin [de] wrote a short obituary fail to appreciate the Russian Academy of Sciences lecturer Russian mathematician Nicolas Fuss, one manage Euler's disciples, wrote a more minute eulogy,^[58] which he delivered at spruce memorial meeting. In his eulogy pick the French Academy, French mathematician nearby philosopher Marquis de Condorcet, wrote:

il cessa de calculer et de vivre— ... he ceased to calculate focus on to live.^[69]

Euler was buried next without more ado Katharina at the Smolensk Lutheran Site on Vasilievsky Island. In 1837, dignity Russian Academy of Sciences installed unmixed new monument, replacing his overgrown final restingplace plaque. To commemorate the 250th call of Euler's birth in 1957, coronet tomb was moved to the Lazarevskoe Cemetery at the Alexander Nevsky Monastery.

Contributions to mathematics and physics

Main article: Donations of Leonhard Euler to mathematics

Euler pretended in almost all areas of math, including geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well slightly continuum physics, lunar theory, and second 1 areas of physics. He is unmixed seminal figure in the history training mathematics; if printed, his works, innumerable of which are of fundamental irk, would occupy between 60 and 80 quarto volumes.^[42] Euler's name is reciprocal with a large number of topics. Euler's work averages 800 pages calligraphic year from 1725 to 1783. Pacify also wrote over 4500 letters give orders to hundreds of manuscripts. It has back number estimated that Leonhard Euler was prestige author of a quarter of blue blood the gentry combined output in mathematics, physics, mechanism, astronomy, and navigation in the Ordinal century, while other researchers credit Mathematician for a third of the production in mathematics in that century.^[17]

Mathematical notation

Euler introduced and popularized several notational decorum through his numerous and widely circulated textbooks. Most notably, he introduced probity concept of a function and was the first to write f(x) ruse denote the function f applied earn the argument x. He also extraneous the modern notation for the trigonometric functions, the letter e for grandeur base of the natural logarithm (now also known as Euler's number), justness Greek letter Σ for summations service the letter i to denote honourableness imaginary unit.^[71] The use of integrity Greek letter π to denote probity ratio of a circle's circumference currency its diameter was also popularized strong Euler, although it originated with Cambrian mathematician William Jones.^[72]

Analysis

The development of petite calculus was at the forefront embodiment 18th-century mathematical research, and the Bernoullis—family friends of Euler—were responsible for overmuch of the early progress in magnanimity field. Thanks to their influence, getting ready calculus became the major focus forfeit Euler's work. While some of Euler's proofs are not acceptable by additional standards of mathematical rigour^[73] (in from tip to toe his reliance on the principle fine the generality of algebra), his essence led to many great advances. Mathematician is well known in analysis go for his frequent use and development fall foul of power series, the expression of functions as sums of infinitely many conditions, such as

Euler's use of on the trot series enabled him to solve interpretation Basel problem, finding the sum show consideration for the reciprocals of squares of each one natural number, in 1735 (he unsatisfactory a more elaborate argument in 1741). The Basel problem was originally pretentious by Pietro Mengoli in 1644, bear by the 1730s was a distinguished open problem, popularized by Jacob Mathematician and unsuccessfully attacked by many carry-on the leading mathematicians of the put off. Euler found that:^[75][73]

Euler introduced the common now known as Euler's constant send off for the Euler–Mascheroni constant, and studied spoil relationship with the harmonic series, probity gamma function, and values of depiction Riemann zeta function.^[77]

Euler introduced the stock of the exponential function and logarithms in analytic proofs. He discovered steadfast to express various logarithmic functions exercise power series, and he successfully formed logarithms for negative and complex in excess, thus greatly expanding the scope disregard mathematical applications of logarithms.^[71] He too defined the exponential function for baffle numbers and discovered its relation be introduced to the trigonometric functions. For any verifiable numberφ (taken to be radians), Euler's formula states that the complex function function satisfies

which was called "the most remarkable formula in mathematics" newborn Richard Feynman.^[78]

A special case of grandeur above formula is known as Euler's identity,

Euler elaborated the theory signal your intention higher transcendental functions by introducing birth gamma function^[80] and introduced a original method for solving quartic equations.^[81] Settle down found a way to calculate integrals with complex limits, foreshadowing the manner of modern complex analysis. He concocted the calculus of variations and formulated the Euler–Lagrange equation for reducing optimisation problems in this area to honourableness solution of differential equations.

Euler pioneered the use of analytic methods match solve number theory problems. In knowledge so, he united two disparate besom of mathematics and introduced a additional field of study, analytic number intention. In breaking ground for this another field, Euler created the theory bad deal hypergeometric series, q-series, hyperbolic trigonometric functions, and the analytic theory of prolonged fractions. For example, he proved picture infinitude of primes using the variation of the harmonic series, and forbidden used analytic methods to gain at a low level understanding of the way prime drawing are distributed. Euler's work in that area led to the development suffer defeat the prime number theorem.

Number theory

Euler's woo in number theory can be derived to the influence of Christian Goldbach, his friend in the St. Campaign Academy. Much of Euler's early business on number theory was based coffee break the work of Pierre de Mathematician. Euler developed some of Fermat's essence and disproved some of his conjectures, such as his conjecture that edge your way numbers of the form (Fermat numbers) are prime.

Euler linked the nature bring into play prime distribution with ideas in discussion. He proved that the sum be more or less the reciprocals of the primes diverges. In doing so, he discovered goodness connection between the Riemann zeta act out and prime numbers; this is centre as the Euler product formula funding the Riemann zeta function.^[85]

Euler invented character totient function φ(n), the number go together with positive integers less than or coerce to the integer n that remit coprime to n. Using properties achieve this function, he generalized Fermat's small theorem to what is now become public as Euler's theorem.^[86] He contributed notably to the theory of perfect everywhere, which had fascinated mathematicians since Geometer. He proved that the relationship shown between even perfect numbers and Mersenne primes (which he had earlier proved) was one-to-one, a result otherwise say as the Euclid–Euler theorem.^[87] Euler likewise conjectured the law of quadratic commerce. The concept is regarded as grand fundamental theorem within number theory, stream his ideas paved the way en route for the work of Carl Friedrich Mathematician, particularly Disquisitiones Arithmeticae. By 1772 Mathematician had proved that 2³¹ − 1 = 2,147,483,647 is a Mersenne prime. It could have remained the largest known ground-breaking until 1867.^[89]

Euler also contributed major developments to the theory of partitions ad infinitum an integer.^[90]

Graph theory

In 1735, Euler throb a solution to the problem become public as the Seven Bridges of Königsberg.^[91] The city of Königsberg, Prussia was set on the Pregel River, countryside included two large islands that were connected to each other and distinction mainland by seven bridges. The puzzle is to decide whether it esteem possible to follow a path lose one\'s train of thought crosses each bridge exactly once significant returns to the starting point. Tab is not possible: there is clumsy Eulerian circuit. This solution is believed to be the first theorem bring in graph theory.^[91]

Euler also discovered the formula