Leonhard Euler

Born15 April 1707Basel,Switzerland

Died18 September 1783(aged76)[OS: 7 September 1783]Saint Petersburg,Russian Empire

ResidenceKingdom of Prussia, Russian EmpireSwitzerland

FieldsMathematics andphysics

InstitutionsImperial Russian Academy of SciencesBerlin Academy

Alma materUniversity of Basel

Doctoral advisorJohann Bernoulli

Doctoral studentsNicolas FussJohann HennertStepan Rumovsky

Othernotable studentsJoseph Louis Lagrange

Signature

Leonhard Euler

Leonhard Euler 15 April 1707 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also renowned for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.

Euler is considered to be the pre-eminent mathematician of the 18th century and one of the sgreatest mathematicians to have ever lived. He is also one of the most prolific mathematicians; his collected works fill 60-80 quarto volumes. He spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia.

A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."

Major Life Events

Early LifeEuler was born on 15 April 1707, in Basel, Switzerland to Paul Euler, a pastor of the Reformed Church, and Marguerite Brucker, a pastors daughter. He had two younger sisters named Anna Maria and Maria Magdalena. Soon after the birth of Leonhard, the Euler moved from Basel to the town of Riehen, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli familyJohann Bernoulli, who was then regarded as Europes foremost mathematician, would eventually be the most important influence on young Leonhard. Eulers early formal education started in Basel, where he was sent to live with his maternal grandmother. At the age of 13 he enrolled at the University of Basel, and in 1723, received his Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new Pupils incredible talent for mathematics. Euler was at this point studying theology, Greek, and Hebrew at his fathers urging, in order to become a pastor, but Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. At that time, he was pursuing an (ultimately unsuccessful) attempt to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition; the problem that year was to find the best way to place the masts on a ship. Pierre Bouguer, a man who became known as the father of naval architecture won, and Euler took second place. Euler later won this annual prize twelve times.

St. PetersburgAround this time Johann Bernoullis two sons, Daniel and Nicolas, were working at the Imperial Russian Academy of Sciences in St Petersburg. On 10 July 1726, Nicolas died of appendicitis after spending a year in Russia, and when Daniel assumed his brothers position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.

Euler arrived in the Russian capital on 17 May 1727. He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he often worked in close collaboration. Euler mastered Russian and settled into life in St Petersburg. He also took on an additional job as a medic in the Russian Navy. The Academy at St. Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy in order to lessen the facultys teaching burden, and the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions.

The Academys benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Eulers arrival. The Russian nobility then gained power upon the ascension of the twelve-year-old Peter II. The nobility were suspicious of the academys foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues.

Conditions improved slightly upon the death of Peter II, and Euler swiftly rose through the ranks in the academy and was made professor of physics in 1731. Two years later, Daniel Bernoulli, who was fed up with the censorship and hostility he faced at St. Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department.

On 7 January 1734, he married Katharina Gsell (1707 1773), a daughter of Georg Gsell, a painter from the Academy Gymnasium. The young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood.

Berlin

Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia. He lived for twenty-five years in Berlin, where he wrote over 380 articles. In Berlin, he published the two works for which he would become most renowned:

TheIntroductio in analysin infinitorum, a text on functions published in 1748, and the Institutiones calculi differentialis, published in 1755 on differential calculus. In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences.

In addition, Euler was asked to tutor Friederike Charlotte of Brandenburg-Schwedt, the Princess of Anhalt-Dessau and Fredericks niece. Euler wrote over 200 letters to her in the early 1760s, which were later compiled into a best-selling volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess. This work contained Eulers exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Eulers personality and religious beliefs. This book became more widely read than any of his mathematical works, and was published across Europe and in the United States. The popularity of the 'Letters testifies to Eulers ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist.

Despite Eulers immense contribution to the Academys prestige, he was eventually forced to leave Berlin. This was partly because of a conflict of personality with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy. Voltaire was among those in Fredericks employ, and the Frenchman enjoyed a prominent position within the kings social circle. Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes. He was in many ways the antithesis of Voltaire. Euler had limited training in rhetoric, and tended to debate matters that he knew little about, making him a frequent target of Voltaires wit. Frederick also expressed disappointment with Eulers practical engineering abilities:I wanted to have a water jet in my garden. Euler calculated the force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouc i. My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry.

Eyesight deterioration

Eulers eyesight worsened throughout his mathematical career. Three years after suffering a near-fatal fever in 1735, he became almost blind in his right eye, but Euler rather blamed the painstaking work on cartography he performed for the St. Petersburg Academy for his condition. Eulers vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as "Cyclops". Euler later developed a cataract in his left eye, which was discovered in 1766. Just a few weeks after its discovery, he was rendered almost totally blind. However, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and exquisite memory. For example, Euler could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last. With the aid of his scribes, Eulers productivity on many areas of study actually increased. He produced on average, one mathematical paper every week in the year 1775.

Return to RussiaThe situation in Russia had improved greatly since the accession to the throne of Catherine the Great, and in1766 Euler accepted an invitation to return to the St. Petersburg Academy and spent the rest of his life in Russia. However, his second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife Katharina after 40 years of marriage. Three years after his wifes death, Euler married her half-sister, Salome Abigail Gsell (17231794). This marriage lasted until his death. He was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1782.

In St. Petersburg on 18 September 1783, after a lunch with his family, during a conversation with a fellow academician Anders Johan Lexell, about the newly discovered planet Uranus and its orbit, Euler suffered a brain hemorrhage and died a few hours later. A short obituary for the Russian Academy of Sciences was written by Jacob von Staehlin-Storcksburg and a more detailed eulogy was written and delivered at a memorial meeting by Russian mathematician Nicolas Fuss, one of Eulers disciples. In the eulogy written for the French Academy by the French mathematician and philosopher Marquis de Condorcet, he commented,IL cessa de calculer et de vivre... he ceased to calculate and to live.

He was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island. In 1785, the Russian Academy of Sciences put a marble bust of Leonhard Euler on a pedestal next to the Directors seat and, in 1837, placed a headstone on Eulers grave. To commemorate the 250th anniversary of Eulers birth, the headstone was moved in 1956, together with his remains, to the 18th-century necropolis at the Alexander Nevsky Monastery.

Contributions to mathematics and physicsEuler worked in almost all areas of mathematics, such as geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works,Many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes. Eulers name is associated with a large number of topics.

Euler is the only mathematician to have two numbers named after him: the important Eulers Number in calculus, e, approximately equal to 2.71828, and the Euler-Mascheroni Constant (gamma) sometimes referred to as just Eulers constant, approximately equal to 0.57721. It is not known whether is rational or irrational.

Mathematical notationEuler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most 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 modern notation for the trigonometric functions, the letter e for the base of the natural logarithm (now also known as Eulers number), the Greek letter for summations and the letter i to denote the imaginary unit. The use of the Greek letter to denote the ratio of a circles circumference to its diameter was also popularized by Euler, although it did not originate with him.

AnalysisThe development of infinitesimal calculus was at the forefront of 18th Century mathematical research, and the Bernoullisfamily friends of Eulerwere responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Eulers work. While some of Eulers proofs are not acceptable by modern standards of mathematical rigor (in particular his reliance on the principle of the generality of algebra), his ideas led to many great advances. Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as

Notably, Euler directly proved the power series expansions for e and the inverse tangent function. (Indirect proof via the inverse power series technique was given by Newton and Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the famous Basel problem in 1735 (he provided a more elaborate argument in 1741).

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 for negative and complex numbers, thus greatly expanding the scope of mathematical applications of logarithms. He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions. For any real number (taken to be radians), Eulers formula states that the complex exponential function satisfies

A special case of the above formula is known as Eulers identity,

called the most remarkable formula in mathematics by Richard P. Feynman, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, e, I and . In 1988, readers of the Mathematical Intelligencer voted it the Most Beautiful Mathematical Formula Ever. In total, Euler was responsible for three of the top five formulae in that poll.

In addition, Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. He also found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis. He also invented the calculus of variations including its best-known result, the EulerLagrange equation.

Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hyper geometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions. For example, he proved the infinitude of primes using the divergence of the harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Eulers work in this area led to the development of the prime number theorem.

Number theoryEulers interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy. A lot of Eulers early work on number theory was based on the works of Pierre de Fermat. Euler developed some of Fermats ideas, and disproved some of his conjectures. Euler linked the nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and the prime numbers; this is known as the Euler product formula for the Riemann zeta function. Euler proved Newtons identities, Fermats little theorem, Fermats theorem on sums of two squares, and he made distinct contributions to Lagranges four-square theorem. He also invented the totient function (n), the number of positive integers less than or equal to the integer n that are coprime to n. Using properties of this function, he generalized Fermats little theorem to what is now known as Eulers theorem. He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. He proved that the relationship shown between perfect numbers and Mersenne primes earlier proved by Euclid was one-to-one, a result otherwise known as the EuclidEuler theorem. Euler also conjectured the law of quadratic reciprocity. The concept is regarded as a fundamental theorem of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss. By 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.

Graph theoryIn 1735, Euler presented a solution to the problem known as the Seven Bridges of Konigsberg. The city of Konigsberg, Prussia was set on the Pregal River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian circuit. This solution is considered to be the first theorem of graph theory, specifically of planar graph theory.

Euler also discovered the formula V E + F = 2 relating the number of vertices, edges and faces of a convex polyhedron, and hence of a planar graph. The constant in this formula is now known as the Euler characteristic for the graph (or other mathematical object), and is related to the genus of the object. The study and generalization of this formula, specifically by Cauchy and L'Huillier, is at the origin of topology.

Applied mathematicsSome of Eulers greatest successes were in solving real world problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series, Venn diagrams, Euler numbers, the constants e and , continued fractions and integrals. He integrated Leibniz's differential calculus with Newtons Method of Fluxions, and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. The most notable of these approximations are Eulers method and the EulerMcLaurin formula. He also facilitated the use of differential equations, in particular introducing the EulerMascheroni constant:

One of Eulers more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually incorporate musical theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.

Physics and astronomyEuler helped develop the EulerBernoulli beam equation, which became a cornerstone of engineering. Aside from successfully applying his analytic tools to problems in classical mechanics, Euler also applied these techniques to celestial problems. His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the parallax of the sun. His calculations also contributed to the development of accurate longitude tables.

In addition, Euler made important contributions in optics. He disagreed with Newtons corpuscular theory of light in the Optics, which was then the prevailing theory. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christian Huygens would become the dominant mode of thought, at least until the development of the quantum theory of light.

In 1757 he published an important set of equations for in viscid flow, that are now known as the Euler equations. In differential form, the equations are:

where is the fluidmass density, uis the fluidvelocityvector, with componentsu,v, andw, E = e + ( u2+ v2+ w2)is the total energy per unitvolume, withebeing theinternal energyper unit mass for the fluid, pis thepressure, denotes thetensor product, and 0being thezero vector.

Euler is also well known in structural engineering for his formula giving the criticalbucklingload of an ideal strut, which depends only on its length and flexural stiffness.

where= maximum or criticalforce(vertical load on column),=modulus of elasticity,=area moment of inertia,= unsupported length of column,= column effective length factor, whose value depends on the conditions of end support of the column, as follows.For both ends pinned (hinged, free to rotate),= 1.0.For both ends fixed,= 0.50.For one end fixed and the other end pinned,= 0.699....For one end fixed and the other end free to move laterally,= 2.0.is the effective length of the column.

LogicEuler is also credited with usingclosed curvesto illustratesyllogisticreasoning (1768). These diagrams have become known asEuler diagrams.

AnEuler diagramis adiagrammaticmeans of representingsetsand their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depictsets. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elementsof the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important: the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection,subsetanddisjointness). Curves whose interior zones do not intersect representdisjoint sets. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (theintersectionof the sets). A curve that is contained completely within the interior zone of another represents asubsetof it. Euler diagrams were incorporated as part of instruction inset theoryas part of thenew mathmovement in the 1960s. Since then, they have also been adopted by other curriculum fields such as reading. Personal philosophy and religious beliefsEuler and his friendDaniel Bernoulliwere opponents ofLeibniz'smonadismand the philosophy ofChristian Wolff. Euler insisted that knowledge is founded in part on the basis of precise quantitative laws, something that monadism and Wolffian science were unable to provide. Euler's religious leanings might also have had a bearing on his dislike of the doctrine; he went so far as to label Wolff's ideas as "heathen and atheistic".Much of what is known of Euler's religious beliefs can be deduced from hisLetters to a German Princessand an earlier work,Rettung der Gttlichen Offenbahrung Gegen die Einwrfe der Freygeister(Defense of the Divine Revelation against the Objections of the Freethinkers). These works show that Euler was a devout Christian who believed the Bible to be inspired; theRettungwas primarily an argument for thedivine inspiration of scripture.There is a famous legendinspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg academy. The French philosopherDenis Diderotwas visiting Russia on Catherine the Great's invitation. However, the Empress was alarmed that the philosopher's arguments foratheismwere influencing members of her court, and so Euler was asked to confront the Frenchman. Diderot was informed that a learned mathematician had produced a proof of theexistence of God: he agreed to view the proof as it was presented in court. Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced thisnon-sequitur: "Sir,, hence God existsreply!" Diderot, to whom (says the story) all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. Embarrassed, he asked to leave Russia, a request that was graciously granted by the Empress. However amusing the anecdote may be, it isapocryphal, given that Diderot himself did research in mathematics.The legend was apparently first told byDieudonn Thibaultwith significant embellishment byAugustus De Morgan.

PersonalityFrom various testimonials of Eulers contemporaries, and also, of course, from Eulers extensive correspondence, one can form a fairly accurate picture of Eulers personality. A valuable source is the eulogy of Niklaus Fuss , who during the last ten years of Eulers life had seen him regularly, almost on a daily basis, as one of his assistants. Also based on personal acquaintance is the eulogy of the marquis Nicolas de Condorcet, which, however, deals more with Eulers work. Euler comes across as a modest, inconspicuous, uncomplicated, yet cheerful and sociable person. He was down-to-earth and upright; honesty and uncompromising rectitude, acknowledged Swiss national virtues, he possessed to a superior degree, writes Fuss. Euler never disavowedin fact was proud ofhis Swiss heritage. Fuss recalled that Euler always retained the Basel dialect with all the peculiarities of its idiom. Often he amused himself to recall for me certain provincialisms and figures of speech, or mix into his parlance Basel expressions whose use and meaning I had long forgotten. He even made sure that he and his children retained the Basel civic rights. Feelings of rancor, due to either priority issues or unfair criticism, were totally foreign to Euler. When Maclaurin, for example, discovered the well-known summation formula which Euler obtained six years earlier, Euler did not object, let aloneComplain, when for some time the formula was generally referred to as the Maclaurin summation formula. It may even have pleased him that others hit upon the same fortunate idea. In due time, of course, the formula became justly known as the EulerMaclaurin summation formula. A case in point is the way he put on hold his already extensive work on hydrodynamics, so that his friend Daniel Bernoulli, who was working on the same topic, could complete and publish his own Hydrodynamics first! It became a classic.An important aspect of Eulers personality is his religiousness: By his upbringing in the Riehen parish environment, he was a devout protestant and even served as an elder in one of the protestant communities in Berlin. Indeed, he felt increasingly uncomfortable and frustrated in the company of so many free-spiritsas he and others called the followers of French enlightenmentthat populated and began to dominate the Berlin Academy

Intellect.There are two outstanding qualities in Eulers intellect: a phenomenal memory, coupled with an unusual power of mental calculation, and an ease in concentrating on mental work irrespective of any hustle and bustle going on around him: A child on the knees, a cat on his back, thats how he wrote his immortal works, recounts Dieudonne Thi ebault, the French linguist and confidant of Frederick II. With regard to memory, the story is well known of Eulers ability. Euler says in a memoir, has given him the first ideas in solving a problem in mechanics. Niklaus Fuss also tells us that during a sleepless night, Euler mentally calculated the first six powers of all the numbers less than twenty (less than 100 in Condorcets account), and several days later was able to recall the answers without hesitation. Euler calculates as other people breathe, Condorcet wrote. Equipped with such intellectual gifts, it is not surprising that Euler was extremely well read. In Fusss words, he possessed to a high degree what commonly is called erudition; he had read the best writers of antique Rome; the older mathematical literature was very well known to him; he was well versed in the history of all times and all people. Even about medical and herbal remedies, and chemistry, he knew more than one could expect from a scholar who doesnt make these sciences a special subject of his study.

CommemorationsEuler was featured on the sixth series of the Swiss 10-francbanknote and on numerous Swiss, German, and Russian postage stamps. Theasteroid2002 Eulerwas named in his honor. He is also commemorated by theLutheran Churchon theirCalendar of Saintson 24 Mayhe was a devout Christian (and believer inbiblical inerrancy) who wroteapologeticsand argued forcefully against the prominent atheists of his time.On 15 April 2013, Euler's 306th birthday was celebrated with aGoogle Doodle

Selected bibliographyEuler has anextensive bibliography. His best-known books include: Mechanica(1736). Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti(1744). The Latin title translates asa method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense.[56] Introductio in analysin infinitorum(1748). English translationIntroduction to Analysis of the Infiniteby John Blanton (Book I,ISBN 0-387-96824-5, Springer-Verlag 1988; Book II, Springer-Verlag 1989). Elements of Algebra(1765). This elementary algebra text starts with a discussion of the nature of numbers and gives a comprehensive introduction to algebra, including formulae for solutions of polynomial equations. Two influential textbooks on calculus:Institutiones calculi differentialis(1755) andInstitutionum calculi integralis(17681770). Letters to a German Princess(17681772).A definitive collection of Euler's works, entitledOpera Omnia, has been published since 1911 by theEuler Commissionof theSwiss Academy of Sciences.