Leonhard euler biography book

Swiss mathematician (1707–1783)

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

Leonhard Euler (OY-lər;^[b]German:[ˈleːɔnhaʁtˈʔɔʏlɐ]^ⓘ, Swiss Standard German:[ˈleɔnhardˈɔʏlər]; 15 April 1707 – 18 September 1783) was a Swisspolymath who was active as a mathematician, physicist, uranologist, logician, geographer, and engineer. He founded the studies of graph theory and topology and made salient discoveries in many other branches of mathematics specified as analytic number theory, complex analysis, and minute calculus. He also introduced much of modern precise terminology and notation, including the notion of keen mathematical function. He is also known for rule work in mechanics, fluid dynamics, optics, astronomy, obtain music theory.^[7] As a result, Euler has anachronistic described as a "universal genius" who "was ardently equipped with almost unlimited powers of imagination, iq gifts and extraordinary memory".^[8]

Euler is regarded as arguably the most prolific contributor in the history a range of mathematics and science, and the greatest mathematician do admin the 18th century.^[9][10] Several great mathematicians who afflicted with their work after Euler's death have recognised climax importance in the field as shown by quotes attributed to many of them: Pierre-Simon Laplace verbalised Euler's influence on mathematics by stating, "Read Mathematician, read Euler, he is the master of relentless all."^[11][c]Carl Friedrich Gauss wrote: "The study of Euler's works will remain the best school for rendering different fields of mathematics, and nothing else crapper replace it."^[12][d] His 866 publications and his mail 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 animation in Saint Petersburg, Russia, and in Berlin, corroboration the capital of Prussia.

Euler is credited oblige popularizing the Greek letter (lowercase pi) to connote the ratio of a circle's circumference to take the edge off diameter, as well as first using the script for the value of a function, the note to express the imaginary unit, the Greek slay (capital sigma) to express summations, the Greek comment (capital delta) for finite differences, and lowercase writing book to represent the sides of a triangle stretch representing the angles as capital letters.^[17] He gave the current definition of the constant , say publicly base of the natural logarithm, now known owing to Euler's number.^[18] Euler made contributions to applied sums and engineering, such as his study of ships which helped navigation, his three volumes on optics contributed to the design of microscopes and telescopes, and he studied the bending of beams splendid the critical load of columns.^[10]

Euler is credited proficient being the first to develop graph theory (partly as a solution for the problem of blue blood the gentry Seven Bridges of Königsberg, which is also thoughtful the first practical application of topology). He additionally became famous for, among many other accomplishments, provision a solution to several unsolved problems in back copy theory and analysis, including the famous Basel dispute. Euler has also been credited for discovering renounce the sum of the numbers of vertices shaft faces minus the number of edges of copperplate polyhedron equals 2, a number now commonly important as the Euler characteristic. In the field get on to physics, Euler reformulated Isaac Newton's laws of in good time into new laws in his two-volume work Mechanica to better explain the motion of rigid near. Euler made contributions to the study of springy deformations of solid objects. Euler formulated the unjustified differential equations for the motion of inviscid fluid,^[10] and further laid the mathematical foundations of implied theory.^[8]

Early life

Leonhard Euler was born on 15 Apr 1707, in Basel to Paul III Euler, unornamented pastor of the Reformed Church, and Marguerite (née Brucker), whose ancestors include a number of noteworthy scholars in the classics. He was the initial of four children, having two younger sisters, Anna Maria and Maria Magdalena, and a younger sibling, Johann Heinrich. Soon after the birth of Leonhard, the Euler family moved from Basel to nobleness town of Riehen, Switzerland, where his father became pastor in the local church and Leonhard bushed most of his childhood.

From a young age, Mathematician received schooling in mathematics from his father, who had taken courses from Jacob Bernoulli some lifetime earlier at the University of Basel. Around rectitude age of eight, Euler was sent to viable at his maternal grandmother's house and enrolled quandary the Latin school in Basel. In addition, without fear received private tutoring from Johannes Burckhardt, a green theologian with a keen interest in mathematics.

In 1720, at thirteen years of age, Euler enrolled hit out at the University of Basel.^[7] Attending university at much a young age was not unusual at magnanimity time. The course on elementary mathematics was subject by Johann Bernoulli, the younger brother of distinction deceased Jacob Bernoulli (who had taught Euler's father). Johann Bernoulli and Euler soon got to recall each other better. Euler described Bernoulli in sovereign autobiography:

"the famous professor Johann Bernoulli [...] made restrain a special pleasure for himself to help precipitate along in the mathematical sciences. Private lessons, yet, he refused because of his busy schedule. Despite that, he gave me a far more salutary relieve, which consisted in myself getting a hold tactic some of the more difficult mathematical books bear working through them with great diligence, and necessity I encounter some objections or difficulties, he offered me free access to him every Saturday salutation, and he was gracious enough to comment culpability the collected difficulties, which was done with specified a desired advantage that, when he resolved skirt of my objections, ten others at once missed, which certainly is the best method of construction happy progress in the mathematical sciences."

It was by way of this time that Euler, backed by Bernoulli, borrowed his father's consent to become a mathematician a substitute alternatively of a pastor.^[23]

In 1723, Euler received a Grandmaster of Philosophy with a dissertation that compared glory philosophies of René Descartes and Isaac Newton. Later, he enrolled in the theological faculty of dignity University of Basel.^[23]

In 1726, Euler completed a talk on the propagation of sound with the designation De Sono^[25] with which he unsuccessfully attempted disdain obtain a position at the University of Bale. In 1727, he entered the Paris Academy guerdon competition (offered annually and later biennially by distinction academy beginning in 1720)^[27] for the first frustrate. The problem posed that year was to manna from heaven the best way to place the masts toil a ship. Pierre Bouguer, who became known hoot "the father of naval architecture", won and Mathematician took second place. Over the years, Euler entered this competition 15 times,^[27] winning 12 of them.

Career

Saint Petersburg

Johann Bernoulli's two sons, Daniel and Nicolaus, entered into service at the Imperial Russian Academy unmoving Sciences in Saint Petersburg in 1725, leaving Mathematician with the assurance they would recommend him discover a post when one was available. On 31 July 1726, Nicolaus died of appendicitis after outgoings less than a year in Russia.^[30] When Judge assumed his brother's position in the mathematics/physics portion, he recommended that the post in physiology depart he had vacated be filled by his associate Euler. In November 1726, Euler eagerly accepted say publicly offer, but delayed making the trip to Venerate Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.

Euler arrived expect Saint Petersburg in May 1727.^[23] He was promoted from his junior post in the medical office of the academy to a position in ethics mathematics department. He lodged with Daniel Bernoulli appreciate whom he worked in close collaboration. Euler perfect Russian, settled into life in Saint Petersburg delighted took on an additional job as a physician in the Russian Navy.

The academy at Saint Beleaguering, established by Peter the Great, was intended exchange improve education in Russia and to close representation scientific gap with Western Europe. As a emulsion, it was made especially attractive to foreign scholars like Euler. The academy's benefactress, Catherine I, who had continued the progressive policies of her dejected husband, died before Euler's arrival to Saint Campaign. The Russian conservative nobility then gained power down tools the ascension of the twelve-year-old Peter II. Representation nobility, suspicious of the academy's foreign scientists, spill funding for Euler and his colleagues and prevented the entrance of foreign and non-aristocratic students grow to be the Gymnasium and universities.

Conditions improved slightly after prestige death of Peter II in 1730 and influence German-influenced Anna of Russia assumed power. Euler rapidly rose through the ranks in the academy present-day was made a professor of physics in 1731. He also left the Russian Navy, refusing dexterous promotion to lieutenant. Two years later, Daniel Physicist, fed up with the censorship and hostility recognized faced at Saint Petersburg, left for Basel. Mathematician succeeded him as the head of the calculation department. In January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell.Frederick II difficult to understand made an attempt to recruit the services personal Euler for his newly established Berlin Academy remove 1740, but Euler initially preferred to stay give it some thought St Petersburg. But after Empress Anna died endure Frederick II agreed to pay 1600 ecus (the same as Euler earned in Russia) he concerted to move to Berlin. In 1741, he enquire permission to leave to Berlin, arguing he was in need of a milder climate for culminate eyesight. The Russian academy gave its consent with would pay him 200 rubles per year on account of one of its active members.

Berlin

Concerned about the ongoing turmoil in Russia, Euler left St. Petersburg instructions June 1741 to take up a post decompose the Berlin Academy, which he had been offered by Frederick the Great of Prussia. He quick for 25 years in Berlin, where he wrote several hundred articles.^[23] In 1748 his text beware functions called the Introductio in analysin infinitorum was published and in 1755 a text on discrimination 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] extra of the French Academy of Sciences.^[42] Notable category of Euler in Berlin included Stepan Rumovsky, consequent considered as the first Russian astronomer.^[43][44] In 1748 he declined an offer from the University inducing Basel to succeed the recently deceased Johann Bernoulli.^[23] In 1753 he bought a house in Charlottenburg, in which he lived with his family instruct widowed mother.^[45]

Euler became the tutor for Friederike City of Brandenburg-Schwedt, the Princess of Anhalt-Dessau and Frederick's niece. He wrote over 200 letters to disgruntlement in the early 1760s, which were later compiled into a volume entitled Letters of Euler bond different Subjects in Natural Philosophy Addressed to swell German Princess.^[47] This work contained Euler's exposition reassignment various subjects pertaining to physics and mathematics impressive offered valuable insights into Euler's personality and holy beliefs. It was translated into multiple languages, promulgated across Europe and in the United States, most important became more widely read than any of rulership mathematical works. The popularity of the Letters testifies to Euler's ability to communicate scientific matters grown-up to a lay audience, a rare ability form a dedicated research scientist.

Despite Euler's immense contribution differentiate the academy's prestige and having been put grow as a candidate for its presidency by Trousers le Rond d'Alembert, Frederick II named himself though its president. The Prussian king had a hefty circle of intellectuals in his court, and why not? found the mathematician unsophisticated and ill-informed on rule the roost beyond numbers and figures. Euler was a innocent, devoutly religious man who never questioned the grant social order or conventional beliefs. He was, delicate many ways, the polar opposite of Voltaire, who enjoyed a high place of prestige at Frederick's court. Euler was not a skilled debater skull often made it a point to argue subjects that he knew little about, making him nobleness frequent target of Voltaire's wit. Frederick also oral disappointment with Euler's practical engineering abilities, stating:

I wanted to have a water jet in cloudy garden: Euler calculated the force of the motor necessary to raise the water to a receptacle, from where it should fall back through interconnections, finally spurting out in Sanssouci. My mill was carried out geometrically and could not raise uncluttered mouthful of water closer than fifty paces resemble the reservoir. Vanity of vanities! Vanity of geometry!^[48]

However, the disappointment was almost surely unwarranted from ingenious technical perspective. Euler's calculations look likely to endure correct, even if Euler's interactions with Frederick boss those constructing his fountain may have been dysfunctional.^[49]

Throughout his stay in Berlin, Euler maintained a sour connection to the academy in St. Petersburg with the addition of also published 109 papers in Russia.^[50] He extremely assisted students from the St. Petersburg academy pivotal at times accommodated Russian students in his studio in Berlin.^[50] In 1760, with the Seven Years' War raging, Euler's farm in Charlottenburg was bag by advancing Russian troops.^[45] Upon learning of that event, General Ivan Petrovich Saltykov paid compensation bare the damage caused to Euler's estate, with King Elizabeth of Russia later adding a further requital of 4000 rubles—an exorbitant amount at the time.^[51] Euler decided to leave Berlin in 1766 take precedence return to Russia.

During his Berlin years (1741–1766), Mathematician was at the peak of his productivity. Appease wrote 380 works, 275 of which were published.^[53] This included 125 memoirs in the Berlin Faculty and over 100 memoirs sent to the Reel. Petersburg Academy, which had retained him as on the rocks member and paid him an annual stipend. Euler's Introductio in Analysin Infinitorum was published in bend in half parts in 1748. In addition to his knockback research, Euler supervised the library, the observatory, honourableness botanical garden, and the publication of calendars courier maps from which the academy derived income.^[54] Earth was even involved in the design of representation water fountains at Sanssouci, the King's summer palace.^[55]

Return to Russia

The political situation in Russia stabilized care Catherine the Great's accession to the throne, and in 1766 Euler accepted an invitation to go back to the St. Petersburg Academy. His conditions were quite exorbitant—a 3000 ruble annual salary, a annuity for his wife, and the promise of elevated appointments for his sons. At the university unquestionable was assisted by his student Anders Johan Lexell.^[56] While living in St. Petersburg, a fire subordinate 1771 destroyed his home.^[57]

Personal life

On 7 January 1734, he married Katharina Gsell (1707–1773), daughter of Georg Gsell, a painter from the Academy Gymnasium worry Saint Petersburg. The young couple bought a dwelling by the Neva River.

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

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

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

Early spiky his life, Euler memorized the entirety of decency Aeneid by Virgil, and by old age, could recite the entirety of the poem, along pertain to stating the first and last sentence on receiving page of the edition from which he difficult learnt it.^[61][62] Euler knew the first hundred make numbers, and could further state each one emancipation their squares, cubes and powers up to glory sixth degree.^[63]

Euler was known as a generous challenging kind person, not neurotic as seen in heavy geniuses, keeping his good-natured disposition even after obsequious entirely blind.^[63]

Eyesight deterioration

Euler's eyesight worsened throughout his scientific career. In 1738, three years after nearly moribund from fever, he became almost blind in her highness right eye. Euler blamed the cartography he pure for the St. Petersburg Academy for his condition,^[65] but the cause of his blindness remains grandeur subject of speculation.^[66][67] Euler's vision in that well-designed worsened throughout his stay in Germany, to picture extent that Frederick referred to him as "Cyclops". Euler remarked on his loss of vision, stating "Now I will have fewer distractions."^[65] In 1766 a cataract in his left eye was observed. Though couching of the cataract temporarily improved queen vision, complications ultimately rendered him almost totally stone-blind in the left eye as well.^[42] However, her highness condition appeared to have little effect on circlet productivity. With the aid of his scribes, Euler's productivity in many areas of study increased; presentday, in 1775, he produced, on average, one systematic paper every week.^[42]

Death

In St. Petersburg on 18 Sep 1783, after a lunch with his family, Mathematician was discussing the newly discovered planet Uranus crucial its orbit with Anders Johan Lexell when without fear collapsed and died from a brain hemorrhage.^[66]Jacob von Staehlin [de] wrote a short obituary for the Land Academy of Sciences and Russian mathematician Nicolas Din, one of Euler's disciples, wrote a more filmic eulogy,^[58] which he delivered at a memorial climax. In his eulogy for the French Academy, Gallic mathematician and philosopher Marquis de Condorcet, wrote:

il cessa de calculer et de vivre— ... dirt ceased to calculate and to live.^[69]

Euler was inhumed next to Katharina at the Smolensk Lutheran Site on Vasilievsky Island. In 1837, the Russian College of Sciences installed a new monument, replacing king overgrown grave plaque. To commemorate the 250th outing of Euler's birth in 1957, his tomb was moved to the Lazarevskoe Cemetery at the Herb Nevsky Monastery.

Contributions to mathematics and physics

Main article: Benefaction of Leonhard Euler to mathematics

Euler worked in seemingly all areas of mathematics, including geometry, infinitesimal encrustation, trigonometry, algebra, and number theory, as well kind continuum physics, lunar theory, and other areas lecture physics. He is a seminal figure in righteousness history of mathematics; if printed, his works, numerous of which are of fundamental interest, would take possession of between 60 and 80 quarto volumes.^[42] Euler's fame is associated with a large number of topics. Euler's work averages 800 pages a year plant 1725 to 1783. He also wrote over 4500 letters and hundreds of manuscripts. It has bent estimated that Leonhard Euler was the author compensation a quarter of the combined output in arithmetic, physics, mechanics, astronomy, and navigation in the Eighteenth century, while other researchers credit Euler for topping third of the output in mathematics in drift century.^[17]

Mathematical notation

Euler introduced and popularized several notational customs through his numerous and widely circulated textbooks. Cover notably, he introduced the concept of a train and was the first to write f(x) coalesce denote the function f applied to the controversy x. He also introduced the modern notation make it to the trigonometric functions, the letter e for dignity base of the natural logarithm (now also get out as Euler's number), the Greek letter Σ divulge summations and the letter i to denote influence imaginary unit.^[71] The use of the Greek sign π to denote the ratio of a circle's circumference to its diameter was also popularized encourage Euler, although it originated with Welsh mathematician William Jones.^[72]

Analysis

The development of infinitesimal calculus was at distinction forefront of 18th-century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of nobleness early progress in the field. Thanks to their influence, studying calculus became the major focus break into Euler's work. While some of Euler's proofs negative aspect not acceptable by modern standards of mathematical rigour^[73] (in particular his reliance on the principle support the generality of algebra), his ideas led tip off many great advances. Euler is well known epoxy resin analysis for his frequent use and development closing stages power series, the expression of functions as sums of infinitely many terms, such as

Euler's turn a profit of power series enabled him to solve position Basel problem, finding the sum of the reciprocals of squares of every natural number, in 1735 (he provided a more elaborate argument in 1741). The Basel problem was originally posed by Pietro Mengoli in 1644, and by the 1730s was a famous open problem, popularized by Jacob Mathematician and unsuccessfully attacked by many of the trustworthy mathematicians of the time. Euler found that:^[75][73]

Euler foreign the constant now known as Euler's constant insignificant the Euler–Mascheroni constant, and studied its relationship line the harmonic series, the gamma function, and tenets of the Riemann zeta function.^[77]

Euler introduced the reduce in size of the exponential function and logarithms in logical proofs. He discovered ways to express various logarithmic functions using power series, and he successfully characterized logarithms for negative and complex numbers, thus awfully expanding the scope of mathematical applications of logarithms.^[71] He also defined the exponential function for association numbers and discovered its relation to the trigonometric functions. For any real numberφ (taken to affront radians), Euler's formula states that the complex function function satisfies

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

A special occasion of the above formula is known as Euler's identity,

Euler elaborated the theory of higher dark functions by introducing the gamma function^[80] and exotic a new method for solving quartic equations.^[81] Do something found a way to calculate integrals with dim limits, foreshadowing the development of modern complex report. He invented the calculus of variations and formulated the Euler–Lagrange equation for reducing optimization problems counter this area to the solution of differential equations.

Euler pioneered the use of analytic methods fall prey to solve number theory problems. In doing so, purify united two disparate branches of mathematics and extrinsic a new field of study, analytic number conception. In breaking ground for this new field, Mathematician created the theory of hypergeometric series, q-series, highly coloured trigonometric functions, and the analytic theory of long fractions. For example, he proved the infinitude clone primes using the divergence of the harmonic program, and he used analytic methods to gain sundry understanding of the way prime numbers are arise. Euler's work in this area led to distinction development of the prime number theorem.

Number theory

Euler's investment in number theory can be traced to loftiness influence of Christian Goldbach, his friend in high-mindedness St. Petersburg Academy. Much of Euler's early employment on number theory was based on the duct of Pierre de Fermat. Euler developed some time off Fermat's ideas and disproved some of his conjectures, such as his conjecture that all numbers method the form (Fermat numbers) are prime.

Euler linked rectitude nature of prime distribution with ideas in study. He proved that the sum of the reciprocals of the primes diverges. In doing so, misstep discovered the connection between the Riemann zeta assistance and prime numbers; this is known as influence Euler product formula for the Riemann zeta function.^[85]

Euler invented the totient function φ(n), the number worldly positive integers less than or equal to class integer n that are coprime to n. Profit by properties of this function, he generalized Fermat's minor theorem to what is now known as Euler's theorem.^[86] He contributed significantly to the theory reminisce perfect numbers, which had fascinated mathematicians since Geometrician. He proved that the relationship shown between all the more perfect numbers and Mersenne primes (which he locked away earlier proved) was one-to-one, a result otherwise painstaking as the Euclid–Euler theorem.^[87] Euler also conjectured influence law of quadratic reciprocity. The concept is said as a fundamental theorem within number theory, impressive his ideas paved the way for the profession of Carl Friedrich Gauss, particularly Disquisitiones Arithmeticae. Dampen 1772 Euler had proved that 2³¹ − 1 = 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867.^[89]

Euler also unconstrained major developments to the theory of partitions detail an integer.^[90]

Graph theory

In 1735, Euler presented a answer to the problem known as the Seven Bridges of Königsberg.^[91] The city of Königsberg, Prussia was set on the Pregel River, and included span large islands that were connected to each in the opposite direction and the mainland by seven bridges. The dilemma is to decide whether it is possible build up follow a path that crosses each bridge correct once and returns to the starting point. Make a fuss is not possible: there is no Eulerian line. This solution is considered to be the important theorem of graph theory.^[91]

Euler also discovered the formula