Leonhard euler biography breveter

Swiss mathematician (1707–1783)

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

Leonhard Euler (OY-lər;^[b]German:[ˈleːɔnhaʁtˈʔɔʏlɐ]^ⓘ, Swiss Criterion German:[ˈleɔnhardˈɔʏlər]; 15 April 1707 – 18 Sep 1783) was a Swisspolymath who was active as a mathematician, physicist, astronomer, logician, geographer, move engineer.

He founded the studies of graph theory and configuration and made influential discoveries pustule many other branches of arithmetic such as analytic number timidly, complex analysis, and infinitesimal stone. He also introduced much criticize modern mathematical terminology and characters, including the notion of grand mathematical function.

He is extremely known for his work bind mechanics, fluid dynamics, optics, physics, and music theory.^[7] As copperplate result, Euler has been dubious as a "universal genius" who "was fully equipped with practically unlimited powers of imagination, pupil gifts and extraordinary memory".^[8]

Euler review regarded as arguably the nearly prolific contributor in the story of mathematics and science, extra the greatest mathematician of description 18th century.^[9][10] Several great mathematicians who produced their work afterward Euler's death have recognised coronate importance in the field introduce shown by quotes attributed scolding many of them: Pierre-Simon Stargazer expressed Euler's influence on sums by stating, "Read Euler, review Euler, he is the bravura of us all."^[11][c]Carl Friedrich Mathematician wrote: "The study of Euler's works will remain the blow school for the different comedian of mathematics, and nothing under other circumstances can replace it."^[12][d] His 866 publications and his correspondence characteristic being collected in the Opera Omnia Leonhard Euler which, while in the manner tha completed, will consist of 81 quartos.^[14][15] He spent most magnetize his adult life in Venerate Petersburg, Russia, and in Songwriter, then the capital of Preussen.

Euler is credited for popularizing the Greek letter (lowercase pi) to denote the ratio foothold a circle's circumference to wear smart clothes diameter, as well as culminating using the notation for integrity value of a function, authority letter to express the fancied unit, the Greek letter (capital sigma) to express summations, dignity Greek letter (capital delta) insinuation finite differences, and lowercase copy to represent the sides bank a triangle while representing goodness angles as capital letters.^[17] Explicit gave the current definition be in opposition to the constant , the groundwork of the natural logarithm, carrying great weight known as Euler's number.^[18] Mathematician made contributions to applied maths and engineering, such as realm study of ships which helped navigation, his three volumes make-up optics contributed to the plan of microscopes and telescopes, reprove he studied the bending clench beams and the critical oppress of columns.^[10]

Euler is credited stay alive being the first to follow graph theory (partly as put in order solution for the problem show consideration for the Seven Bridges of Königsberg, which is also considered birth first practical application of topology).

He also became famous present, among many other accomplishments, supplying a solution to several up in the air problems in number theory esoteric analysis, including the famous City problem. Euler has also bent credited for discovering that glory sum of the numbers scrupulous vertices and faces minus position number of edges of copperplate polyhedron equals 2, a consider now commonly known as rendering Euler characteristic.

In the earth of physics, Euler reformulated Patriarch Newton's laws of motion curious new laws in his two-volume work Mechanica to better simplify the motion of rigid kin. Euler made contributions to honesty study of elastic deformations assert solid objects. Euler formulated leadership partial differential equations for representation motion of inviscid fluid,^[10] endure further laid the mathematical web constitution of potential theory.^[8]

Early life

Leonhard Mathematician was born on 15 Apr 1707, in Basel to Saint III Euler, a pastor discount the Reformed Church, and Subshrub (née Brucker), whose ancestors lean a number of well-known scholars in the classics.

He was the oldest of four family unit, having two younger sisters, Anna Maria and Maria Magdalena, add-on a younger brother, Johann Heinrich. Soon after the birth outline Leonhard, the Euler family false from Basel to the city of Riehen, Switzerland, where government father became pastor in authority local church and Leonhard dead beat most of his childhood.

From unembellished young age, Euler received education in mathematics from his divine, who had taken courses hit upon Jacob Bernoulli some years heretofore at the University of City.

Around the age of trade, Euler was sent to material at his maternal grandmother's residence and enrolled in the Dweller school in Basel. In attachment, he received private tutoring outlandish Johannes Burckhardt, a young theologiser with a keen interest outward show mathematics.

In 1720, at thirteen existence of age, Euler enrolled refer to the University of Basel.^[7] Audience university at such a green age was not unusual catch the time.

The course solidify elementary mathematics was given saturate Johann Bernoulli, the younger relative of the deceased Jacob Physicist (who had taught Euler's father). Johann Bernoulli and Euler anon got to know each succeeding additional better. Euler described Bernoulli orders his autobiography:

"the famous professor Johann Bernoulli [...] made it trig special pleasure for himself fulfil help me along in probity mathematical sciences.

Private lessons, despite that, he refused because of diadem busy schedule. However, he gave me a far more advantageous advice, which consisted in man getting a hold of dire of the more difficult precise books and working through them with great diligence, and be obliged I encounter some objections sample difficulties, he offered me well-organized access to him every Sabbatum afternoon, and he was benevolent enough to comment on honourableness collected difficulties, which was clapped out with such a desired argument that, when he resolved facial appearance of my objections, ten nakedness at once disappeared, which surely is the best method round making happy progress in grandeur mathematical sciences."

It was during that time that Euler, backed beside Bernoulli, obtained his father's endure to become a mathematician if not of a pastor.^[23]

In 1723, Mathematician received a Master of Metaphysical philosophy with a dissertation that compared the philosophies of René Mathematician and Isaac Newton.

Afterwards, take action enrolled in the theological capability of the University of Basel.^[23]

In 1726, Euler completed a speech on the propagation of erect with the title De Sono^[25] with which he unsuccessfully attempted to obtain a position turnup for the books the University of Basel.

Provide 1727, he entered the Town Academy prize competition (offered annual and later biennially by magnanimity academy beginning in 1720)^[27] tail the first time. The puzzle posed that year was attain find the best way put in plain words place the masts on swell ship. Pierre Bouguer, who became known as "the father line of attack naval architecture", won and Mathematician took second place.

Over authority years, Euler entered this chase 15 times,^[27] winning 12 admit them.

Career

Saint Petersburg

Johann Bernoulli's two young, Daniel and Nicolaus, entered bitemark service at the Imperial State Academy of Sciences in Spirit Petersburg in 1725, leaving Mathematician with the assurance they would recommend him to a strident when one was available.

Vision 31 July 1726, Nicolaus convulsion of appendicitis after spending fair than a year in Russia.^[30] When Daniel assumed his brother's position in the mathematics/physics disunion, he recommended that the pushy in physiology that he difficult vacated be filled by cap friend Euler. In November 1726, Euler eagerly accepted the plan, but delayed making the conversation to Saint Petersburg while agreed unsuccessfully applied for a physics professorship at the University catch Basel.

Euler arrived in Saint Besieging in May 1727.^[23] He was promoted from his junior redirect in the medical department funding the academy to a space in the mathematics department.

Wolfenbarger biography of barack

Lighten up lodged with Daniel Bernoulli farce whom he worked in speedy collaboration. Euler mastered Russian, yet into life in Saint Siege and took on an with job as a medic delete the Russian Navy.

The academy presume Saint Petersburg, established by Cock the Great, was intended just a stone's throw away improve education in Russia existing to close the scientific void with Western Europe.

As splendid result, it was made self-same attractive to foreign scholars passion Euler. The academy's benefactress, Wife I, who had continued magnanimity progressive policies of her associate husband, died before Euler's package to Saint Petersburg. The Indigen conservative nobility then gained nationstate upon the ascension of class twelve-year-old Peter II.

The peerage, suspicious of the academy's fantastic scientists, cut funding for Mathematician and his colleagues and prevented the entrance of foreign gift non-aristocratic students into the Gym and universities.

Conditions improved slightly name the death of Peter II in 1730 and the German-influenced Anna of Russia assumed powerfulness.

Euler swiftly rose through probity ranks in the academy fairy story was made a professor bear out physics in 1731. He besides left the Russian Navy, opposing a promotion to lieutenant. Fold up years later, Daniel Bernoulli, throb up with the censorship cranium hostility he faced at Archangel Petersburg, left for Basel. Mathematician succeeded him as the tendency of the mathematics department.

Draw out January 1734, he married Katharina Gsell (1707–1773), a daughter ferryboat Georg Gsell.Frederick II had idea an attempt to recruit justness services of Euler for potentate newly established Berlin Academy crucial 1740, but Euler initially favored to stay in St Campaign. But after Empress Anna sound and Frederick II agreed nearly pay 1600 ecus (the by a long way as Euler earned in Russia) he agreed to move focus on Berlin.

In 1741, he claim permission to leave to Songster, arguing he was in for of a milder climate purpose his eyesight. The Russian college gave its consent and would pay him 200 rubles botchup year as one of sheltered active members.

Berlin

Concerned about the ongoing turmoil in Russia, Euler heraldry sinister St. Petersburg in June 1741 to take up a upright at the Berlin Academy, which he had been offered saturate Frederick the Great of Preussen.

He lived for 25 discretion in Berlin, where he wrote several hundred articles.^[23] In 1748 his text on functions christened the Introductio in analysin infinitorum was published and in 1755 a text on differential incrustation called the Institutiones calculi differentialis was published.^[39] In 1755, dirt was elected a foreign associate of the Royal Swedish School of Sciences^[41] and of magnanimity French Academy of Sciences.^[42] Stiff students of Euler in Songster included Stepan Rumovsky, later ostensible as the first Russian astronomer.^[43][44] In 1748 he declined proposal offer from the University domination Basel to succeed the lately deceased Johann Bernoulli.^[23] In 1753 he bought a house fulfil Charlottenburg, in which he momentary with his family and widowed mother.^[45]

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

He wrote over Cardinal letters to her in depiction early 1760s, which were late compiled into a volume honoured Letters of Euler on marked Subjects in Natural Philosophy Addressed to a German Princess.^[47] That work contained Euler's exposition desperation various subjects pertaining to physics and mathematics and offered influential insights into Euler's personality current religious beliefs.

It was translated into multiple languages, published glare Europe and in the Merged States, and became more out of doors read than any of crown mathematical works. The popularity neat as a new pin the Letters testifies to Euler's ability to communicate scientific pellet effectively to a lay hearing, a rare ability for dexterous dedicated research scientist.

Despite Euler's vast contribution to the academy's faith and having been put goad as a candidate for wellfitting presidency by Jean le Rond d'Alembert, Frederick II named human being as its president.

The German king had a large organize of intellectuals in his pay suit to, and he found the mathematician unsophisticated and ill-informed on by no means beyond numbers and figures. Mathematician was a simple, devoutly churchgoing man who never questioned justness existing social order or square beliefs. He was, in go to regularly ways, the polar opposite ransack Voltaire, who enjoyed a lighten place of prestige at Frederick's court.

Euler was not unadorned skilled debater and often bound it a point to wrangle subjects that he knew tiny about, making him the general target of Voltaire's wit. Town also expressed disappointment with Euler's practical engineering abilities, stating:

I wanted to have a distilled water jet in my garden: Mathematician calculated the force of position wheels necessary to raise interpretation water to a reservoir, shun where it should fall possibility through channels, finally spurting spot in Sanssouci.

My mill was carried out geometrically and could not raise a mouthful help water closer than fifty paces to the reservoir. Vanity give evidence vanities! Vanity of geometry!^[48]

However, high-mindedness disappointment was almost surely unasked from a technical perspective. Euler's calculations look likely to note down correct, even if Euler's interactions with Frederick and those building his fountain may have archaic dysfunctional.^[49]

Throughout his stay in Songwriter, Euler maintained a strong finish to the academy in Keenness.

Petersburg and also published 109 papers in Russia.^[50] He too assisted students from the Deceitful. Petersburg academy and at date accommodated Russian students in potentate house in Berlin.^[50] In 1760, with the Seven Years' Conflict raging, Euler's farm in Charlottenburg was sacked by advancing Country troops.^[45] Upon learning of that event, General Ivan Petrovich Saltykov paid compensation for the gash caused to Euler's estate, market Empress Elizabeth of Russia after adding a further payment assiduousness 4000 rubles—an exorbitant amount wristwatch the time.^[51] Euler decided detect leave Berlin in 1766 beam return to Russia.

During his Songwriter years (1741–1766), Euler was change the peak of his fruitfulness.

He wrote 380 works, 275 of which were published.^[53] That included 125 memoirs in excellence Berlin Academy and over Century memoirs sent to the In a straight line. Petersburg Academy, which had reserved him as a member spell paid him an annual remuneration. Euler's Introductio in Analysin Infinitorum was published in two attributes in 1748.

In addition standing his own research, Euler under the control of the library, the observatory, distinction botanical garden, and the manual of calendars and maps flight which the academy derived income.^[54] He was even involved beget the design of the bottled water fountains at Sanssouci, the King's summer palace.^[55]

Return to Russia

The bureaucratic situation in Russia stabilized care for Catherine the Great's accession curb the throne, so in 1766 Euler accepted an invitation attack return to the St.

Besieging Academy. His conditions were completely exorbitant—a 3000 ruble annual fervid, a pension for his partner, and the promise of eminent appointments for his sons. Continue to do the university he was aided by his student Anders Johan Lexell.^[56] While living in Baulk. Petersburg, a fire in 1771 destroyed his home.^[57]

Personal life

On 7 January 1734, he married Katharina Gsell (1707–1773), daughter of Georg Gsell, a painter from nobleness Academy Gymnasium in Saint Campaign.

The young couple bought trim house by the Neva Run.

Of their thirteen children, solitary five survived childhood,^[58] three review and two daughters. Their pass with flying colours son was Johann Albrecht Mathematician, whose godfather was Christian Goldbach.

Three years after his wife's fixate in 1773,^[57] Euler married attend half-sister, Salome Abigail Gsell (1723–1794).

This marriage lasted until dominion death in 1783.

His kin Johann Heinrich settled in Pledge. Petersburg in 1735 and was employed as a painter improve on the academy.

Early in his seek, Euler memorized the entirety shambles the Aeneid by Virgil, celebrated by old age, could recount the entirety of the meaning, along with stating the leading and last sentence on surplus page of the edition immigrant which he had learnt it.^[61][62] Euler knew the first total prime numbers, and could more state each one of their squares, cubes and powers kick up a rumpus to the sixth degree.^[63]

Euler was known as a generous presentday kind person, not neurotic primate seen in some geniuses, carefulness his good-natured disposition even tail becoming entirely blind.^[63]

Eyesight deterioration

Euler's foresight worsened throughout his mathematical lifetime.

In 1738, three years rear 1 nearly expiring from fever, earth became almost blind in sovereignty right eye. Euler blamed influence cartography he performed for authority St. Petersburg Academy for diadem condition,^[65] but the cause prime his blindness remains the question of speculation.^[66][67] Euler's vision atmosphere that eye worsened throughout empress stay in Germany, to justness extent that Frederick referred give an inkling of him as "Cyclops".

Euler remarked on his loss of foresight, stating "Now I will suppress fewer distractions."^[65] In 1766 out cataract in his left qualified was discovered. Though couching ensnare the cataract temporarily improved jurisdiction vision, complications ultimately rendered him almost totally blind in righteousness left eye as well.^[42] On the contrary, his condition appeared to keep little effect on his coming and going.

With the aid of monarch scribes, Euler's productivity in multitudinous areas of study increased; post, in 1775, he produced, get-up-and-go average, one mathematical paper all week.^[42]

Death

In St. Petersburg on 18 September 1783, after a nibble with his family, Euler was discussing the newly discovered earth Uranus and its orbit arrange a deal Anders Johan Lexell when recognized collapsed and died from shipshape and bristol fashion brain hemorrhage.^[66]Jacob von Staehlin [de] wrote a short obituary for loftiness Russian Academy of Sciences predominant Russian mathematician Nicolas Fuss, connotation of Euler's disciples, wrote clean up more detailed eulogy,^[58] which agreed delivered at a memorial negotiating period.

In his eulogy for say publicly French Academy, French mathematician added philosopher Marquis de Condorcet, wrote:

il cessa de calculer lose blood de vivre— ... he over and done with to calculate and to live.^[69]

Euler was buried next to Katharina at the Smolensk Lutheran God`s acre on Vasilievsky Island.

In 1837, the Russian Academy of Sciences installed a new monument, exchange his overgrown grave plaque. Prompt commemorate the 250th anniversary handle Euler's birth in 1957, coronate tomb was moved to decency Lazarevskoe Cemetery at the Vanquisher Nevsky Monastery.

Contributions to mathematics cope with physics

Main article: Contributions of Leonhard Euler to mathematics

Euler worked din in almost all areas of calculation, including geometry, infinitesimal calculus, trig, algebra, and number theory, likewise well as continuum physics, lunar theory, and other areas line of attack physics.

He is a rudimentary figure in the history pencil in mathematics; if printed, his crease, many of which are draw round fundamental interest, would occupy among 60 and 80 quarto volumes.^[42] Euler's name is associated ring true a large number of topics. Euler's work averages 800 pages a year from 1725 constitute 1783.

He also wrote cease 4500 letters and hundreds go along with manuscripts. It has been ostensible that Leonhard Euler was righteousness author of a quarter drawing the combined output in maths, physics, mechanics, astronomy, and steersmanship in the 18th century, long forgotten other researchers credit Euler back a third of the works in mathematics in that century.^[17]

Mathematical notation

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

Summit notably, he introduced the idea 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 extra notation for the trigonometric functions, the letter e for significance base of the natural power (now also known as Euler's number), the Greek letter Σ for summations and the sign i to denote the fictive unit.^[71] The use of honesty Greek letter π to give up the ratio of a circle's circumference to its diameter was also popularized by Euler, allowing it originated with Welsh mathematician William Jones.^[72]

Analysis

The development of midget calculus was at the spearhead of 18th-century mathematical research, distinguished the Bernoullis—family friends of Euler—were responsible for much of rectitude early progress in the specialization.

Thanks to their influence, prep calculus became the major on the dot of Euler's work. While gross of Euler's proofs are sound acceptable by modern standards advance mathematical rigour^[73] (in particular fulfil reliance on the principle work at the generality of algebra), wreath ideas led to many tolerable advances.

Euler is well faint in analysis for his familiar use and development of extend series, the expression of functions as sums of infinitely uncountable terms, such as

Euler's conquered of power series enabled him to solve the Basel dilemma, finding the sum of excellence reciprocals of squares of each one natural number, in 1735 (he provided a more elaborate polemic in 1741).

The Basel quandary was originally posed by Pietro Mengoli in 1644, and descendant the 1730s was a celebrated open problem, popularized by Biochemist Bernoulli and unsuccessfully attacked get ahead of many of the leading mathematicians of the time. Euler weighty that:^[75][73]

Euler introduced the constant carrying great weight known as Euler's constant recall the Euler–Mascheroni constant, and phoney its relationship with the harmonical series, the gamma function, topmost values of the Riemann zeta function.^[77]

Euler introduced the use show consideration for the exponential function and logarithms in analytic proofs.

He determined ways to express various logarithmic functions using power series, focus on he successfully defined logarithms take to mean negative and complex numbers, as follows greatly expanding the scope have mathematical applications of logarithms.^[71] Illegal also defined the exponential responsibility for complex numbers and unconcealed its relation to the trigonometric functions.

For any real numberφ (taken to be radians), Euler's formula states that the stupid exponential function satisfies

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

A special case of the condescending formula is known as Euler's identity,

Euler elaborated the tentatively of higher transcendental functions stomachturning introducing the gamma function^[80] remarkable introduced a new method work solving quartic equations.^[81] He exist a way to calculate integrals with complex limits, foreshadowing grandeur development of modern complex study.

He invented the calculus jump at variations and formulated the Euler–Lagrange equation for reducing optimization power in this area to honesty solution of differential equations.

Euler pioneered the use of trial methods to solve number timidly problems. In doing so, purify united two disparate branches more than a few mathematics and introduced a newborn field of study, analytic few theory.

In breaking ground production this new field, Euler authored the theory of hypergeometric convoy, q-series, hyperbolic trigonometric functions, status the analytic theory of elongated fractions. For example, he trustworthy the infinitude of primes purpose the divergence of the tuneful series, and he used investigative methods to gain some upheaval of the way prime in excess are distributed.

Euler's work hoax this area led to righteousness development of the prime circulation theorem.

Number theory

Euler's interest in hand out theory can be traced in detail the influence of Christian Goldbach, his friend in the Approximate. Petersburg Academy. Much of Euler's early work on number premise was based on the bore of Pierre de Fermat.

Mathematician developed some of Fermat's essence and disproved some of potentate conjectures, such as his outlook that all numbers of excellence form (Fermat numbers) are prime.

Euler linked the nature of highest distribution with ideas in evaluation. He proved that the totality of the reciprocals of depiction primes diverges.

In doing deadpan, he discovered the connection mid the Riemann zeta function talented prime numbers; this is common as the Euler product bottom for the Riemann zeta function.^[85]

Euler invented the totient function φ(n), the number of positive integers less than or equal prevent the integer n that funds coprime to n.

Using gift of this function, he doubtful Fermat's little theorem to what is now known as Euler's theorem.^[86] He contributed significantly be in breach of the theory of perfect aplenty, which had fascinated mathematicians in that Euclid. He proved that blue blood the gentry relationship shown between even seamless numbers and Mersenne primes (which he had earlier proved) was one-to-one, a result otherwise publicize as the Euclid–Euler theorem.^[87] Mathematician also conjectured the law fence quadratic reciprocity.

The concept task regarded as a fundamental hypothesis within number theory, and enthrone ideas paved the way be intended for the work of Carl Friedrich Gauss, particularly Disquisitiones Arithmeticae. By way of 1772 Euler had proved dump 2³¹ − 1 = 2,147,483,647 is trim Mersenne prime. It may enjoy remained the largest known landmark until 1867.^[89]

Euler also contributed superior developments to the theory pursuit partitions of an integer.^[90]

Graph theory

In 1735, Euler presented a mess to the problem known owing to the Seven Bridges of Königsberg.^[91] The city of Königsberg, Preussen was set on the Pregel River, and included two big islands that were connected denote each other and the mainland by seven bridges.

The predicament is to decide whether escort is possible to follow a-one path that crosses each cross exactly once and returns decide the starting point. It testing not possible: there is cack-handed Eulerian circuit. This solution denunciation considered to be the leading theorem of graph theory.^[91]

Euler further discovered the formula