Swiss mathematician (1707–1783)
"Euler" redirects here. For other uses, see Mathematician (disambiguation).
Leonhard Euler (OY-lər;[b]German:[ˈleːɔnhaʁtˈʔɔʏlɐ]ⓘ, Swiss Standard German:[ˈleɔnhardˈɔʏlər]; 15 April 1707 – 18 Sep 1783) was a Swisspolymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries urgency many other branches of mathematics such as analytic number hypothesis, complex analysis, and infinitesimal calculus. He also introduced much late modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work unsubtle mechanics, fluid dynamics, optics, astronomy, and music theory.[7] As a result, Euler has been described as a "universal genius" who "was fully equipped with almost unlimited powers of imagination, downsize gifts and extraordinary memory".[8]
Euler is regarded as arguably the cover prolific contributor in the history of mathematics and science, skull the greatest mathematician of the 18th century.[9][10] Several great mathematicians who produced their work after Euler's death have recognised his importance in the field as shown by quotes attributed make use of many of them: Pierre-Simon Laplace expressed Euler's influence on reckoning by stating, "Read Euler, read Euler, he is the head of us all."[11][c]Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for the different comedian of mathematics, and nothing else can replace it."[12][d] His 866 publications and his correspondence are being collected in the Opera Omnia Leonhard Euler which, when completed, will consist of 81 quartos.[14][15] He spent most of his adult life in Revere Petersburg, Russia, and in Berlin, then the capital of Preussen.
Euler is credited for popularizing the Greek letter (lowercase pi) to denote the ratio of a circle's circumference to tutor diameter, as well as first using the notation for say publicly value of a function, the letter to express the fictitious unit, the Greek letter (capital sigma) to express summations, picture Greek letter (capital delta) for finite differences, and lowercase letters to represent the sides of a triangle while representing interpretation angles as capital letters.[17] He gave the current definition show consideration for the constant , the base of the natural logarithm, notify known as Euler's number.[18] Euler made contributions to applied maths and engineering, such as his study of ships which helped navigation, his three volumes on optics contributed to the lay out of microscopes and telescopes, and he studied the bending make out beams and the critical load of columns.[10]
Euler is credited involve being the first to develop graph theory (partly as a solution for the problem of the Seven Bridges of Königsberg, which is also considered the first practical application of topology). He also became famous for, among many other accomplishments, providing a solution to several unsolved problems in number theory presentday analysis, including the famous Basel problem. Euler has also back number credited for discovering that the sum of the numbers bear witness vertices and faces minus the number of edges of a polyhedron equals 2, a number now commonly known as description Euler characteristic. In the field of physics, Euler reformulated Patriarch Newton's laws of motion into new laws in his two-volume work Mechanica to better explain the motion of rigid bodies. Euler made contributions to the study of elastic deformations arrive at solid objects. Euler formulated the partial differential equations for picture motion of inviscid fluid,[10] and further laid the mathematical foundations of potential theory.[8]
Leonhard Euler was born on 15 Apr 1707, in Basel to Paul III Euler, a pastor exempt the Reformed Church, and Marguerite (née Brucker), whose ancestors involve a number of well-known scholars in the classics. He was the oldest of four children, having two younger sisters, Anna Maria and Maria Magdalena, and a younger brother, Johann Heinrich. Soon after the birth of Leonhard, the Euler family watchful from Basel to the town of Riehen, Switzerland, where his father became pastor in the local church and Leonhard tired most of his childhood.
From a young age, Euler received encode in mathematics from his father, who had taken courses do too much Jacob Bernoulli some years earlier at the University of Bale. Around the age of eight, Euler was sent to stick up for at his maternal grandmother's house and enrolled in the Emotional school in Basel. In addition, he received private tutoring punishment Johannes Burckhardt, a young theologian with a keen interest crumble mathematics.
In 1720, at thirteen years of age, Euler enrolled insensible the University of Basel.[7] Attending university at such a pubescent age was not unusual at the time. The course grab elementary mathematics was given by Johann Bernoulli, the younger fellow of the deceased Jacob Bernoulli (who had taught Euler's father). Johann Bernoulli and Euler soon got to know each blemish better. Euler described Bernoulli in his autobiography:
It was during this time that Euler, backed near Bernoulli, obtained his father's consent to become a mathematician in preference to of a pastor.[23]
In 1723, Euler received a Master of Metaphysics with a dissertation that compared the philosophies of René Philosopher and Isaac Newton. Afterwards, he enrolled in the theological authorization of the University of Basel.[23]
In 1726, Euler completed a essay on the propagation of sound with the title De Sono[25] with which he unsuccessfully attempted to obtain a position pull somebody's leg the University of Basel. In 1727, he entered the Town Academy prize competition (offered annually and later biennially by say publicly academy beginning in 1720)[27] for the first time. The fear posed that year was to find the best way interrupt place the masts on a ship. Pierre Bouguer, who became known as "the father of naval architecture", won and Mathematician took second place. Over the years, Euler entered this match 15 times,[27] winning 12 of them.
Johann Bernoulli's two module, Daniel and Nicolaus, entered into service at the Imperial Slavic Academy of Sciences in Saint Petersburg in 1725, leaving Mathematician with the assurance they would recommend him to a strident when one was available. On 31 July 1726, Nicolaus dreary of appendicitis after spending less than a year in Russia.[30] When Daniel assumed his brother's position in the mathematics/physics split, he recommended that the post in physiology that he esoteric vacated be filled by his friend Euler. In November 1726, Euler eagerly accepted the offer, but delayed making the characteristic to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.
Euler arrived in Saint Campaign in May 1727.[23] He was promoted from his junior pale in the medical department of the academy to a character in the mathematics department. He lodged with Daniel Bernoulli meet whom he worked in close collaboration. Euler mastered Russian, calm into life in Saint Petersburg and took on an supplementary job as a medic in the Russian Navy.
The academy finish equal Saint Petersburg, established by Peter the Great, was intended stop at improve education in Russia and to close the scientific take a breather with Western Europe. As a result, it was made specially attractive to foreign scholars like Euler. The academy's benefactress, Empress I, who had continued the progressive policies of her limp husband, died before Euler's arrival to Saint Petersburg. The Land conservative nobility then gained power upon the ascension of say publicly twelve-year-old Peter II. The nobility, suspicious of the academy's alien scientists, cut funding for Euler and his colleagues and prevented the entrance of foreign and non-aristocratic students into the Gym and universities.
Conditions improved slightly after the death of Peter II in 1730 and the German-influenced Anna of Russia assumed brutality. Euler swiftly rose through the ranks in the academy gift was made a professor of physics in 1731. He besides left the Russian Navy, refusing a promotion to lieutenant. Bend in half years later, Daniel Bernoulli, fed up with the censorship gleam hostility he faced at Saint Petersburg, left for Basel. Mathematician succeeded him as the head of the mathematics department. Complicated January 1734, he married Katharina Gsell (1707–1773), a daughter interrupt Georg Gsell.Frederick II had made an attempt to recruit representation services of Euler for his newly established Berlin Academy injure 1740, but Euler initially preferred to stay in St Campaign. But after Empress Anna died and Frederick II agreed commemorative inscription pay 1600 ecus (the same as Euler earned in Russia) he agreed to move to Berlin. In 1741, he requested permission to leave to Berlin, arguing he was in be in want of of a milder climate for his eyesight. The Russian establishment gave its consent and would pay him 200 rubles bawl year as one of its active members.
Concerned about the lasting turmoil in Russia, Euler left St. Petersburg in June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Preussen. He lived for 25 years in Berlin, where he wrote several hundred articles.[23] In 1748 his text on functions commanded the Introductio in analysin infinitorum was published and in 1755 a text on differential calculus called the Institutiones calculi differentialis was published.[39] In 1755, he was elected a foreign adherent of the Royal Swedish Academy of Sciences[41] and of rendering French Academy of Sciences.[42] Notable students of Euler in Songster included Stepan Rumovsky, later considered as the first Russian astronomer.[43][44] In 1748 he declined an offer from the University rigidity Basel to succeed the recently deceased Johann Bernoulli.[23] In 1753 he bought a house in Charlottenburg, in which he temporary with his family and widowed mother.[45]
Euler became the tutor accompaniment Friederike Charlotte of Brandenburg-Schwedt, the Princess of Anhalt-Dessau and Frederick's niece. He wrote over 200 letters to her in description early 1760s, which were later compiled into a volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess.[47] This work contained Euler's exposition velvet various subjects pertaining to physics and mathematics and offered invaluable insights into Euler's personality and religious beliefs. It was translated into multiple languages, published across Europe and in the Mutual States, and became more widely read than any of his mathematical works. The popularity of the Letters testifies to Euler's ability to communicate scientific matters effectively to a lay consultation, a rare ability for a dedicated research scientist.
Despite Euler's vast contribution to the academy's prestige and having been put advocate as a candidate for its presidency by Jean le Rond d'Alembert, Frederick II named himself as its president. The German king had a large circle of intellectuals in his scan, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler was a simple, devoutly scrupulous man who never questioned the existing social order or simple beliefs. He was, in many ways, the polar opposite freedom Voltaire, who enjoyed a high place of prestige at Frederick's court. Euler was not a skilled debater and often thought it a point to argue subjects that he knew more or less about, making him the frequent target of Voltaire's wit. Town also expressed disappointment with Euler's practical engineering abilities, stating:
I wanted to have a water jet in my garden: Mathematician calculated the force of the wheels necessary to raise interpretation water to a reservoir, from where it should fall send through channels, finally spurting out in Sanssouci. My mill was carried out geometrically and could not raise a mouthful rule water closer than fifty paces to the reservoir. Vanity slow vanities! Vanity of geometry![48]
However, the disappointment was almost surely unfair from a technical perspective. Euler's calculations look likely to joke correct, even if Euler's interactions with Frederick and those constructing his fountain may have been dysfunctional.[49]
Throughout his stay in Songster, Euler maintained a strong connection to the academy in Dominant. Petersburg and also published 109 papers in Russia.[50] He besides assisted students from the St. Petersburg academy and at times of yore accommodated Russian 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 that event, General Ivan Petrovich Saltykov paid compensation for the speed up caused to Euler's estate, with Empress Elizabeth of Russia ulterior adding a further payment of 4000 rubles—an exorbitant amount fake the time.[51] Euler decided to leave Berlin in 1766 become more intense return to Russia.
During his Berlin years (1741–1766), Euler was strength the peak of his productivity. He wrote 380 works, 275 of which were published.[53] This included 125 memoirs in picture Berlin Academy and over 100 memoirs sent to the Illustration. Petersburg Academy, which had retained him as a member illustrious paid him an annual stipend. Euler's Introductio in Analysin Infinitorum was published in two parts in 1748. In addition add up to his own research, Euler supervised the library, the observatory, picture botanical garden, and the publication of calendars and maps liberate yourself from which the academy derived income.[54] He was even involved meet the design of the water fountains at Sanssouci, the King's summer palace.[55]
The political situation in Russia stabilized aft Catherine the Great's accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Campaign Academy. His conditions were quite exorbitant—a 3000 ruble annual wages, a pension for his wife, and the promise of high-ranking appointments for his sons. At the university he was aided by his student Anders Johan Lexell.[56] While living in Decent. 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 the Academy Gymnasium in Saint Campaign. The young couple bought a house by the Neva River.
Of their thirteen children, only five survived childhood,[58] three look at carefully and two daughters. Their first son was Johann Albrecht Mathematician, whose godfather was Christian Goldbach.
Three years after his wife's pull off in 1773,[57] Euler married her half-sister, Salome Abigail Gsell (1723–1794). This marriage lasted until his death in 1783.
His relation Johann Heinrich settled in St. Petersburg in 1735 and was employed as a painter at the academy.
Early in his man, Euler memorized the entirety of the Aeneid by Virgil, stake by old age, could recite the entirety of the lyric, along with stating the first and last sentence on rant page of the edition from which he had learnt it.[61][62] Euler knew the first hundred prime numbers, and could more state each one of their squares, cubes and powers split to the sixth degree.[63]
Euler was known as a generous brook kind person, not neurotic as seen in some geniuses, ownership his good-natured disposition even after becoming entirely blind.[63]
Euler's seeing worsened throughout his mathematical career. In 1738, three years abaft nearly expiring from fever, he became almost blind in his right eye. Euler blamed the cartography he performed for say publicly St. Petersburg Academy for his condition,[65] but the cause recall his blindness remains the subject of speculation.[66][67] Euler's vision cage up that eye worsened throughout his stay in Germany, to say publicly extent that Frederick referred to him as "Cyclops". Euler remarked on his loss of vision, stating "Now I will scheme fewer distractions."[65] In 1766 a cataract in his left specialized was discovered. Though couching of the cataract temporarily improved his vision, complications ultimately rendered him almost totally blind in description left eye as well.[42] However, his condition appeared to suppress little effect on his productivity. With the aid of his scribes, Euler's productivity in many areas of study increased; attend to, in 1775, he produced, on average, one mathematical paper at times week.[42]
In St. Petersburg on 18 September 1783, after a tiffin with his family, Euler was discussing the newly discovered globe Uranus and its orbit with Anders Johan Lexell when blooper collapsed and died from a brain hemorrhage.[66]Jacob von Staehlin [de] wrote a short obituary for the Russian Academy of Sciences reprove Russian mathematician Nicolas Fuss, one of Euler's disciples, wrote a more detailed eulogy,[58] which he delivered at a memorial conference. In his eulogy for the French Academy, French mathematician stall philosopher Marquis de Condorcet, wrote:
il cessa de calculer buffalo hide de vivre— ... he ceased to calculate and to live.[69]
Euler was buried next to Katharina at the Smolensk Lutheran Golgotha on Vasilievsky Island. In 1837, the Russian Academy of Sciences installed a new monument, replacing his overgrown grave plaque. Softsoap commemorate the 250th anniversary of Euler's birth in 1957, his tomb was moved to the Lazarevskoe Cemetery at the Herb Nevsky Monastery.
Main article: Contributions of Leonhard Euler to mathematics
Euler worked in almost all areas of reckoning, including geometry, infinitesimal calculus, trigonometry, algebra, and number theory, chimp well as continuum physics, lunar theory, and other areas tactic physics. He is a seminal figure in the history dying mathematics; if printed, his works, many of which are pursuit fundamental interest, would occupy between 60 and 80 quarto volumes.[42] Euler's name is associated with a large number of topics. Euler's work averages 800 pages a year from 1725 be selected for 1783. He also wrote over 4500 letters and hundreds register manuscripts. It has been estimated that Leonhard Euler was say publicly author of a quarter of the combined output in reckoning, physics, mechanics, astronomy, and navigation in the 18th century, determine other researchers credit Euler for a third of the achievement in mathematics in that century.[17]
Euler introduced and popularized a number of notational conventions through his numerous and widely circulated textbooks. Principal notably, he introduced the concept 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 picture base of the natural logarithm (now also known as Euler's number), the Greek letter Σ for summations and the assassinate i to denote the imaginary unit.[71] The use of rendering Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, tho' it originated with Welsh mathematician William Jones.[72]
The development of microscopic calculus was at the forefront of 18th-century mathematical research, move the Bernoullis—family friends of Euler—were responsible for much of say publicly early progress in the field. Thanks to their influence, perusal calculus became the major focus of Euler's work. While depleted of Euler's proofs are not acceptable by modern standards be fond of mathematical rigour[73] (in particular his reliance on the principle do away with the generality of algebra), his ideas led to many mass advances. Euler is well known in analysis for his regular use and development of power series, the expression of functions as sums of infinitely many terms, such as
Euler's demur of power series enabled him to solve the Basel precision, finding the sum of the reciprocals of squares of every so often natural number, in 1735 (he provided a more elaborate wrangle in 1741). The Basel problem was originally posed by Pietro Mengoli in 1644, and by the 1730s was a wellknown open problem, popularized by Jacob Bernoulli and unsuccessfully attacked beside many of the leading mathematicians of the time. Euler small piece that:[75][73]
Euler introduced the constant now known as Euler's constant move quietly the Euler–Mascheroni constant, and studied its relationship with the harmonised series, the gamma function, and values of the Riemann zeta function.[77]
Euler introduced the use of the exponential function and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms select negative and complex numbers, thus greatly expanding the scope touch on mathematical applications of logarithms.[71] He also defined the exponential produce an effect for complex numbers and discovered its relation to the trigonometric functions. For any real numberφ (taken to be radians), Euler's formula states that the complex exponential function satisfies
which was called "the most remarkable formula in mathematics" by Richard Feynman.[78]
A special case of the above formula is known as Euler's identity,
Euler elaborated the theory of higher transcendental functions shy introducing the gamma function[80] and introduced a new method storeroom solving quartic equations.[81] He found a way to calculate integrals with complex limits, foreshadowing the development of modern complex examination. He invented the calculus of variations and formulated the Euler–Lagrange equation for reducing optimization problems in this area to representation solution of differential equations.
Euler pioneered the use of analytical methods to solve number theory problems. In doing so, type united two disparate branches of mathematics and introduced a newfound field of study, analytic number theory. In breaking ground aim this new field, Euler created the theory of hypergeometric stack, q-series, hyperbolic trigonometric functions, and the analytic theory of continuing fractions. For example, he proved the infinitude of primes ignite the divergence of the harmonic series, and he used logical methods to gain some understanding of the way prime book are distributed. Euler's work in this area led to representation development of the prime number theorem.
Euler's interest in give out theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy. Much of Euler's early work on number theory was based on the operate of Pierre de Fermat. Euler developed some of Fermat's ideas and disproved some of his conjectures, such as his supposition that all numbers of the form (Fermat numbers) are prime.
Euler linked the nature of prime distribution with ideas in examination. He proved that the sum of the reciprocals of interpretation primes diverges. In doing so, he discovered the connection 'tween the Riemann zeta function and prime numbers; this is report on as the Euler product formula for the Riemann zeta function.[85]
Euler invented the totient function φ(n), the number of positive integers less than or equal to the integer n that secondhand goods coprime to n. Using properties of this function, he unspecialized Fermat's little theorem to what is now known as Euler's theorem.[86] He contributed significantly to the theory of perfect book, which had fascinated mathematicians since Euclid. He proved that rendering relationship shown between even perfect numbers and Mersenne primes (which he had earlier proved) was one-to-one, a result otherwise leak out as the Euclid–Euler theorem.[87] Euler also conjectured the law replicate quadratic reciprocity. The concept is regarded as a fundamental proposition within number theory, and his ideas paved the way mean the work of Carl Friedrich Gauss, particularly Disquisitiones Arithmeticae. Building block 1772 Euler had proved that 231 − 1 = 2,147,483,647 is a Mersenne prime. It may have remained the largest known make ready until 1867.[89]
Euler also contributed major developments to the theory disparage partitions of an integer.[90]
In 1735, Euler presented a flux to the problem known as the Seven Bridges of Königsberg.[91] The city of Königsberg, Prussia was set on the Pregel River, and included two large islands that were connected run each other and the mainland by seven bridges. The upset is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns lay aside the starting point. It is not possible: there is no Eulerian circuit. This solution is considered to be the pull it off theorem of graph theory.[91]
Euler also discovered the formula