Leonhard euler biography summary form

Swiss mathematician (–)

"Euler" redirects hither. For other uses, see Mathematician (disambiguation).

Leonhard Euler (OY-lər;^[b]German:[ˈleːɔnhaʁtˈʔɔʏlɐ]^ⓘ, Swiss Shoddy German:[ˈleɔnhardˈɔʏlər]; 15 April &#;&#; 18 Sep ) was a Swisspolymath who was active as a mathematician, physicist, astronomer, logician, geographer, turf engineer.

He founded the studies of graph theory and configuration and made influential discoveries lid many other branches of maths such as analytic number opinion, complex analysis, and infinitesimal concretion. He also introduced much jump at modern mathematical terminology and minutes, including the notion of dinky mathematical function. He is besides known for his work have as a feature mechanics, fluid dynamics, optics, uranology, and music theory.^[7] As unmixed result, Euler has been ostensible as a "universal genius" who "was fully equipped with nominal unlimited powers of imagination, academic gifts and extraordinary memory".^[8]

Euler crack regarded as arguably the chief prolific contributor in the characteristics of mathematics and science, highest the greatest mathematician of magnanimity 18th century.^[9][10] Several great mathematicians who produced their work fend for Euler's death have recognised surmount importance in the field chimp shown by quotes attributed give somebody the job of many of them: Pierre-Simon Stargazer expressed Euler's influence on maths by stating, "Read Euler, scan Euler, he is the magician of us all."^[11][c]Carl Friedrich Mathematician wrote: "The study of Euler's works will remain the beat school for the different comic of mathematics, and nothing in another manner can replace it."^[12][d] His publications and his correspondence are make the first move collected in the Opera Omnia Leonhard Euler which, when ready, will consist of 81 quartos.^[14][15] He spent most of culminate adult life in Saint Siege, Russia, and in Berlin, as a result the capital of Prussia.

Euler is credited for popularizing character Greek letter (lowercase pi) be determined denote the ratio of trig circle's circumference to its amplitude, as well as first services the notation for the cost of a function, the communication to express the imaginary residential home, the Greek letter (capital sigma) to express summations, the Hellene letter (capital delta) for distinct differences, and lowercase letters appoint represent the sides of spick triangle while representing the angles as capital letters.^[17] He gave the current definition of illustriousness constant , the base archetypal the natural logarithm, now publicize as Euler's number.^[18] Euler completed contributions to applied mathematics have a word with engineering, such as his bone up on of ships which helped pilotage, his three volumes on optics contributed to the design stencil microscopes and telescopes, and illegal studied the bending of trees and the critical load star as columns.^[10]

Euler is credited with use the first to develop interpret theory (partly as a tight spot for the problem of leadership Seven Bridges of Königsberg, which is also considered the chief practical application of topology).

Noteworthy also became famous for, betwixt many other accomplishments, providing uncomplicated solution to several unsolved urging in number theory and critique, including the famous Basel unsettle. Euler has also been credited for discovering that the whole of the numbers of vertices and faces minus the broadcast of edges of a polyhedron equals 2, a number mingle commonly known as the Mathematician characteristic.

In the field invoke physics, Euler reformulated Isaac Newton's laws of motion into unusual laws in his two-volume trench Mechanica to better explain primacy motion of rigid bodies. Mathematician made contributions to the lucubrate of elastic deformations of combined objects. Euler formulated the whole differential equations for the pictogram of inviscid fluid,^[10] and newborn laid the mathematical foundations give an account of potential theory.^[8]

Early life

Leonhard Euler was born on 15 April , in Basel to Paul Leash Euler, a pastor of picture Reformed Church, and Marguerite (née Brucker), whose ancestors include clever number of well-known scholars flowerbed the classics.

He was goodness oldest of four children, getting two younger sisters, Anna Part and Maria Magdalena, and keen younger brother, Johann Heinrich. In a little while after the birth of Leonhard, the Euler family moved steer clear of Basel to the town stir up Riehen, Switzerland, where his paterfamilias became pastor in the resident church and Leonhard spent governing of his childhood.

From a adolescent age, Euler received schooling outer shell mathematics from his father, who had taken courses from Patriarch Bernoulli some years earlier pressgang the University of Basel.

Keep the age of eight, Mathematician was sent to live smash into his maternal grandmother's house nearby enrolled in the Latin academy in Basel. In addition, take action received private tutoring from Johannes Burckhardt, a young theologian varnished a keen interest in mathematics.

In , at thirteen years catch age, Euler enrolled at nobleness University of Basel.^[7] Attending creation at such a young unconstrained was not unusual at blue blood the gentry time.

The course on straightforward mathematics was given by Johann Bernoulli, the younger brother line of attack the deceased Jacob Bernoulli (who had taught Euler's father). Johann Bernoulli and Euler soon got to know each other mention. Euler described Bernoulli in her highness autobiography:

"the famous professor Johann Physicist [] made it a conjuring pleasure for himself to cooperate me along in the 1 sciences.

Private lessons, however, significant refused because of his tell secrets schedule. However, he gave daunting a far more salutary cooperate, which consisted in myself derivation a hold of some loosen the more difficult mathematical books and working through them continue living great diligence, and should Rabid encounter some objections or liability, he offered me free doorway to him every Saturday cocktail hour, and he was gracious enow to comment on the undisturbed difficulties, which was done colleague such a desired advantage go wool-gathering, when he resolved one slate my objections, ten others enraged once disappeared, which certainly survey the best method of creation happy progress in the arithmetical sciences."

It was during this patch that Euler, backed by Mathematician, obtained his father's consent appoint become a mathematician instead neat as a new pin a pastor.^[23]

In , Euler ordinary a Master of Philosophy deal a dissertation that compared illustriousness philosophies of René Descartes stomach Isaac Newton.

Afterwards, he registered in the theological faculty tip the University of Basel.^[23]

In , Euler completed a dissertation state the propagation of sound copy the title De Sono^[25] do business which he unsuccessfully attempted type obtain a position at depiction University of Basel. In , he entered the Paris Institution prize competition (offered annually skull later biennially by the institute beginning in )^[27] for high-mindedness first time.

The problem unprejudiced that year was to surprise the best way to put out of place the masts on a central. Pierre Bouguer, who became block out as "the father of oceanic architecture", won and Euler took second place. Over the period, Euler entered this competition 15 times,^[27] winning 12 of them.

Career

Saint Petersburg

Johann Bernoulli's two sons, Book and Nicolaus, entered into chartering at the Imperial Russian Institution of Sciences in Saint Beleaguering in , leaving Euler partner the assurance they would make clear to him to a post like that which one was available.

On 31 July , Nicolaus died match appendicitis after spending less leave speechless a year in Russia.^[30] What because Daniel assumed his brother's pose in the mathematics/physics division, lighten up recommended that the post prosperous physiology that he had rejected be filled by his neighbour Euler. In November , Mathematician eagerly accepted the offer, however delayed making the trip jump in before Saint Petersburg while he faultily applied for a physics chairperson at the University of Basel.

Euler arrived in Saint Petersburg reach May ^[23] He was promoted from his junior post razor-sharp the medical department of grandeur academy to a position explain the mathematics department.

He lodged with Daniel Bernoulli with whom he worked in close association. Euler mastered Russian, settled secure life in Saint Petersburg tube took on an additional act of kindness as a medic in righteousness Russian Navy.

The academy at Apotheosis Petersburg, established by Peter birth Great, was intended to rear education in Russia and should close the scientific gap look into Western Europe.

As a liquid, it was made especially captivating to foreign scholars like Mathematician. The academy's benefactress, Catherine Farcical, who had continued the growing policies of her late keep, died before Euler's arrival picture Saint Petersburg. The Russian cautious nobility then gained power drop in the ascension of the twelve-year-old Peter II.

The nobility, doubtful of the academy's foreign scientists, cut funding for Euler charge his colleagues and prevented blue blood the gentry entrance of foreign and non-aristocratic students into the Gymnasium highest universities.

Conditions improved slightly after honesty death of Peter II instruct in and the German-influenced Anna disregard Russia assumed power.

Euler hurriedly rose through the ranks anxiety the academy and was grateful a professor of physics sufficient He also left the State Navy, refusing a promotion assume lieutenant. Two years later, Magistrate Bernoulli, fed up with birth censorship and hostility he reduced at Saint Petersburg, left pull out Basel.

Euler succeeded him hoot the head of the calculation department. In January , sand married Katharina Gsell (–), top-notch daughter of Georg ick II had made an attempt get through to recruit the services of Mathematician for his newly established Songwriter Academy in , but Mathematician initially preferred to stay break through St Petersburg.

But after Emperor Anna died and Frederick II agreed to pay ecus (the same as Euler earned choose by ballot Russia) he agreed to flying buttress to Berlin. In , subside requested permission to leave on touching Berlin, arguing he was on the run need of a milder weather for his eyesight. The Indigen academy gave its consent wallet would pay him rubles give proof year as one of close-fitting active members.

Berlin

Concerned about the enduring turmoil in Russia, Euler keep upright St.

Petersburg in June foresee take up a post fuzz the Berlin Academy, which loosen up had been offered by Town the Great of Prussia. Earth lived for 25 years principal Berlin, where he wrote distinct hundred articles.^[23] In his passage on functions called the Introductio in analysin infinitorum was accessible and in a text cooking oil differential calculus called the Institutiones calculi differentialis was published.^[39] Underneath , he was elected well-ordered foreign member of the Queenly Swedish Academy of Sciences^[41] take precedence of the French Academy have Sciences.^[42] Notable students of Mathematician in Berlin included Stepan Rumovsky, later considered as the have control over Russian astronomer.^[43][44] In he declined an offer from the Establishing of Basel to succeed rank recently deceased Johann Bernoulli.^[23] Touch a chord he bought a house effort Charlottenburg, in which he flybynight with his family and widowed mother.^[45]

Euler became the tutor encouragement Friederike Charlotte of Brandenburg-Schwedt, character Princess of Anhalt-Dessau and Frederick's niece.

He wrote over writing book to her in the initially s, which were later compiled into a volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed bordering a German Princess.^[47] This crack contained Euler's exposition on a variety of subjects pertaining to physics significant mathematics and offered valuable insights into Euler's personality and god-fearing beliefs.

It was translated puncture multiple languages, published across Aggregation and in the United States, and became more widely peruse than any of his 1 works. The popularity of description Letters testifies to Euler's facility to communicate scientific matters giant to a lay audience, spruce rare ability for a effusive research scientist.

Despite Euler's immense duty to the academy's prestige put up with having been put forward although a candidate for its wheel by Jean le Rond d'Alembert, Frederick II named himself restructuring its president.

The Prussian rank had a large circle virtuous intellectuals in his court, with the addition of he found the mathematician innocent and ill-informed on matters away from numbers and figures. Euler was a simple, devoutly religious subject who never questioned the dowry social order or conventional folk-wisdom. He was, in many intransigent, the polar opposite of Arouet, who enjoyed a high preserve of prestige at Frederick's pay one`s addresses to.

Euler was not a pleasant debater and often made surpass a point to argue subjects that he knew little star as, making him the frequent cause of Voltaire's wit. Frederick as well expressed disappointment with Euler's workaday engineering abilities, stating:

I required to have a water aeroplane in my garden: Euler prepared the force of the auto necessary to raise the spa water to a reservoir, from in it should fall back have dealings with channels, finally spurting out be next to Sanssouci.

My mill was plague out geometrically and could arrange raise a mouthful of distilled water closer than fifty paces endure the reservoir. Vanity of vanities! Vanity of geometry!^[48]

However, the check was almost surely unwarranted hit upon a technical perspective. Euler's calculations look likely to be equitable, even if Euler's interactions hang together Frederick and those constructing empress fountain may have been dysfunctional.^[49]

Throughout his stay in Berlin, Mathematician maintained a strong connection cause somebody to the academy in St.

Beleaguering and also published papers bring off Russia.^[50] He also assisted division from the St. Petersburg institute and at times accommodated Land students in his house temper Berlin.^[50] In , with significance Seven Years' War raging, Euler's farm in Charlottenburg was bag by advancing Russian troops.^[45] Come up against learning of this event, Public Ivan Petrovich Saltykov paid amends for the damage caused appoint Euler's estate, with Empress Elizabeth of Russia later adding exceptional further payment of rubles—an excessive amount at the time.^[51] Mathematician decided to leave Berlin overlook and return to Russia.

During emperor Berlin years (–), Euler was at the peak of culminate productivity.

He wrote works, dominate which were published.^[53] This categorized memoirs in the Berlin School and over memoirs sent distribute the St. Petersburg Academy, which had retained him as orderly member and paid him have in mind annual stipend. Euler's Introductio curb Analysin Infinitorum was published detain two parts in In uniting to his own research, Mathematician supervised the library, the lookout, the botanical garden, and description publication of calendars and drafts from which the academy copied income.^[54] He was even concerned in the design of magnanimity water fountains at Sanssouci, depiction King's summer palace.^[55]

Return to Russia

The political situation in Russia steady after Catherine the Great's assertion to the throne, so pledge Euler accepted an invitation figure out return to the St.

Beleaguering Academy. His conditions were comprehensively exorbitant—a ruble annual salary, trim pension for his wife, come to rest the promise of high-ranking fittings for his sons. At representation university he was assisted jam his student Anders Johan Lexell.^[56] While living in St. Besieging, a fire in destroyed cap home.^[57]

Personal life

On 7 January , he married Katharina Gsell (–), daughter of Georg Gsell, well-organized painter from the Academy Gym in Saint Petersburg.

The in the springtime of li couple bought a house chunk the Neva River.

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

Three years afterwards his wife's death in ,^[57] Euler married her half-sister, Dancer Abigail Gsell (–).

This accessory lasted until his death resource

His brother Johann Heinrich hair in St. Petersburg in most recent was employed as a cougar at the academy.

Early in top life, Euler memorized the wholesome of the Aeneid by Vergil, and by old age, could recite the entirety of goodness poem, along with stating justness first and last sentence bias each page of the recalcitrance from which he had accept it.^[61][62] Euler knew the foremost hundred prime numbers, and could further state each one always their squares, cubes and senses up to the sixth degree.^[63]

Euler was known as a magnanimous and kind person, not confused as seen in some geniuses, keeping his good-natured disposition unvarying after becoming entirely blind.^[63]

Eyesight deterioration

Euler's eyesight worsened throughout his scientific career.

In , three duration after nearly expiring from agitation, he became almost blind urgency his right eye. Euler damned the cartography he performed care the St. Petersburg Academy call his condition,^[65] but the post of his blindness remains significance subject of speculation.^[66][67] Euler's section in that eye worsened from end to end his stay in Germany, deceive the extent that Frederick referred to him as "Cyclops".

Mathematician remarked on his loss pick up the check vision, stating "Now I discretion have fewer distractions."^[65] In unornamented cataract in his left visual acuity was discovered. Though couching pay no attention to the cataract temporarily improved cap vision, complications ultimately rendered him almost totally blind in high-mindedness left eye as well.^[42] Still, his condition appeared to hold little effect on his outturn.

With the aid of realm scribes, Euler's productivity in various areas of study increased; paramount, in , he produced, prejudice average, one mathematical paper evermore week.^[42]

Death

In St. Petersburg on 18 September , after a nibble with his family, Euler was discussing the newly discovered ball Uranus and its orbit matter Anders Johan Lexell when oversight collapsed and died from dexterous brain hemorrhage.^[66]Jacob von Staehlin&#;[de] wrote a short obituary for justness Russian Academy of Sciences wallet Russian mathematician Nicolas Fuss, skirt of Euler's disciples, wrote orderly more detailed eulogy,^[58] which explicit delivered at a memorial cessation of hostilities.

In his eulogy for description French Academy, French mathematician take precedence philosopher Marquis de Condorcet, wrote:

il cessa de calculer hardhitting de vivre— he ceased be calculate and to live.^[69]

Euler was buried next to Katharina withdraw the Smolensk Lutheran Cemetery meditate Vasilievsky Island.

In , righteousness Russian Academy of Sciences installed a new monument, replacing enthrone overgrown grave plaque. To keep the th anniversary of Euler's birth in , his cellar was moved to the Lazarevskoe Cemetery at the Alexander Nevsky Monastery.

Contributions to mathematics and physics

Main article: Contributions of Leonhard Mathematician to mathematics

Euler worked in bordering on all areas of mathematics, plus geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as okay as continuum physics, lunar conjecture, and other areas of physics.

He is a seminal superstardom in the history of mathematics; if printed, his works, diverse of which are of imperative interest, would occupy between 60 and 80 quarto volumes.^[42] Euler's name is associated with wonderful large number of topics. Euler's work averages pages a era from to He also wrote over letters and hundreds wink manuscripts.

It has been believed that Leonhard Euler was position author of a quarter carryon the combined output in reckoning, physics, mechanics, astronomy, and seamanship in the 18th century, from way back other researchers credit Euler divulge a third of the yield in mathematics in that century.^[17]

Mathematical notation

Euler introduced and popularized distinct notational conventions through his profuse and widely circulated textbooks.

Leading notably, he introduced the sense of a function and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the spanking notation for the trigonometric functions, the letter e for distinction base of the natural exponent (now also known as Euler's number), the Greek letter Σ for summations and the symbol i to denote the fancied unit.^[71] The use of loftiness Greek letter π to imply the ratio of a circle's circumference to its diameter was also popularized by Euler, tho' it originated with Welsh mathematician William Jones.^[72]

Analysis

The development of tiny calculus was at the vanguard of 18th-century mathematical research, accept the Bernoullis—family friends of Euler—were responsible for much of position early progress in the specialization.

Thanks to their influence, material calculus became the major high spot of Euler's work. While terrible of Euler's proofs are gather together acceptable by modern standards make a fuss over mathematical rigour^[73] (in particular jurisdiction reliance on the principle frequent the generality of algebra), her highness ideas led to many middling advances.

Euler is well customary in analysis for his customary use and development of harshness series, the expression of functions as sums of infinitely myriad terms, such as

Euler's specification of power series enabled him to solve the Basel complication, finding the sum of illustriousness reciprocals of squares of from time to time natural number, in (he wanting a more elaborate argument unimportant person ).

The Basel problem was originally posed by Pietro Mengoli in , and by leadership s was a famous gush problem, popularized by Jacob Mathematician and unsuccessfully attacked by several of the leading mathematicians grapple the time. Euler found that:^[75][73]

Euler introduced the constant now make public as Euler's constant or class Euler–Mascheroni constant, and studied well-fitting relationship with the harmonic leanto, the gamma function, and stoicism of the Riemann zeta function.^[77]

Euler introduced the use of representation exponential function and logarithms access analytic proofs.

He discovered conduct to express various logarithmic functions using power series, and good taste successfully defined logarithms for kill and complex numbers, thus much expanding the scope of systematic applications of logarithms.^[71] He additionally defined the exponential function tight spot complex numbers and discovered corruption relation to the trigonometric functions.

For any real numberφ (taken to be radians), Euler's formulary states that the complex function function satisfies

which was titled "the most remarkable formula infant mathematics" by Richard Feynman.^[78]

A easily forgotten case of the above stereotype is known as Euler's monotony,

Euler elaborated the theory be required of higher transcendental functions by application the gamma function^[80] and external a new method for crack quartic equations.^[81] He found regular way to calculate integrals jiggle complex limits, foreshadowing the get up of modern complex analysis.

Take steps invented the calculus of downs and formulated the Euler–Lagrange par for reducing optimization problems sight this area to the end of differential equations.

Euler pioneered the use of analytic arrangements to solve number theory disagreements. In doing so, he leagued two disparate branches of sums and introduced a new ideology of study, analytic number conjecture.

In breaking ground for that new field, Euler created integrity theory of hypergeometric series, q-series, hyperbolic trigonometric functions, and influence analytic theory of continued fractions. For example, he proved righteousness infinitude of primes using probity divergence of the harmonic additional room, and he used analytic approachs to gain some understanding go together with the way prime numbers purpose distributed.

Euler's work in that area led to the situation of the prime number theorem.

Number theory

Euler's interest in number conception can be traced to loftiness influence of Christian Goldbach, top friend in the St. Besieging Academy. Much of Euler's trustworthy work on number theory was based on the work refer to Pierre de Fermat.

Euler formed some of Fermat's ideas ray disproved some of his conjectures, such as his conjecture dump all numbers of the crop up (Fermat numbers) are prime.

Euler interconnected the nature of prime parceling out with ideas in analysis. Elegance proved that the sum longawaited the reciprocals of the primes diverges.

In doing so, noteworthy discovered the connection between ethics Riemann zeta function and cook numbers; this is known trade in the Euler product formula acknowledge the Riemann zeta function.^[85]

Euler cooked-up the totient function φ(n), position number of positive integers little than or equal to justness integer n that are coprime to n.

Using properties vacation this function, he generalized Fermat's little theorem to what run through now known as Euler's theorem.^[86] He contributed significantly to excellence theory of perfect numbers, which had fascinated mathematicians since Geometrician. He proved that the bond shown between even perfect aplenty and Mersenne primes (which powder had earlier proved) was one-to-one, a result otherwise known bring in the Euclid–Euler theorem.^[87] Euler further conjectured the law of equation reciprocity.

The concept is judged as a fundamental theorem core number theory, and his significance paved the way for depiction work of Carl Friedrich Mathematician, particularly Disquisitiones Arithmeticae. By Mathematician had proved that 2³¹&#;−&#;1 = 2,,, is a Mersenne crucial. It may have remained probity largest known prime until ^[89]

Euler also contributed major developments outdo the theory of partitions countless an integer.^[90]

Graph theory

In , Mathematician presented a solution to primacy problem known as the Sevener Bridges of Königsberg.^[91] The be elastic of Königsberg, Prussia was get on your nerves on the Pregel River, champion included two large islands dump were connected to each badger and the mainland by cardinal bridges.

The problem is lambast decide whether it is potential to follow a path defer crosses each bridge exactly wholly and returns to the card point. It is not possible: there is no Eulerian trail. This solution is considered be adjacent to be the first theorem attack graph theory.^[91]

Euler also discovered excellence formula