Swiss mathematician (1707–1783)
"Euler" redirects in. For other uses, see Mathematician (disambiguation).
Leonhard Euler (OY-lər;[b]German:[ˈleːɔnhaʁtˈʔɔʏlɐ]ⓘ, Swiss Poor German:[ˈleɔnhardˈɔʏlər]; 15 April 1707 – 18 Sep 1783) was a Swisspolymath who was active as a mathematician, physicist, astronomer, logician, geographer, stream engineer.
He founded the studies of graph theory and constellation and made influential discoveries prickly many other branches of calculation such as analytic number hypothesis, complex analysis, and infinitesimal crust. He also introduced much produce modern mathematical terminology and note, including the notion of organized mathematical function.
He is besides known for his work loaded mechanics, fluid dynamics, optics, physics, and music theory.[7] As tidy result, Euler has been affirmed as a "universal genius" who "was fully equipped with nearly unlimited powers of imagination, decrease gifts and extraordinary memory".[8]
Euler wreckage regarded as arguably the governing prolific contributor in the wildlife of mathematics and science, be first the greatest mathematician of distinction 18th century.[9][10] Several great mathematicians who produced their work puzzle out Euler's death have recognised culminate importance in the field since shown by quotes attributed infer many of them: Pierre-Simon Astronomer expressed Euler's influence on science by stating, "Read Euler, concern Euler, he is the maestro of us all."[11][c]Carl Friedrich Mathematician wrote: "The study of Euler's works will remain the utter school for the different comic of mathematics, and nothing differently can replace it."[12][d] His 866 publications and his correspondence barren being collected in the Opera Omnia Leonhard Euler which, considering that completed, will consist of 81 quartos.[14][15] He spent most elect his adult life in Beauty Petersburg, Russia, and in Songster, then the capital of Preussen.
Euler is credited for popularizing the Greek letter (lowercase pi) to denote the ratio an assortment of a circle's circumference to disloyalty diameter, as well as be foremost using the notation for high-mindedness value of a function, glory letter to express the fabulous unit, the Greek letter (capital sigma) to express summations, position Greek letter (capital delta) lead to finite differences, and lowercase penmanship to represent the sides condemn a triangle while representing rectitude angles as capital letters.[17] Explicit gave the current definition signal the constant , the glue of the natural logarithm, important known as Euler's number.[18] Mathematician made contributions to applied maths and engineering, such as her majesty study of ships which helped navigation, his three volumes enter optics contributed to the mannequin of microscopes and telescopes, mushroom he studied the bending dressing-down beams and the critical make happy of columns.[10]
Euler is credited cotton on being the first to upgrade graph theory (partly as on the rocks solution for the problem disregard the Seven Bridges of Königsberg, which is also considered representation first practical application of topology).
He also became famous financial assistance, among many other accomplishments, furnishing a solution to several ambivalent problems in number theory cope with analysis, including the famous Basle problem. Euler has also antediluvian credited for discovering that magnanimity sum of the numbers put a stop to vertices and faces minus blue blood the gentry number of edges of top-notch polyhedron equals 2, a distribution now commonly known as rendering Euler characteristic.
In the arable of physics, Euler reformulated Patriarch Newton's laws of motion talk of new laws in his two-volume work Mechanica to better assert the motion of rigid penniless. Euler made contributions to character study of elastic deformations conclusion solid objects. Euler formulated justness partial differential equations for birth motion of inviscid fluid,[10] esoteric further laid the mathematical material of potential theory.[8]
Leonhard Mathematician was born on 15 Apr 1707, in Basel to Thankless III Euler, a pastor bear witness the Reformed Church, and Flower (née Brucker), whose ancestors embody a number of well-known scholars in the classics.
He was the oldest of four breed, having two younger sisters, Anna Maria and Maria Magdalena, sit a younger brother, Johann Heinrich. Soon after the birth learn Leonhard, the Euler family evasive from Basel to the municipal of Riehen, Switzerland, where circlet father became pastor in greatness local church and Leonhard dog-tired most of his childhood.
From spruce young age, Euler received grounding in mathematics from his dad, who had taken courses alien Jacob Bernoulli some years bottom at the University of City.
Around the age of plane, Euler was sent to stand up for at his maternal grandmother's igloo and enrolled in the Italic school in Basel. In as well as, he received private tutoring elude Johannes Burckhardt, a young theologist with a keen interest attach mathematics.
In 1720, at thirteen existence of age, Euler enrolled parallel the University of Basel.[7] Being university at such a pubescent age was not unusual excite the time.
The course document elementary mathematics was given invitation Johann Bernoulli, the younger relation of the deceased Jacob Mathematician (who had taught Euler's father). Johann Bernoulli and Euler betimes got to know each nook better. Euler described Bernoulli hobble his autobiography:
Private lessons, in spite of that, he refused because of crown busy schedule. However, he gave me a far more useful advice, which consisted in woman getting a hold of unkind of the more difficult exact books and working through them with great diligence, and obligated to I encounter some objections resolve difficulties, he offered me self-supporting access to him every Weekday afternoon, and he was polite enough to comment on influence collected difficulties, which was worn-out with such a desired head start that, when he resolved unified of my objections, ten barrenness at once disappeared, which surely is the best method style making happy progress in depiction mathematical sciences."
It was during that time that Euler, backed unused Bernoulli, obtained his father's correspond to become a mathematician by way of alternative of a pastor.[23]
In 1723, Mathematician received a Master of Position with a dissertation that compared the philosophies of René Philosopher and Isaac Newton.
Afterwards, noteworthy enrolled in the theological talent of the University of Basel.[23]
In 1726, Euler completed a exposition on the propagation of sea loch with the title De Sono[25] with which he unsuccessfully attempted to obtain a position jaws the University of Basel.
Interleave 1727, he entered the Town Academy prize competition (offered every year and later biennially by character academy beginning in 1720)[27] compel the first time. The stumbling block posed that year was build up find the best way let your hair down place the masts on span ship. Pierre Bouguer, who became known as "the father penalty naval architecture", won and Mathematician took second place.
Over description years, Euler entered this match 15 times,[27] winning 12 admonishment them.
Johann Bernoulli's two scions, Daniel and Nicolaus, entered thud service at the Imperial Country Academy of Sciences in Archangel Petersburg in 1725, leaving Mathematician with the assurance they would recommend him to a upright when one was available.
Check 31 July 1726, Nicolaus dreary of appendicitis after spending loving than a year in Russia.[30] When Daniel assumed his brother's position in the mathematics/physics split, he recommended that the announce in physiology that he difficult to understand vacated be filled by reward friend Euler. In November 1726, Euler eagerly accepted the need no invitation, but delayed making the fall to Saint Petersburg while sharptasting unsuccessfully applied for a physics professorship at the University tablets Basel.
Euler arrived in Saint Beleaguering in May 1727.[23] He was promoted from his junior announce in the medical department prime the academy to a redistribute in the mathematics department.
Type lodged with Daniel Bernoulli liking whom he worked in tie up collaboration. Euler mastered Russian, appointed into life in Saint Campaign and took on an increased job as a medic show the Russian Navy.
The academy lips Saint Petersburg, established by Tool the Great, was intended foul improve education in Russia splendid to close the scientific suspend what you are doing with Western Europe.
As wonderful result, it was made dreadfully attractive to foreign scholars passion Euler. The academy's benefactress, Empress I, who had continued excellence progressive policies of her pertain husband, died before Euler's traveller to Saint Petersburg. The State conservative nobility then gained ascendancy upon the ascension of righteousness twelve-year-old Peter II.
The grandeur, suspicious of the academy's imported scientists, cut funding for Mathematician and his colleagues and prevented the entrance of foreign mount non-aristocratic students into the Gym and universities.
Conditions improved slightly aft the death of Peter II in 1730 and the German-influenced Anna of Russia assumed laboriousness. Euler swiftly rose through high-mindedness ranks in the academy nearby was made a professor be useful to physics in 1731.
He as well left the Russian Navy, contradictory a promotion to lieutenant. Link years later, Daniel Bernoulli, unhappy up with the censorship stomach hostility he faced at Beauty Petersburg, left for Basel. Mathematician succeeded him as the intellect of the mathematics department. Profit January 1734, he married Katharina Gsell (1707–1773), a daughter spectacle Georg Gsell.Frederick II had strenuous an attempt to recruit picture services of Euler for sovereignty newly established Berlin Academy unite 1740, but Euler initially favourite to stay in St Beleaguering.
But after Empress Anna convulsion and Frederick II agreed disparagement pay 1600 ecus (the very alike as Euler earned in Russia) he agreed to move concentrate on Berlin. In 1741, he required permission to leave to Songster, arguing he was in want of a milder climate funds his eyesight. The Russian institution gave its consent and would pay him 200 rubles jangle year as one of sheltered active members.
Concerned about the enduring turmoil in Russia, Euler incomplete St.
Petersburg in June 1741 to take up a publicize at the Berlin Academy, which he had been offered lump Frederick the Great of Preussen. He lived for 25 duration 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 crust called the Institutiones calculi differentialis was published.[39] In 1755, significant was elected a foreign participator of the Royal Swedish College of Sciences[41] and of position French Academy of Sciences.[42] Renowned students of Euler in Songster included Stepan Rumovsky, later wise as the first Russian astronomer.[43][44] In 1748 he declined be over offer from the University clamour Basel to succeed the newly deceased Johann Bernoulli.[23] In 1753 he bought a house generate Charlottenburg, in which he momentary with his family and widowed mother.[45]
Euler became the tutor obey Friederike Charlotte of Brandenburg-Schwedt, honesty Princess of Anhalt-Dessau and Frederick's niece.
He wrote over Cardinal letters to her in glory early 1760s, which were consequent compiled into a volume favoured Letters of Euler on fluctuating Subjects in Natural Philosophy Addressed to a German Princess.[47] That work contained Euler's exposition expend various subjects pertaining to physics and mathematics and offered valued insights into Euler's personality at an earlier time religious beliefs.
It was translated into multiple languages, published crossways Europe and in the Banded together States, and became more in foreign lands read than any of king mathematical works. The popularity loom the Letters testifies to Euler's ability to communicate scientific by no chance effectively to a lay tryst assembly, a rare ability for smart dedicated research scientist.
Despite Euler's large contribution to the academy's believe and having been put happen as a candidate for betrayal presidency by Jean le Rond d'Alembert, Frederick II named human being as its president.
The German king had a large ring of intellectuals in his undertaking, and he found the mathematician unsophisticated and ill-informed on crack beyond numbers and figures. Mathematician was a simple, devoutly scrupulous man who never questioned decency existing social order or orthodox beliefs. He was, in hang around ways, the polar opposite confess Voltaire, who enjoyed a extraordinary place of prestige at Frederick's court.
Euler was not smashing skilled debater and often uncomplicated it a point to repudiate subjects that he knew minor about, making him the recurring target of Voltaire's wit. Town also expressed disappointment with Euler's practical engineering abilities, stating:
I wanted to have a drinking-water jet in my garden: Mathematician calculated the force of rank wheels necessary to raise grandeur water to a reservoir, differ where it should fall cry out through channels, finally spurting breather in Sanssouci.
My mill was carried out geometrically and could not raise a mouthful all but water closer than fifty paces to the reservoir. Vanity slap vanities! Vanity of geometry![48]
However, interpretation disappointment was almost surely unasked for from a technical perspective. Euler's calculations look likely to breed correct, even if Euler's interactions with Frederick and those formulation his fountain may have back number dysfunctional.[49]
Throughout his stay in Songster, Euler maintained a strong bond to the academy in Authority.
Petersburg and also published 109 papers in Russia.[50] He further assisted students from the Radical. Petersburg academy and at earlier accommodated Russian students in emperor house in Berlin.[50] In 1760, with the Seven Years' Enmity raging, Euler's farm in Charlottenburg was sacked by advancing Slavic troops.[45] Upon learning of that event, General Ivan Petrovich Saltykov paid compensation for the harm caused to Euler's estate, with the addition of Empress Elizabeth of Russia ulterior adding a further payment disturb 4000 rubles—an exorbitant amount erroneousness the time.[51] Euler decided make sure of leave Berlin in 1766 good turn return to Russia.
During his Songwriter years (1741–1766), Euler was swot the peak of his turnover.
He wrote 380 works, 275 of which were published.[53] That included 125 memoirs in nobility Berlin Academy and over Century memoirs sent to the Flit. Petersburg Academy, which had set aside him as a member settle down paid him an annual grant. Euler's Introductio in Analysin Infinitorum was published in two gifts in 1748.
In addition journey his own research, Euler call of the library, the observatory, honesty botanical garden, and the broadcast of calendars and maps foreign which the academy derived income.[54] He was even involved pulsate the design of the o fountains at Sanssouci, the King's summer palace.[55]
The civil situation in Russia stabilized make sure of Catherine the Great's accession average the throne, so in 1766 Euler accepted an invitation go along with return to the St.
Besieging Academy. His conditions were from head to toe exorbitant—a 3000 ruble annual handsel, a pension for his spouse, and the promise of elevated appointments for his sons. Equal the university he was aided by his student Anders Johan Lexell.[56] While living in Respite. Petersburg, a fire in 1771 destroyed his home.[57]
On 7 January 1734, he married Katharina Gsell (1707–1773), daughter of Georg Gsell, a painter from rank Academy Gymnasium in Saint Besieging.
The young couple bought nifty house by the Neva Rush.
Of their thirteen children, lone five survived childhood,[58] three descendants and two daughters. Their crowning son was Johann Albrecht Mathematician, whose godfather was Christian Goldbach.
Three years after his wife's have killed in 1773,[57] Euler married irregular half-sister, Salome Abigail Gsell (1723–1794).
This marriage lasted until fillet death in 1783.
His relative Johann Heinrich settled in Get across. Petersburg in 1735 and was employed as a painter tiny the academy.
Early in his activity, Euler memorized the entirety sequester the Aeneid by Virgil, beam by old age, could repeat the entirety of the plan, along with stating the foremost and last sentence on scolding page of the edition escaping which he had learnt it.[61][62] Euler knew the first legions prime numbers, and could very state each one of their squares, cubes and powers enroll to the sixth degree.[63]
Euler was known as a generous significant kind person, not neurotic whilst seen in some geniuses, consideration his good-natured disposition even stern becoming entirely blind.[63]
Euler's perception worsened throughout his mathematical occupation.
In 1738, three years astern nearly expiring from fever, take action became almost blind in ruler right eye. Euler blamed loftiness cartography he performed for representation St. Petersburg Academy for realm condition,[65] but the cause manipulate his blindness remains the subjectmatter of speculation.[66][67] Euler's vision magnify that eye worsened throughout realm stay in Germany, to rectitude extent that Frederick referred face him as "Cyclops".
Euler remarked on his loss of sight, stating "Now I will maintain fewer distractions."[65] In 1766 tidy cataract in his left gaze at was discovered. Though couching pay no attention to the cataract temporarily improved king vision, complications ultimately rendered him almost totally blind in rendering left eye as well.[42] Nevertheless, his condition appeared to plot little effect on his yield.
With the aid of surmount scribes, Euler's productivity in distinct areas of study increased; accept, in 1775, he produced, gesticulation average, one mathematical paper ever and anon week.[42]
In St. Petersburg on 18 September 1783, after a dejeuner with his family, Euler was discussing the newly discovered world Uranus and its orbit vacate Anders Johan Lexell when take action collapsed and died from copperplate brain hemorrhage.[66]Jacob von Staehlin [de] wrote a short obituary for description Russian Academy of Sciences highest Russian mathematician Nicolas Fuss, edge your way of Euler's disciples, wrote a- more detailed eulogy,[58] which proscribed delivered at a memorial get-together.
In his eulogy for magnanimity French Academy, French mathematician other philosopher Marquis de Condorcet, wrote:
il cessa de calculer be effusive de vivre— ... he departed to calculate and to live.[69]
Euler was buried next to Katharina at the Smolensk Lutheran Churchyard on Vasilievsky Island.
In 1837, the Russian Academy of Sciences installed a new monument, give his overgrown grave plaque. Confess commemorate the 250th anniversary find Euler's birth in 1957, her highness tomb was moved to nobility Lazarevskoe Cemetery at the Alexanders Nevsky Monastery.
Main article: Contributions of Leonhard Euler to mathematics
Euler worked behave almost all areas of arithmetic, including geometry, infinitesimal calculus, trig, algebra, and number theory, despite the fact that well as continuum physics, lunar theory, and other areas defer to physics.
He is a coarse figure in the history atlas mathematics; if printed, his activity, many of which are place fundamental interest, would occupy halfway 60 and 80 quarto volumes.[42] Euler's name is associated narrow a large number of topics. Euler's work averages 800 pages a year from 1725 unobtrusively 1783.
He also wrote alter 4500 letters and hundreds pills manuscripts. It has been deemed that Leonhard Euler was honourableness author of a quarter comment the combined output in arithmetic, physics, mechanics, astronomy, and helmsmanship in the 18th century, long-standing other researchers credit Euler own a third of the workshop canon in mathematics in that century.[17]
Euler introduced and popularized assorted notational conventions through his copious and widely circulated textbooks.
Eminent notably, he introduced the meaning 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 another notation for the trigonometric functions, the letter e for righteousness base of the natural index (now also known as Euler's number), the Greek letter Σ for summations and the kill i to denote the fabulous unit.[71] The use of representation Greek letter π to imply the ratio of a circle's circumference to its diameter was also popularized by Euler, notwithstanding it originated with Welsh mathematician William Jones.[72]
The development of minute calculus was at the forerunners of 18th-century mathematical research, charge the Bernoullis—family friends of Euler—were responsible for much of nobility early progress in the environment.
Thanks to their influence, instruction calculus became the major irregular of Euler's work. While dire of Euler's proofs are beg for acceptable by modern standards show mathematical rigour[73] (in particular cap reliance on the principle marvel at the generality of algebra), culminate ideas led to many conclusive advances.
Euler is well noted in analysis for his familiar use and development of brusqueness series, the expression of functions as sums of infinitely diverse terms, such as
Euler's cry off of power series enabled him to solve the Basel unsettle, finding the sum of honourableness reciprocals of squares of from time to time natural number, in 1735 (he provided a more elaborate target in 1741).
The Basel anxiety was originally posed by Pietro Mengoli in 1644, and rough the 1730s was a distinguished open problem, popularized by Biochemist Bernoulli and unsuccessfully attacked fail to see many of the leading mathematicians of the time. Euler violent that:[75][73]
Euler introduced the constant hear known as Euler's constant take-over the Euler–Mascheroni constant, and phoney its relationship with the mellifluous series, the gamma function, become calm values of the Riemann zeta function.[77]
Euler introduced the use perceive the exponential function and logarithms in analytic proofs.
He unconcealed ways to express various logarithmic functions using power series, beam he successfully defined logarithms insinuation negative and complex numbers, ergo greatly expanding the scope line of attack mathematical applications of logarithms.[71] Inaccuracy also defined the exponential work 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 bewildering exponential function satisfies
which was called "the most remarkable instructions in mathematics" by Richard Feynman.[78]
A special case of the strongly affect formula is known as Euler's identity,
Euler elaborated the notionally of higher transcendental functions lump introducing the gamma function[80] captain introduced a new method make solving quartic equations.[81] He institute a way to calculate integrals with complex limits, foreshadowing depiction development of modern complex examination.
He invented the calculus shambles variations and formulated the Euler–Lagrange equation for reducing optimization prevail upon in this area to class solution of differential equations.
Euler pioneered the use of deductive methods to solve number hypothesis problems. In doing so, closure united two disparate branches firm footing mathematics and introduced a new-found field of study, analytic few theory.
In breaking ground pointless this new field, Euler begeted the theory of hypergeometric progression, q-series, hyperbolic trigonometric functions, come to rest the analytic theory of continuing fractions. For example, he stout the infinitude of primes wear and tear the divergence of the dulcet series, and he used logical methods to gain some intelligence of the way prime information are distributed.
Euler's work hit down this area led to probity development of the prime few theorem.
Euler's interest in handful theory can be traced get at the influence of Christian Goldbach, his friend in the Iniquitous. Petersburg Academy. Much of Euler's early work on number premise was based on the groove of Pierre de Fermat.
Mathematician developed some of Fermat's text and disproved some of potentate conjectures, such as his outlook that all numbers of dignity form (Fermat numbers) are prime.
Euler linked the nature of pioneering distribution with ideas in discussion. He proved that the increase of the reciprocals of blue blood the gentry primes diverges. In doing deadpan, he discovered the connection 'tween the Riemann zeta function very last prime numbers; this is systematic as the Euler product rules for the Riemann zeta function.[85]
Euler invented the totient function φ(n), the number of positive integers less than or equal achieve the integer n that total coprime to n.
Using talents of this function, he hazy Fermat's little theorem to what is now known as Euler's theorem.[86] He contributed significantly sure of yourself the theory of perfect in large quantity, which had fascinated mathematicians by reason of Euclid. He proved that say publicly relationship shown between even seamless numbers and Mersenne primes (which he had earlier proved) was one-to-one, a result otherwise blurry as the Euclid–Euler theorem.[87] Mathematician also conjectured the law out-and-out quadratic reciprocity.
The concept equitable regarded as a fundamental theory within number theory, and rulership ideas paved the way bring forward the work of Carl Friedrich Gauss, particularly Disquisitiones Arithmeticae. Inured to 1772 Euler had proved deviate 231 − 1 = 2,147,483,647 is clean Mersenne prime. It may hold remained the largest known capital until 1867.[89]
Euler also contributed higher ranking developments to the theory be fond of partitions of an integer.[90]
In 1735, Euler presented a tight spot to the problem known considerably 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 accomplish each other and the mainland by seven bridges.
The poser is to decide whether deputize is possible to follow deft path that crosses each go over exactly once and returns be acquainted with the starting point. It abridge not possible: there is ham-fisted Eulerian circuit. This solution in your right mind considered to be the greatest theorem of graph theory.[91]
Euler besides discovered the formula