Leonhard Euler

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Leonhard Euler
Leonhard Euler.jpg
Portrait by Jakob Emanuel Handmann (1753)
Born(1707-04-15)15 April 1707
Basel, Switzerland
Died18 September 1783(1783-09-18) (aged 76)
[OS: 7 September 1783]
Alma mater University of Basel (MPhil)
Known for Contributions
Spouse(s)Katharina Gsell (1734–1773)
Salome Abigail Gsell (1776–1783)
Scientific career
FieldsMathematics and physics
Institutions Imperial Russian Academy of Sciences
Berlin Academy
Thesis Dissertatio physica de sono (Physical dissertation on sound)  (1726)
Doctoral advisor Johann Bernoulli
Doctoral students Johann Hennert
Other notable students Nicolas Fuss
Stepan Rumovsky
Joseph-Louis Lagrange (epistolary correspondent)
Anders Johan Lexell
Euler's signature.svg
He is the father of the mathematician Johann Euler.
He is listed by an academic genealogy as the equivalent to the doctoral advisor of Joseph Louis Lagrange. [1]

Leonhard Euler ( /ˈɔɪlər/ OY-lər; [2] German: [ˈɔʏlɐ] ( Loudspeaker.svg listen ); [3] 15 April 1707 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the study of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. [4] He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory.


Euler is held to be one of the greatest mathematicians in history. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." [5] [6] Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." [7] Euler is also widely considered to be the most prolific, as his collected works fill 92 volumes, [8] more than anyone else in the field. He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.

Amongst his many discoveries and developments, Euler is credited for, among other things, popularizing the Greek letter π (lowercase pi) to denote Archimedes' constant (the ratio of a circle's circumference to its diameter), as well as first employing the term f(x) to describe a function's y-axis, the letter i to express the imaginary unit equivalent to √-1, and the Greek letter Σ (uppercase sigma) to express summations. He gave the current definition of the constant e, the base of the natural logarithm, still known as Euler's number. [9]

Euler also revolutionized the field of physics by reformulating Newton's classic laws of physics into new laws that could explain the motion of rigid bodies more easily, and made significant contributions to the study of elastic deformations of solid objects.

Early life

Leonhard Euler's Alma mater, the University of Basel 11-11-24-basel-by-ralfr-035.jpg
Leonhard Euler's Alma mater, the University of Basel

Leonhard Euler was born on 15 April 1707, in Basel, Switzerland, to Paul III Euler, a pastor of the Reformed Church, and Marguerite (née 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. [10] [8] Soon after the birth of Leonhard, the Euler family moved from Basel to the town of Riehen, Switzerland, where his father became pastor in the local church and Leonhard spent most of his childhood. [8] Paul was a friend of the Bernoulli family, [11] interested in mathematics and took classes from Jacob Bernoulli. [9] Johann Bernoulli, then regarded as Europe's foremost mathematician, would eventually be an important influence on young Leonhard. [11]

Euler's formal education started in Basel, where he was sent to live with his maternal grandmother. [8] In 1720, at only thirteen years of age, he enrolled at the University of Basel. [8] In 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. [8] During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. [12] [8] 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. [13]

In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono [14] [15] with which he unsuccessfully attempted to obtain a position at the University of Basel. [16] 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, who became known as "the father of naval architecture", won and Euler took second place. Euler later won this annual prize twelve times. [17]


1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler. Euler-USSR-1957-stamp.jpg
1957 Soviet Union stamp commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.

Saint Petersburg

Around this time Johann Bernoulli's two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia. [18] [19] 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. [16] 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. [16]

Euler arrived in Saint Petersburg on 17 May 1727. [16] 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. [20] Euler mastered Russian, settled into life in Saint Petersburg and took on an additional job as a medic in the Russian Navy. [21]

The Academy at Saint 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 to lessen the faculty's teaching burden. The academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions. [17]

The Academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died before Euler's arrival to St.Petersburg. The Russian nobility then gained power upon the ascension of the twelve-year-old Peter II. The nobility, suspicious of the academy's foreign scientists, cut funding and caused other difficulties for Euler and his colleagues. [22]

Conditions improved slightly after the death of Peter II, and Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731 while leaving the Russian Navy, refusing a promotion to a lieutenant. [23] Two years later, Daniel Bernoulli, who was 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. [24]

On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell, a painter from the Academy Gymnasium. [25] The young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood. [26]


Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, in English written as "v - e + f = 2". Euler GDR stamp.jpg
Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. Across the centre it shows his polyhedral formula, in English written as "v  e + f = 2".

Concerned 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. [8] He lived for 25 years in Berlin, where he wrote over 380 articles. In Berlin, he published the two works for which he would become most renowned: the Introductio in analysin infinitorum , a text on functions published in 1748, and the Institutiones calculi differentialis , [27] published in 1755 on differential calculus. [28] In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences [29] and of the French Academy of Sciences. [30] Notable students of Euler in Berlin included Stepan Rumovsky, later considered as the first Russian astronomer. [31] [32] In 1753 he bought a house in Charlottenburg, in which he lived with his family and widowed mother. [33]

In addition, Euler was asked to tutor Friederike Charlotte of Brandenburg-Schwedt, the Princess of Anhalt-Dessau and Frederick's niece. Euler 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 . [34] This work contained Euler's exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler's personality and religious beliefs. This book became more widely read than any of his mathematical works. Translated into multiple languages, it was published across Europe and in the United States. 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. [28]

Despite Euler's immense contribution to the Academy's prestige and having been put forward as a candidate for its presidency by Jean le Rond d'Alembert, Frederick II named himself as its president. [8] After several further misunderstandings Euler decided to leave Berlin in 1766. [8] 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 Voltaire, 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. [28] Frederick also expressed disappointment with Euler's 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 Sanssouci. 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! [35]

Personal life

Eyesight deterioration

Euler's eyesight worsened throughout his mathematical career. In 1738, three years after nearly expiring from fever, [36] he became almost blind in his right eye. Euler rather blamed the painstaking work on cartography he performed for the St. Petersburg Academy for his condition, [37] but the cause of his blindness remains the subject of speculation. [38] Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as "Cyclops". Euler remarked on his loss of vision, "Now I will have fewer distractions." [37] He later developed a cataract in his left eye, which was discovered in 1766. Just a few weeks after its discovery, a failed surgical restoration rendered him almost totally blind. He was 59 years old then. However, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and exceptional 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, Euler's productivity in many areas of study actually increased. [39] He produced, on average, one mathematical paper every week in the year 1775. [30] The Eulers bore a double name, Euler-Schölpi, the latter of which derives from schelb and schief, signifying squint-eyed, cross-eyed, or crooked. This suggests that the Eulers had a susceptibility to eye problems. [40]

Return to Russia and death

In 1760, with the Seven Years' War raging, Euler's farm in Charlottenburg was sacked by advancing Russian troops. [33] Upon learning of this event, General Ivan Petrovich Saltykov paid compensation for the damage caused to Euler's estate, with Empress Elizabeth of Russia later adding a further payment of 4000 roubles—an exorbitant amount at the time. [41] The political situation in Russia stabilized after Catherine the Great's 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. All of these requests were granted. At the university he was assisted by his student Anders Johan Lexell. [42] He 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. [43]

Three years after his wife's death, Euler married her half-sister, Salome Abigail Gsell (1723–1794). [44] This marriage lasted until his death. In 1782 he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences. [45]

Euler's grave at the Alexander Nevsky Monastery Euler Grave at Alexander Nevsky Monastry.jpg
Euler's grave at the Alexander Nevsky Monastery

In St. Petersburg on 18 September 1783, after a lunch with his family, Euler was discussing the newly discovered planet Uranus and its orbit with a fellow academician Lexell, when he collapsed and died from a brain hemorrhage. [38] Jacob von Staehlin  [ de ] wrote a short obituary for the Russian Academy of Sciences and Russian mathematician Nicolas Fuss, one of Euler's disciples, wrote a more detailed eulogy, [26] which he delivered at a memorial meeting. In his eulogy for the French Academy, French mathematician and philosopher Marquis de Condorcet, wrote:

il cessa de calculer et de vivre— ... he ceased to calculate and to live. [46]

Euler was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island. In 1837, the Russian Academy of Sciences 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 Lazarevskoe Cemetery at the Alexander Nevsky Monastery. [47]

Contributions to mathematics and physics

Euler 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. [30] Euler's name is associated with a large number of topics.

Mathematical notation

Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function [4] 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 Euler's number), the Greek letter Σ for summations and the letter i to denote the imaginary unit. [48] The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it originated with Welsh mathematician William Jones. [49]


The development of infinitesimal calculus was at the forefront of 18th-century mathematical research, and the 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 rigour [50] (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, [51] 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): [50]

He introduced the constant

now known as Euler's constant or the Euler–Mascheroni constant, and studied its relationship with the harmonic series, the gamma function, and values of the Riemann zeta function. [52]

A geometric interpretation of Euler's formula Euler's formula.svg
A geometric interpretation of Euler's formula

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. [48] He also defined the exponential function for complex numbers and discovered its relation to the trigonometric functions. For any real number φ (taken to be radians), Euler's formula states that the 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, e, i and π. [53]

Euler elaborated the theory of higher transcendental functions by introducing the gamma function [54] [55] and introduced a new method for solving quartic equations. [56] He found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis. He invented the calculus of variations and formulated the Euler–Lagrange equation for reducing optimization problems in this area to the solution of differential equations.

Euler 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 hypergeometric 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. Euler's work in this area led to the development of the prime number theorem. [57]

Number theory

Euler's interest in number theory can be traced to the influence of Christian Goldbach, [58] his friend in the St. Petersburg Academy. [8] Much of Euler's early work on number theory was based on the works 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. [59]

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. [60]

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. [61] 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. [62] 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, particularly Disquisitiones Arithmeticae . [63] 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. [64]

Graph theory

Map of Konigsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges. Konigsberg bridges.png
Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges.

In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg. [65] The city of Königsberg, Prussia was set on the 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 circuit. This solution is considered to be the first theorem of graph theory, specifically of planar graph theory. [65]

Euler also discovered the formula relating the number of vertices, edges and faces of a convex polyhedron, [66] 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. [67] The study and generalization of this formula, specifically by Cauchy [68] and L'Huilier, [69] is at the origin of topology. [66]

Physics, astronomy, and engineering

Some 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 e and π, continued fractions and integrals. He integrated Leibniz's differential calculus 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 [70] and the Euler–Maclaurin formula. [71] [72] [73]

Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering. [74] 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 parallax of the Sun. His calculations contributed to the development of accurate longitude tables. [75]

Euler made important contributions in optics. [76] He disagreed with Newton's corpuscular theory of light, [77] 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 quantum theory of light. [78]

In 1757 he published an important set of equations for inviscid flow in fluid dynamics, that are now known as the Euler equations. [79]

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. [8]


Euler is credited with using closed curves to illustrate syllogistic reasoning (1768). These diagrams have become known as Euler diagrams. [80]

Euler's diagram Euler Diagram.svg
Euler's diagram

An Euler diagram is a diagrammatic means of representing sets and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict sets. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the 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, subset and 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 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. [81] Since then, they have come into wide use as a way of visualizing combinations of characteristics. [82]


One of Euler's 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. [83] Even when dealing with music, Euler's approach is mainly mathematical, [84] including for instance the introduction of binary logarithms as a way of describing numerically the subdivision of octaves into fractional parts. [85] 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. [84]

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" [86] ), 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 diatonic, chromatic and enharmonic, respectively, of the Ancients. Genre 18 (2m.33.52) is the "diatonico-chromatic", "used generally in all compositions", [87] and which turns out to be identical with the system described by Johann Mattheson. [88] Euler later envisaged the possibility of describing genres including the prime number 7. [89]

Euler devised a specific graph, the Speculum musicum, [90] 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 above). The device drew renewed interest as the Tonnetz in neo-Riemannian theory (see also Lattice (music)). [91]

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. [92] Formulas have been proposed extending this system to any number of prime numbers, e.g. in the form

where pi are prime numbers and ki their exponents. [93]

Personal philosophy and religious beliefs

Euler opposed the concepts of Leibniz's monadism and the philosophy of Christian Wolff. [94] 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". [95]

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 divine inspiration of scripture. [96]

There is a famous legend [97] 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: "Sir, a+bn/n=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 apocryphal, given that Diderot himself did research in mathematics. [98] The legend was apparently first told by Dieudonné Thiébault with embellishment by Augustus De Morgan. [97]


Euler portrait on the sixth series of the 10 Franc banknote Euler-10 Swiss Franc banknote (front).jpg
Euler portrait on the sixth series of the 10 Franc banknote
Euler portrait on the seventh series of the 10 Franc banknote CHF10 7 front horizontal.jpg
Euler portrait on the seventh series of the 10 Franc banknote

Euler was featured on both the sixth [99] and seventh [100] series of the Swiss 10-franc banknote and on numerous Swiss, German, and Russian postage stamps. The asteroid 2002 Euler was named in his honour. [101] He is also commemorated by the Lutheran Church on their Calendar of Saints on 24 May—he was a devout Christian (and believer in biblical inerrancy) who wrote apologetics and argued forcefully against the prominent atheists of his time. [96]

Selected bibliography

Euler has an extensive bibliography. His books include:

The first collection of Euler's work was made by Paul Heinrich von Fuss, Euler's great-grandson and Nicolas Fuss's son, in 1862. [108] A definitive collection of Euler's works, entitled Opera Omnia, has been published since 1911 by the Euler Commission of the Swiss Academy of Sciences. [109] A chronological catalog of Euler's works was compiled by Swedish mathematician Gustaf Eneström and published from 1910 to 1913, [110] and Euler's works are often cited by their number in the Eneström index, from E1 to E866. [111] Full text, open access versions of many of Euler's papers are available in the original language and English translations at the Euler Archive, hosted by University of the Pacific. The Euler Archive was started at Dartmouth College [112] before moving to the Mathematical Association of America [113] and, most recently, to University of the Pacific in 2017. [114]

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Jacob Bernoulli was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy. He is known for his numerous contributions to calculus, and along with his brother Johann, was one of the founders of the calculus of variations. He also discovered the fundamental mathematical constant e. However, his most important contribution was in the field of probability, where he derived the first version of the law of large numbers in his work Ars Conjectandi.

Joseph-Louis Lagrange Italian-French mathematician and astronomer

Joseph-Louis Lagrange, also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian mathematician and astronomer, later naturalized French. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.

Jakob Hermann

Jakob Hermann was a mathematician who worked on problems in classical mechanics. He is the author of Phoronomia, an early treatise on Mechanics in Latin, which has been translated by Ian Bruce in 2015-16. In 1729, he proclaimed that it was as easy to graph a locus on the polar coordinate system as it was to graph it on the Cartesian coordinate system.

Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Wilhelm Leibniz independently developed the theory of infinitesimal calculus in the later 17th century. By the end of the 17th century, both Leibniz and Newton claimed that the other had stolen his work, and the Leibniz–Newton calculus controversy continued until the death of Leibniz in 1716.

Anders Johan Lexell

Anders Johan Lexell was a Finnish-Swedish astronomer, mathematician, and physicist who spent most of his life in Imperial Russia, where he was known as Andrei Ivanovich Leksel.

Nicolaus II Bernoulli

Nicolaus II Bernoulli, a.k.a. Niklaus Bernoulli, Nikolaus Bernoulli was a Swiss mathematician as were his father Johann Bernoulli and one of his brothers, Daniel Bernoulli. He was one of the many prominent mathematicians in the Bernoulli family.

Differential equation Mathematical equation involving derivatives of an unknown function

In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

The 18th-century Swiss mathematician Leonhard Euler (1707–1783) is among the most prolific and successful mathematicians in the history of the field. His seminal work had a profound impact in numerous areas of mathematics and he is widely credited for introducing and popularizing modern notation and terminology.

1 + 2 + 3 + 4 + ⋯ Divergent series

The infinite series whose terms are the natural numbers 1 + 2 + 3 + 4 + ⋯ is a divergent series. The nth partial sum of the series is the triangular number

<i>Introductio in analysin infinitorum</i>

Introductio in analysin infinitorum is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin and published in 1748, the Introductio contains 18 chapters in the first part and 22 chapters in the second. It has Eneström numbers E101 and E102.


  1. Leonhard Euler at the Mathematics Genealogy Project
  2. The pronunciation /ˈjuːlər/ is incorrect. See:
  3. However, in the Swiss variety of Standard German with audible /r/: German pronunciation: [ˈoʏlɛr]
  4. 1 2 Dunham 1999, p. 17.
  5. Dunham 1999, p. xiii "Lisez Euler, lisez Euler, c'est notre maître à tous."
  6. The quote appeared in Gugliemo Libri's review of a recently published collection of correspondence among eighteenth-century mathematicians: Libri, Gugliemo (January 1846). "Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIe siècle, ..." [Mathematical and physical correspondence of some famous geometers of the eighteenth century, ...]. Journal des Savants (in French): 51."... nous rappellerions que Laplace lui même, ... ne cessait de répéter aux jeunes mathématiciens ces paroles mémorables que nous avons entendues de sa propre bouche : 'Lisez Euler, lisez Euler, c'est notre maître à tous.' " [... we would recall that Laplace himself, ... never ceased to repeat to young mathematicians these memorable words that we heard from his own mouth: 'Read Euler, read Euler, he is our master in everything.]
  7. Grinstein, Louise; Lipsey, Sally I. (2001). "Euler, Leonhard (1707–1783)". Encyclopedia of Mathematics Education. Routledge. p. 235. ISBN   9780415763684.
  8. 1 2 3 4 5 6 7 8 9 10 11 12 Gautschi, Walter (2008). "Leonhard Euler: His Life, the Man, and His Works". SIAM Review . 50 (1): 3–33. Bibcode:2008SIAMR..50....3G. CiteSeerX . doi:10.1137/070702710. ISSN   0036-1445. JSTOR   20454060.
  9. 1 2 "Leonhard Euler". Encyclopedia Britannica. Retrieved 27 May 2021.
  10. Calinger 2016, p. 11.
  11. 1 2 Calinger 1996, pp. 124–125.
  12. James, Ioan (2002). Remarkable Mathematicians: From Euler to von Neumann . Cambridge University Press. p.  2. ISBN   978-0-521-52094-2.
  13. Calinger 1996, p. 124.
  14. Calinger 2016, p. 32.
  15. Euler, Leonhard (1727). Dissertatio physica de sono [Physical dissertation on sound] (in Latin). Basel: E. and J. R. Thurnisiorum via Euler archive. Translated into English as Bruce, Ian. "Euler's Dissertation De Sono : E002" (PDF). Some Mathematical Works of the 17th & 18th Centuries, including Newton's Principia, Euler's Mechanica, Introductio in Analysin, etc., translated mainly from Latin into English. Retrieved 12 June 2021.
  16. 1 2 3 4 Calinger 1996, p. 125.
  17. 1 2 Calinger 1996, p. 156.
  18. Calinger 1996, pp. 121–166.
  19. O'Connor, John J.; Robertson, Edmund F. "Nicolaus (II) Bernoulli". MacTutor History of Mathematics archive . University of St Andrews.
  20. Calinger 1996, pp. 126–127.
  21. Calinger 1996, p. 127.
  22. Calinger 1996, p. 126.
  23. Calinger 1996, p. 128.
  24. Calinger 1996, pp. 128–29.
  25. Gekker & Euler 2007, p.  402.
  26. 1 2 Fuss, Nicolas (1783). "Éloge de M. Léonhard Euler" [Eulogy for Leonhard Euler]. Nova Acta Academiae Scientiarum Imperialis Petropolitanae (in French). 1: 159–212 via Bioheritage Diversity Library. Translated into English as "Eulogy of Leonhard Euler by Nicolas Fuss". MacTutor History of Mathematics Archive. Translated by Glaus, John S. D. St Andrews University. Retrieved 30 August 2006.
  27. Euler, Leonhard (1787). Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum [Foundations of Differential Calculus, with Applications to Finite Analysis and Series] (in Latin). 1. Petri Galeatii via Euler archive.
  28. 1 2 3 4 Dunham 1999, pp. xxiv–xxv.
  29. Stén, Johan C.-E. (2014). "Academic events in Saint Petersburg". A Comet of the Enlightenment. Vita Mathematica. 17. Birkhäuser. pp. 119–135. doi:10.1007/978-3-319-00618-5_7. See in particular footnote 37, p. 131.
  30. 1 2 3 Finkel, B. F. (1897). "Biography – Leonard Euler". The American Mathematical Monthly . 4 (12): 297–302. doi:10.2307/2968971. JSTOR   2968971. MR   1514436.
  31. Biographical Encyclopedia of Astronomers. Springer. 18 September 2007. p. 992. ISBN   978-0-387-30400-7.
  32. Clark, William; Golinski, Jan; Schaffer, Simon (1 July 1999). The Sciences in Enlightened Europe. University of Chicago Press. p. 395. ISBN   978-0-226-10940-4.
  33. 1 2 Knobloch, Eberhard (2007). ""Leonhard Euler 1707-1783. Zum 300. Geburtstag eines langjährigen Wahlberliners"". Mitteilungen der Deutschen Mathematiker-Vereinigung. 15: 276–288.
  34. Euler, Leonhard (1802). Letters of Euler on Different Subjects of Physics and Philosophy, Addressed to a German Princess. Translated by Hunter, Henry (2nd ed.). London via Internet Archive.
  35. Frederick II of Prussia (1927). Letters of Voltaire and Frederick the Great, Letter H 7434, 25 January 1778. Richard Aldington. New York: Brentano's.
  36. Gautschi 2008, p. 6.
  37. 1 2 Eves, Howard W. (1969). "Euler's blindness". In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes, Quadrants III and IV. Prindle, Weber, & Schmidt. p. 48. Also quoted by Richeson (2012), p. 17, cited to Eves.
  38. 1 2 Asensi, Victor; Asensi, Jose M. (March 2013). "Euler's right eye: the dark side of a bright scientist". Clinical Infectious Diseases . 57 (1): 158–159. doi:10.1093/cid/cit170.
  39. Gautschi 2008, pp. 9–10.
  40. Calinger 2016, p. 8.
  41. Gindikin, Simon (2007). "Leonhard Euler". Tales of Mathematicians and Physicists. Springer. pp. 171–212. doi:10.1007/978-0-387-48811-0_7. ISBN   978-0-387-48811-0. See in particular p. 182.
  42. Maehara, Hiroshi; Martini, Horst (2017). "On Lexell's Theorem". The American Mathematical Monthly. 124 (4): 337–344. doi:10.4169/amer.math.monthly.124.4.337. ISSN   0002-9890.
  43. Thiele, Rüdiger (2005). "The mathematics and science of Leonhard Euler". In Kinyon, Michael; van Brummelen, Glen (eds.). Mathematics and the Historian's Craft: The Kenneth O. May Lectures. Springer. pp. 81–140. ISBN   978-0-387-25284-1.
  44. Gekker & Euler 2007, p.  405.
  45. "E" (PDF). Members of the American Academy of Arts & Sciences, 1780-2017. American Academy of Arts and Sciences. pp. 164–179. Entry for Euler is on p. 177.
  46. Marquis de Condorcet. "Eulogy of Euler – Condorcet" . Retrieved 30 August 2006.
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  84. 1 2 Pesic, Peter (2014). "Euler: the mathematics of musical sadness; Euler: from sound to light". Music and the Making of Modern Science. MIT Press. pp. 133–150, 151–160. ISBN   9780262027274.
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  89. See:
    • Perret, Wilfrid (1926). Some Questions of Musical Theory. Cambridge: W. Heffer & Sons. pp. 60–62.
    • "What is an Euler-Fokker genus?". Microtonality. Hugens-Fokker Foundation. Retrieved 12 June 2015.
  90. Leonhard Euler,Tentamen novae theoriae musicae, St Petersburg, 1739, p. 147; De harmoniae veris principiis, St Petersburg, 1774, p. 350.
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Further reading