Early lifeLeonhard Euler was born on 15 April 1707, in , Switzerland, to Paul III Euler, a pastor of the , and Marguerite ( Brucker), another pastor's daughter. 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 moved from Basel to the town of , Switzerland, where his father became pastor in the local church and Leonhard spent most of his childhood. Paul was a friend of the , interested in mathematics, and took classes from . , then regarded as Europe's foremost mathematician, would eventually be an important influence on young Leonhard. Euler's formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, at only thirteen years of age, he enrolled at the . In 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of and . Afterwards he enrolled in the theological faculty of the University of Basel. He was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered Euler's talent for mathematics. It was during this time that Euler, encouraged by the results of Johann Bernoulli's tutorial, obtained his father's consent to become a mathematician instead of a pastor. In 1726, Euler completed a dissertation on the propagation of sound with the title ''De Sono'' with which he unsuccessfully attempted to obtain a position at the University of Basel. In 1727, he entered the Paris Academy prize competition (offered annually and later biennially by the academy beginning in 1720) for the first time. The problem that year was to find the best way to place the masts on a ship. , who became known as "the father of naval architecture", won and Euler took second place. Euler eventually entered this competition 15 times, winning 12 of them.
Saint PetersburgJohann Bernoulli's two sons, and , entered into service at the Imperial Russian Academy of Sciences in in 1725, leaving Euler with the assurance they'd recommend him to a post when one was available. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia. Retrieved 2 July 2021. When Daniel assumed his brother's 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 Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg in 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 worked in close collaboration. Euler mastered , settled into life in Saint Petersburg and took on an additional job as a medic in the . The Academy at Saint Petersburg, established by , 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's benefactress, , who had continued the progressive policies of her late husband, died before Euler's arrival to Saint Petersburg. The Russian conservative nobility then gained power upon the ascension of the twelve-year-old . The nobility, suspicious of the academy's foreign scientists, cut funding for Euler and his colleagues and prevented the entrance of foreign and non-aristocratic students into the Gymnasium and Universities. Conditions improved slightly after the death of Peter II in 1730 and the German-influenced assumed. Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731. He also left the Russian Navy, refusing a promotion to a . Two years later, Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. In January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell. Frederick II has made an attempt to recruit the services of Euler for his newly established Berlin Academy in 1740, but Euler initially preferred to stay in St Petersburg. But after Emperor Anna died and Frederick II agreed to pay 1600 Ecus (the same as Euler earned in Russia) he agreed to move to Berlin. In 1741, he requested for permission to leave to Berlin, arguing he was in need for a milder climate for his eyesight. The Russian academy gave its consent and would pay him 200 Rubles per year as one of its active members.
BerlinConcerned about the continuing 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 Prussia. He lived for 25 years in , where he wrote several hundred articles. In 1748 his text on functions called the '' '' was published and in 1755 a text on called the '' '' was published. In 1755, he was elected a foreign member of the and of the Notable students of Euler in Berlin included , later considered as the first Russian astronomer. In 1748 he declined an offer from the University of Basel to succeed the recently deceased Johann Bernoulli. In 1753 he bought a house in , in which he lived with his family and widowed mother. Euler became the tutor for Friederike Charlotte of Brandenburg-Schwedt, the Princess of and Frederick's niece. He wrote over 200 letters to her in the early 1760s, which were later compiled into a volume entitled '' Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess''. This work contained Euler's exposition on various subjects pertaining to physics and mathematics and offered valuable insights into Euler's personality and religious beliefs. It was translated into multiple languages, published across Europe and in the United 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 audience, a rare ability for a dedicated research scientist. Despite Euler's immense contribution to the Academy's prestige and having been put forward as a candidate for its presidency by , named himself as its president. The Prussian king had a large circle of intellectuals in his court, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs, in many ways the polar opposite of , who enjoyed a high place of prestige at Frederick's court. Euler was not a skilled debater and often made it a point to argue subjects that he knew little about, making him the frequent target of Voltaire's wit. Frederick also expressed disappointment with Euler's practical engineering abilities, stating: Throughout his stay in Berlin, he maintained a strong connection to the Academy in St. Petersburg and also published 109 papers in Russia. He also assisted students from the Academy in St. Petersburg and at times accommodated Russian students in his house in Berlin. In 1760, with the raging, Euler's farm in Charlottenburg was sacked by advancing Russian troops. Upon learning of this event, General Ivan Petrovich Saltykov paid compensation for the damage caused to Euler's estate, with of Russia later adding a further payment of 4000 roubles—an exorbitant amount at the time. Euler decided to leave Berlin in 1766 and return to Russia.
Return to RussiaThe political situation in Russia stabilized after accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Petersburg Academy. His conditions were quite exorbitant—a 3000 ruble annual salary, a pension for his wife, and the promise of high-ranking appointments for his sons. At the university he was assisted by his student . While living in St. Petersburg, a fire in 1771 destroyed his home.
Personal lifeOn 7 January 1734, he married Katharina Gsell (1707–1773), daughter of Georg Gsell, a painter from the Academy Gymnasium in Saint Petersburg. The young couple bought a house by the . Of their thirteen children, only five survived childhood, three sons and two daughters. Their first son was Johann Albrecht Euler, whose godfather was . Three years after his wife's death in 1773, Euler married her half-sister, Salome Abigail Gsell (1723–1794). This marriage lasted until his death in 1783. His brother Johann Heinrich settled in the St. Petersburg in 1735 and was employed as a painter at the Academy.
Eyesight deteriorationEuler's worsened throughout his mathematical career. In 1738, three years after nearly expiring from fever, he became almost blind in his right eye. Euler blamed the he performed for the St. Petersburg Academy for his condition, but the cause of his blindness remains the subject of speculation. Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as " ". Euler remarked on his loss of vision, stating "Now I will have fewer distractions." In 1766 a in his left eye was discovered, and a few weeks later a failed surgical restoration rendered him almost totally blind. However, his condition appeared to have little effect on his productivity. With the aid of his scribes, Euler's productivity in many areas of study increased and in 1775 he produced, on average, one mathematical paper every week.
DeathIn St. Petersburg on 18 September 1783, after a lunch with his family, Euler was discussing the newly discovered planet and its with Lexell when he collapsed and died from a . wrote a short obituary for the and Russian mathematician , one of Euler's disciples, wrote a more detailed eulogy, which he delivered at a memorial meeting. In his eulogy for the French Academy, French mathematician and philosopher , wrote: Euler was buried next to Katharina at the on . In 1837, the installed a new monument, replacing his overgrown grave plaque. To commemorate the 250th anniversary of Euler's birth in 1957, his tomb was moved to the at the .
Contributions to mathematics and physicsEuler worked in almost all areas of mathematics, such as , , , , and , 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 (text), quarto volumes. Euler's name is associated with a List of topics named after Leonhard Euler, large number of topics.
Mathematical notationEuler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function (mathematics), 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 for the base of the (now also known as Euler's number), the Greek letter Sigma, Σ for summations and the letter to denote the imaginary unit. The use of the Greek letter ''pi (letter), π'' to denote the pi, ratio of a circle's circumference to its diameter was also popularized by Euler, although it originated with Welsh people, Welsh mathematician William Jones (mathematician), William Jones.
AnalysisThe development of was at the forefront of 18th-century mathematical research, and the Bernoulli family, Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigor, mathematical rigour (in particular his reliance on the principle of the generality of algebra), his ideas led to many great advances. Euler is well known in Mathematical analysis, analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as Euler's use of power series enabled him to solve the famous Basel problem in 1735 (he provided a more elaborate argument in 1741): He introduced the constant now known as Euler's constant or the Euler–Mascheroni constant, and studied its relationship with the harmonic series (mathematics), harmonic series, the gamma function, and values of the Riemann zeta function. 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), Euler's formula states that the Exponential function#On the complex plane, complex exponential function satisfies A special case of the above formula is known as Euler's 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, , and . Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. He found a way to calculate integrals with complex limits, foreshadowing the development of modern
Number theoryEuler's interest in number theory can be traced to the influence of , his friend in the St. Petersburg Academy. Much of Euler's early work on number theory was based on the work of Pierre de Fermat. Euler developed some of Fermat's ideas and disproved some of his conjectures, such as his conjecture that all numbers of the form (Fermat numbers) are prime. Euler linked the nature of prime distribution with ideas in analysis. He proved that Proof that the sum of the reciprocals of the primes diverges, 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 Proof of the Euler product formula for the Riemann zeta function, Euler product formula for the Riemann zeta function. Euler invented the totient function φ(''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 Fermat's little theorem to what is now known as Euler's theorem. He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. He proved that the relationship shown between even perfect numbers and Mersenne primes earlier proved by Euclid was one-to-one, a result otherwise known as the Euclid–Euler 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 Euler contributed major developments to the theory of Partition (number theory), partitions of an integer.
Graph theoryIn 1735, Euler presented a solution to the problem known as the . The city of Königsberg, was set on the Pregolya, Pregel 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 path, Eulerian circuit. This solution is considered to be the first theorem of . Euler also discovered the Planar graph#Euler's formula, formula relating the number of vertices, edges and faces of a Convex polytope, convex polyhedron, and hence of a planar graph. The constant in this formula is now known as the for the graph (or other mathematical object), and is related to the genus (mathematics), genus of the object. The study and generalization of this formula, specifically by Augustin-Louis Cauchy, Cauchy and Simon Antoine Jean L'Huilier, L'Huilier, is at the origin of .
Physics, astronomy, and engineeringSome of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series, Euler numbers, the constants and pi, , continued fractions and integrals. He integrated Gottfried Leibniz, Leibniz's with Newton's 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 Euler's method and the Euler–Maclaurin formula. Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering. Besides successfully applying his analytic tools to problems in classical mechanics, Euler applied these techniques to celestial problems. His work in astronomy was recognized by multiple 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 solar parallax, parallax of the Sun. His calculations contributed to the development of accurate History of longitude, longitude tables. Euler made important contributions in He disagreed with Newton's corpuscular theory of light, which was then the prevailing theory. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christiaan Huygens would become the dominant mode of thought, at least until the development of the wave-particle duality, quantum theory of light. In , Euler was the first to predict the phenomenon of cavitation, in 1754, long before its first observation in the late 19th century, and the Euler number (physics), Euler number used in fluid flow calculations comes from his related work on the efficiency of turbines. In 1757 he published an important set of equations for inviscid flow in , that are now known as the Euler equations (fluid dynamics), Euler equations. Euler is well known in structural engineering for his formula giving Euler's critical load, the critical buckling load of an ideal strut, which depends only on its length and flexural stiffness.
LogicEuler is credited with using closed curves to illustrate syllogism, syllogistic reasoning (1768). These diagrams have become known as Euler diagrams. An Euler diagram is a diagrammatic means of representing Set (mathematics), sets and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict Set (mathematics), sets. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the element (mathematics), elements of 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 (set theory), intersection, subset and Disjoint sets, disjointness). Curves whose interior zones do not intersect represent disjoint 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 (the intersection (set theory), intersection of the sets). A curve that is contained completely within the interior zone of another represents a subset of it. Euler diagrams (and their refinement to Venn diagrams) were incorporated as part of instruction in set theory as part of the new math movement in the 1960s. Since then, they have come into wide use as a way of visualizing combinations of characteristics.
MusicOne of Euler's more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the ''Tentamen novae theoriae musicae'' (''Attempt at a New Theory of Music''), 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. Even when dealing with music, Euler's approach is mainly mathematical, including for instance the introduction of binary logarithms as a way of describing numerically the subdivision of octaves into fractional parts. His writings on music are not particularly numerous (a few hundred pages, in his total production of about thirty thousand pages), but they reflect an early preoccupation and one that did not leave him throughout his life. A first point of Euler's musical theory is the definition of "genres", i.e. of possible divisions of the octave using the prime numbers 3 and 5. Euler describes 18 such genres, with the general definition 2mA, where A is the "exponent" of the genre (i.e. the sum of the exponents of 3 and 5) and 2m (where "m is an indefinite number, small or large, so long as the sounds are perceptible"), expresses that the relation holds independently of the number of octaves concerned. The first genre, with A = 1, is the octave itself (or its duplicates); the second genre, 2m.3, is the octave divided by the fifth (fifth + fourth, C–G–C); the third genre is 2m.5, major third + minor sixth (C–E–C); the fourth is 2m.32, two-fourths and a tone (C–F–B–C); the fifth is 2m.3.5 (C–E–G–B–C); etc. Genres 12 (2m.33.5), 13 (2m.32.52) and 14 (2m.3.53) are corrected versions of the Genus (music), diatonic, chromatic and enharmonic, respectively, of the Ancients. Genre 18 (2m.33.52) is the "diatonico-chromatic", "used generally in all compositions", and which turns out to be identical with the system described by Johann Mattheson. Euler later envisaged the possibility of describing genres including the prime number 7. Euler devised a specific graph, the ''Speculum musicum'', to illustrate the diatonico-chromatic genre, and discussed paths in this graph for specific intervals, recalling his interest in the Seven Bridges of Königsberg (see #Graph theory, above). The device drew renewed interest as the Tonnetz in neo-Riemannian theory (see also Lattice (music)). Euler further used the principle of the "exponent" to propose a derivation of the ''gradus suavitatis'' (degree of suavity, of agreeableness) of intervals and chords from their prime factors – one must keep in mind that he considered just intonation, i.e. 1 and the prime numbers 3 and 5 only. Formulas have been proposed extending this system to any number of prime numbers, e.g. in the form where ''p''''i'' are prime numbers and ''k''''i'' their exponents.
Personal philosophy and religious beliefsEuler opposed the concepts of Gottfried Leibniz, Leibniz's monadism and the philosophy of Christian Wolff (philosopher), Christian 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". Euler stayed a religious person throughout his life. Much of what is known of Euler's religious beliefs can be deduced from his ''Letters to a German Princess'' and an earlier work, ''Rettung der Göttlichen Offenbahrung gegen die Einwürfe 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; the ''Rettung'' was primarily an argument for the Biblical inspiration, divine inspiration of scripture. There is a famous legend inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg Academy. The French philosopher Denis Diderot was visiting Russia on Catherine the Great's invitation. However, the Empress was alarmed that the philosopher's arguments for atheism were 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 the existence 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 this Non sequitur (literary device), non-sequitur: "Sir, =''x'', hence God exists—reply!" 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 is wikt:apocryphal, apocryphal, given that Diderot himself did research in mathematics. The legend was apparently first told by Dieudonné Thiébault with embellishment by Augustus De Morgan.
CommemorationsEuler was featured on both the sixth and seventh series of the Swiss 10-Swiss franc, franc banknote and on numerous Swiss, German, and Russian postage stamps. In 1782 he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences. The asteroid 2002 Euler was named in his honour.
Selected bibliographyEuler has an Contributions of Leonhard Euler to mathematics#Works, extensive bibliography. His books include: * ''Mechanica'' (1736).
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