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Leonhard Euler | |
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Portrait by Jakob Emanuel Handmann (1753) | |

Born | Basel, Switzerland | 15 April 1707

Died | 18 September 1783 76) [OS: 7 September 1783] | (aged

Residence | Kingdom of Prussia Russian Empire Switzerland |

Alma mater | University of Basel (MPhil) |

Known for | See full list |

Scientific career | |

Fields | Mathematics 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) |

Signature | |

Notes | |

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: [ˈɔɪlər] (^{ [3] } He is also known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.^{ [4] }

A **mathematician** is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems.

A **physicist** is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate causes of phenomena, and usually frame their understanding in mathematical terms. Physicists work across a wide range of research fields, spanning all length scales: from sub-atomic and particle physics, through biological physics, to cosmological length scales encompassing the universe as a whole. The field generally includes two types of physicists: experimental physicists who specialize in the observation of physical phenomena and the analysis of experiments, and theoretical physicists who specialize in mathematical modeling of physical systems to rationalize, explain and predict natural phenomena. Physicists can apply their knowledge towards solving practical problems or to developing new technologies.

An **astronomer** is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets, and galaxies – in either observational or theoretical astronomy. Examples of topics or fields astronomers study include planetary science, solar astronomy, the origin or evolution of stars, or the formation of galaxies. Related but distinct subjects like physical cosmology, which studies the Universe as a whole.

- Life
- Early years
- Saint Petersburg
- Berlin
- Eyesight deterioration
- Return to Russia and death
- Contributions to mathematics and physics
- Mathematical notation
- Analysis
- Number theory
- Graph theory
- Applied mathematics
- Physics and astronomy
- Logic
- Music
- Personal philosophy and religious beliefs
- Commemorations
- Selected bibliography
- See also
- References
- Sources
- Further reading
- External links

Euler was one of the most eminent mathematicians of the 18th century and is held to be one of the greatest in history. He is also widely considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes,^{ [5] } more than anybody in the field. He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.

**Saint Petersburg** is Russia's second-largest city after Moscow, with 5 million inhabitants in 2012, part of the Saint Petersburg agglomeration with a population of 6.2 million (2015). An important Russian port on the Baltic Sea, it has a status of a federal subject.

The **Russian Empire**, also known as **Imperial Russia** or simply **Russia**, was an empire that existed across Eurasia and North America from 1721, following the end of the Great Northern War, until the Republic was proclaimed by the Provisional Government that took power after the February Revolution of 1917.

**Berlin** is the capital and largest city of Germany by both area and population. Its 3,748,148 (2018) inhabitants make it the second most populous city proper of the European Union after London. The city is one of Germany's 16 federal states. It is surrounded by the state of Brandenburg, and contiguous with its capital, Potsdam. The two cities are at the center of the Berlin-Brandenburg capital region, which is, with about six million inhabitants and an area of more than 30,000 km², Germany's third-largest metropolitan region after the Rhine-Ruhr and Rhine-Main regions.

A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."^{ [6] }^{ [7] }

**Pierre-Simon, marquis de Laplace** was a French scholar whose work was important to the development of engineering, mathematics, statistics, physics and astronomy. He summarized and extended the work of his predecessors in his five-volume *Mécanique Céleste* (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the Bayesian interpretation of probability was developed mainly by Laplace.

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, a pastor's daughter. He had two younger sisters: Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich.^{ [8] } Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli family; Johann Bernoulli was then regarded as Europe's foremost mathematician, and would eventually be the most important influence on young Leonhard.

**Basel** is a city in northwestern Switzerland on the river Rhine. Basel is Switzerland's third-most-populous city with about 180,000 inhabitants.

**Riehen** is a municipality in the canton of Basel-Stadt in Switzerland. Together with the city of Basel and Bettingen, Riehen is one of three municipalities in the canton.

The **Bernoulli** family of Basel was a patrician family, notable for having produced eight mathematically gifted academics who, among them, contributed substantially to the development of mathematics and physics during the early modern period. Originally from Antwerp, a branch of the family relocated to Basel in 1620. This branch later became related by marriage to the prominent French academic dynasty, the Curie family, through Johann Bernoulli (1667–1748). While their origin in Antwerp is certain, proposed earlier connections with the Dutch family *Bornouilla* (*Bernoullie*), or with the Castilian family *de Bernuy*, are uncertain.

Euler's formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, and in 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics.^{ [9] } At that time Euler's main studies included theology, Greek, and Hebrew at his father's urging in order to become a pastor, but Bernoulli convinced his father that Leonhard was destined to become a great mathematician.

The **University of Basel** is located in Basel, Switzerland. Founded on 4 April 1460, it is Switzerland's oldest university and among the world's oldest surviving universities. The university is traditionally counted among the leading institutions of higher learning in the country.

**René Descartes** was a French philosopher, mathematician, and scientist. A native of the Kingdom of France, he spent about 20 years (1629–1649) of his life in the Dutch Republic after serving for a while in the Dutch States Army of Maurice of Nassau, Prince of Orange and the Stadtholder of the United Provinces. He is generally considered one of the most notable intellectual figures of the Dutch Golden Age.

**Sir Isaac Newton** was an English mathematician, physicist, astronomer, theologian, and author who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution. His book *Philosophiæ Naturalis Principia Mathematica*, first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

In 1726, Euler completed a dissertation on the propagation of sound with the title *De Sono*.^{ [10] } At that time, he was unsuccessfully attempting 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, who became known as "the father of naval architecture", won and Euler took second place. Euler later won this annual prize twelve times.^{ [11] }

The **speed of sound** is the distance travelled per unit time by a sound wave as it propagates through an elastic medium. At 20 °C (68 °F), the speed of sound in air is about 343 meters per second, or a kilometre in 2.9 s or a mile in 4.7 s. It depends strongly on temperature, but also varies by several meters per second, depending on which gases exist in the medium through which a soundwave is propagating.

The **French Academy of Sciences** is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at the forefront of scientific developments in Europe in the 17th and 18th centuries, and is one of the earliest Academies of Sciences.

The **mast** of a sailing vessel is a tall spar, or arrangement of spars, erected more or less vertically on the centre-line of a ship or boat. Its purposes include carrying sail, spars, and derricks, and giving necessary height to a navigation light, look-out position, signal yard, control position, radio aerial or signal lamp. Large ships have several masts, with the size and configuration depending on the style of ship. Nearly all sailing masts are guyed.

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,^{ [12] }^{ [13] } and 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.^{ [14] }

Euler arrived in Saint Petersburg 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 Saint Petersburg. He also took on an additional job as a medic in the Russian Navy.^{ [15] }

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 in order to lessen the faculty's teaching burden, and the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions.^{ [11] }

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

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. 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.^{ [16] }

On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell, a painter from the Academy Gymnasium.^{ [17] } The young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood.^{ [18] }

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 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 *,^{ [19] } published in 1755 on differential calculus.^{ [20] } 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 Frederick's 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 *.^{ [21] } 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 and 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.^{ [20] }

Despite Euler's immense contribution to the Academy's prestige, he eventually incurred the ire of Frederick and ended up having to leave Berlin. 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.^{ [20] } 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!

^{ [22] }

Euler's eyesight worsened throughout his mathematical career. In 1738, three years after nearly expiring from fever, 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. 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."^{ [24] } He 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 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 on many areas of study actually increased. He produced, on average, one mathematical paper every week in the year 1775.^{ [5] } 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 may have had a susceptibility to eye problems.^{ [25] }

In 1760, with the Seven Years' War raging, Euler's farm in Charlottenburg was ransacked by advancing Russian troops. Upon learning of this event, General Ivan Petrovich Saltykov paid compensation for the damage caused to Euler's estate, later Empress Elizabeth of Russia added a further payment of 4000 roubles—an exorbitant amount at the time.^{ [26] } 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. 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.

Three years after his wife's death, Euler married her half-sister, Salome Abigail Gsell (1723–1794).^{ [27] } This marriage lasted until his death. In 1782 he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences.^{ [28] }

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 Anders Johan Lexell, when he collapsed from a brain hemorrhage. He died a few hours later.^{ [29] } Jacob von Staehlin-Storcksburg 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,^{ [30] } 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.^{ [31] }

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

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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.^{ [5] } Euler's name is associated with a large number of topics.

Euler is the only mathematician to have *two* numbers named after him: the important Euler's number in calculus, *e*, approximately equal to 2.71828, and the Euler–Mascheroni constant γ (gamma) sometimes referred to as just "Euler's constant", approximately equal to 0.57721. It is not known whether γ is rational or irrational.^{ [32] }

Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function ^{ [3] } 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.^{ [33] } 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.^{ [34] }

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 ^{ [35] } (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):^{ [35] }

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.^{ [33] } 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.^{ [36] } In 1988, readers of the * Mathematical Intelligencer * voted it "the Most Beautiful Mathematical Formula Ever".^{ [37] } In total, Euler was responsible for three of the top five formulae in that poll.^{ [37] }

De Moivre's formula is a direct consequence of Euler's formula.

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 Euler–Lagrange 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 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.^{ [38] }

Euler's interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy. A lot 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.

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 Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and he made distinct contributions to Lagrange's four-square theorem. He also 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 Carl Friedrich Gauss.^{ [39] } By 1772 Euler had proved that 2^{31} − 1 = 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867.^{ [40] }

In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg.^{ [41] } 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.^{ [41] }

Euler also discovered the formula relating the number of vertices, edges and faces of a convex polyhedron,^{ [42] } 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.^{ [43] } The study and generalization of this formula, specifically by Cauchy ^{ [44] } and L'Huilier,^{ [45] } is at the origin of topology.

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 and the Euler–Maclaurin formula. He also facilitated the use of differential equations, in particular introducing the Euler–Mascheroni constant:

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.^{ [46] }

In 1911, almost 130 years after Euler's death, Alfred J. Lotka used Euler's work to derive the Euler–Lotka equation for calculating rates of population growth for age-structured populations, a fundamental method that is commonly used in population biology and ecology.

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Euler helped develop the Euler–Bernoulli 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.^{ [47] }

In addition, Euler made important contributions in optics. He disagreed with Newton's corpuscular theory of light in the * Opticks *, 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.^{ [48] }

In 1757 he published an important set of equations for inviscid flow, that are now known as the Euler equations.^{ [49] } In differential form, the equations are:

where

*ρ*is the fluid mass density,is the fluid velocity vector, with components**u***u*,*v*, and*w*,*E*=*ρ e*+ ½*ρ*(*u*^{2}+*v*^{2}+*w*^{2}) is the total energy per unit volume, with*e*being the internal energy per unit mass for the fluid,*p*is the pressure,*⊗*denotes the tensor product, and**0**being the zero vector.

Euler is also well known in structural engineering for his formula giving the critical buckling load of an ideal strut, which depends only on its length and flexural stiffness:^{ [50] }

where

*F*= maximum or critical force (vertical load on column),*E*= modulus of elasticity,*I*= area moment of inertia,*L*= unsupported length of column,*K*= 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),
*K*= 1.0. - For both ends fixed,
*K*= 0.50. - For one end fixed and the other end pinned,
*K*= 0.699… - For one end fixed and the other end free to move laterally,
*K*= 2.0.

- For both ends pinned (hinged, free to rotate),

*K L*is the effective length of the column.

Euler is also credited with using closed curves to illustrate syllogistic reasoning (1768). These diagrams have become known as Euler diagrams.^{ [51] }

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 generalization in 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 also been adopted by other curriculum fields such as reading.^{ [52] }

Even when dealing with music, Euler's approach is mainly mathematical. 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.^{ [53] }

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 2^{m}A, where A is the "exponent" of the genre (i.e. the sum of the exponents of 3 and 5) and 2^{m} (where "m is an indefinite number, small or large, so long as the sounds are perceptible"^{ [54] }), 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, 2^{m}.3, is the octave divided by the fifth (fifth + fourth, C–G–C); the third genre is 2^{m}.5, major third + minor sixth (C–E–C); the fourth is 2^{m}.3^{2}, two fourths and a tone (C–F–B♭–C); the fifth is 2^{m}.3.5 (C–E–G–B–C); etc. Genres 12 (2^{m}.3^{3}.5), 13 (2^{m}.3^{2}.5^{2}) and 14 (2^{m}.3.5^{3}) are corrected versions of the diatonic, chromatic and enharmonic, respectively, of the Ancients. Genre 18 (2^{m}.3^{3}.5^{2}) is the "diatonico-chromatic", "used generally in all compositions",^{ [55] } and which turns out to be identical with the system described by Johann Mattheson.^{ [56] } Euler later envisaged the possibility of describing genres including the prime number 7.^{ [57] }

Euler devised a specific graph, the *Speculum musicum*,^{ [58] } 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)).^{ [59] }

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.^{ [60] } Formulas have been proposed extending this system to any number of prime numbers, e.g. in the form

*ds*= Σ*(k*+ 1_{i}p_{i}– k_{i})

where *p _{i}* are prime numbers and

Euler and his friend Daniel Bernoulli were opponents of Leibniz's monadism and the philosophy of 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".^{ [62] }

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.^{ [63] }

There is a famous legend^{ [64] } 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+b ^{n}*/

Euler was featured on the sixth series of the Swiss 10-franc banknote and on numerous Swiss, German, and Russian postage stamps. The asteroid 2002 Euler was named in his honor. 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.^{ [63] }

- Euler, Leonhard (2015).
*Elements of Algebra*. ISBN 978-1-5089-0118-1. (A translation of Euler's*Vollständige Anleitung zur 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.)

**Euler has an extensive 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 as*a method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense*.^{ [69] } *Introductio in analysin infinitorum*(1748). English translation*Introduction to Analysis of the Infinite*by John Blanton (Book I, ISBN 0-387-96824-5, Springer-Verlag 1988; Book II, ISBN 0-387-97132-7, Springer-Verlag 1989).

- Two influential textbooks on calculus:
*Institutiones calculi differentialis*(1755) and*Institutionum calculi integralis*(1768–1770). *Letters to a German Princess*(1768–1772).

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. A complete chronological list of Euler's works is available at the following page: * The Eneström Index * (PDF).

- Euler's number,
*e*≈ 2.71828, the base of the natural logarithm, also known as*Napier's constant* - Martin Knutzen
- List of things named after Leonhard Euler

In mathematics, the **Bernoulli numbers***B*_{n} are a sequence of rational numbers which occur frequently in number theory. The values of the first 20 Bernoulli numbers are given in the adjacent table. For every even *n* other than 0, *B*_{n} is negative if *n* is divisible by 4 and positive otherwise. For every odd *n* other than 1, *B*_{n} = 0.

**Euler's formula**, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:

The number **e** is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to **2.71828**, and is the limit of (1 + 1/*n*)^{n} as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series

In mathematics, the **Euler–Maclaurin formula** is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence.

In mathematics, **Euler's identity** is the equality

**Daniel Bernoulli** FRS was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics. His name is commemorated in the Bernoulli's principle, a particular example of the conservation of energy, which describes the mathematics of the mechanism underlying the operation of two important technologies of the 20th century: the carburetor and the airplane wing.

In number theory, **Euler's totient function** counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as *φ*(*n*) or *ϕ*(*n*), and may also be called **Euler's phi function**. It can be defined more formally as the number of integers k in the range 1 ≤ *k* ≤ *n* for which the greatest common divisor gcd(*n*, *k*) is equal to 1. The integers k of this form are sometimes referred to as totatives of n.

**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** was an Italian Enlightenment Era mathematician and astronomer. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.

Calculus, known in its early history as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Wilhelm Leibniz independently discovered calculus in the mid-17th century. However, both inventors claimed that the other had stolen his work, and the Leibniz-Newton calculus controversy continued until the end of their lives.

**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**, 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.

In mathematics, the **Leibniz formula for π**, named after Gottfried Leibniz, states that

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.

**Trigonometry** is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. In particular, 3rd-century astronomers first noted that the ratio of the lengths of two sides of a right-angled triangle depends only of one acute angles of the triangle. These dependencies are now called trigonometric functions.

A timeline of **calculus** and **mathematical analysis**.

- ↑ Leonhard Euler at the Mathematics Genealogy Project
- ↑ The pronunciation /ˈjuːlər/ is incorrect. "Euler", Oxford English Dictionary, second edition, Oxford University Press, 1989 "Euler", Merriam–Webster's Online Dictionary, 2009. "Euler, Leonhard", The American Heritage Dictionary of the English Language, fifth edition, Houghton Mifflin Company, Boston, 2011. Peter M. Higgins (2007).
*Nets, Puzzles, and Postmen: An Exploration of Mathematical Connections*. Oxford University Press. p. 43. - 1 2 Dunham 1999, p. 17
- ↑ Saint Petersburg (1739). "Tentamen novae theoriae musicae ex certissimis harmoniae principiis dilucide expositae".
- 1 2 3 Finkel, B.F. (1897). "Biography – Leonard Euler".
*The American Mathematical Monthly*.**4**(12): 297–302. doi:10.2307/2968971. JSTOR 2968971. - ↑ Dunham 1999, p. xiii "Lisez Euler, lisez Euler, c'est notre maître à tous."
- ↑ The quote appeared in Gugliemo Libri's review of a recently published collection of correspondence among eighteenth-century mathematicians: Gugliemo Libri (January 1846), Book review: "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*, p. 51. From p. 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.) - ↑ Calinger, Ronald S. (2015).
*Leonhard Euler: Mathematical Genius in the Enlightenment*. Princeton University Press. p. 11. ISBN 978-0-691-11927-4. - ↑ James, Ioan (2002).
*Remarkable Mathematicians: From Euler to von Neumann*. Cambridge. p. 2. ISBN 978-0-521-52094-2. - ↑ Ian Bruce. "Euler's Dissertation De Sono : E002. Translated & Annotated" (PDF). 17centurymaths.com. Retrieved 14 September 2011.
- 1 2 Calinger 1996, p. 156
- ↑ Ronald Calinger. "Leonhard Euler: The First St. Petersburg Years (1727–1741)".
*Historia Mathematica*23, 2 (1996), 121–66, read online - ↑ O'Connor, John J.; Robertson, Edmund F. "Nicolaus (II) Bernoulli".
*MacTutor History of Mathematics archive*. University of St Andrews . Retrieved 2016-01-24. - ↑ Calinger 1996, p. 125
- ↑ Calinger 1996, p. 127
- ↑ Calinger 1996, pp. 128–29
- ↑ Gekker & Euler 2007, p. 402
- ↑ Fuss, Nicolas. "Eulogy of Euler by Fuss" . Retrieved 30 August 2006.
- ↑ "E212 – Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum". Dartmouth.
- 1 2 3 Dunham 1999, pp. xxiv–xxv
- ↑ Euler, Leonhard. "Letters to a German Princess on Diverse Subjects of Natural Philosophy". Internet Archive, Digitzed by Google. Retrieved 15 April 2013.
- ↑ Frederick II of Prussia (1927).
*Letters of Voltaire and Frederick the Great, Letter H 7434, 25 January 1778*. Richard Aldington. New York: Brentano's. - ↑ Calinger 1996, pp. 154–55
- ↑ David S. Richeson (2012).
*Euler's Gem: The Polyhedron Formula and the Birth of Topology*. Princeton University Press. p. 17. ISBN 978-1-4008-3856-1. Quoted from Howard W. Eves (1969).*In Mathematical Circles: A Selection of Mathematical Stories and Anecdotes*. Prindle, Weber, & Schmidt. p. 48. - ↑ Calinger, Ronald (2016).
*Leonhard Euler mathematical genius in the Enlightenment*. Princeton University Press. p. 8. ISBN 978-1-4008-6663-2. - ↑ Gindikin, S.G., Гиндикин С. Г., МЦНМО, НМУ, 2001, с. 217.
- ↑ Gekker & Euler 2007, p. 405
- ↑ Leonhard Euler, in the Book of members of the AAAS.
- ↑ A. Ya. Yakovlev (1983).
*Leonhard Euler*. M.: Prosvesheniye. - ↑ "Eloge de M. Leonhard Euler. Par M. Fuss".
*Nova Acta Academiae Scientiarum Imperialis Petropolitanae*.**1**: 159–212. 1783. - ↑ Marquis de Condorcet. "Eulogy of Euler – Condorcet" . Retrieved 30 August 2006.
- ↑ Derbyshire, John (2003).
*Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics*. Washington, D.C.: Joseph Henry Press. p. 422. - 1 2 Boyer, Carl B.; Merzbach, Uta C. (1991).
*A History of Mathematics*. John Wiley & Sons. pp. 439–45. ISBN 978-0-471-54397-8. - ↑ Wolfram, Stephen. "Mathematical Notation: Past and Future" . Retrieved 23 September 2014.
- 1 2 Wanner, Gerhard; Hairer, Ernst (2005).
*Analysis by its history*(1st ed.). Springer. p. 63. - ↑ Feynman, Richard (1970). "Chapter 22: Algebra".
*The Feynman Lectures on Physics*.**I**. p. 10. - 1 2 Wells, David (1990). "Are these the most beautiful?".
*Mathematical Intelligencer*.**12**(3): 37–41. doi:10.1007/BF03024015.

Wells, David (1988). "Which is the most beautiful?".*Mathematical Intelligencer*.**10**(4): 30–31. doi:10.1007/BF03023741. - ↑ Dunham 1999, Ch. 3, Ch. 4
- ↑ Dunham 1999, Ch. 1, Ch. 4
- ↑ Caldwell, Chris.
*The largest known prime by year* - 1 2 Alexanderson, Gerald (July 2006). "Euler and Königsberg's bridges: a historical view".
*Bulletin of the American Mathematical Society*.**43**(4): 567. Bibcode:1994BAMaS..30..205W. doi:10.1090/S0273-0979-06-01130-X. - ↑ Cromwell, Peter R. (1999).
*Polyhedra*. Cambridge University Press. pp. 189–90. ISBN 978-0-521-66405-9. - ↑ Gibbons, Alan (1985).
*Algorithmic Graph Theory*. Cambridge University Press. p. 72. ISBN 978-0-521-28881-1. - ↑ Cauchy, A.L. (1813). "Recherche sur les polyèdres – premier mémoire".
*Journal de l'École Polytechnique*. 9 (Cahier 16): 66–86. - ↑ L'Huillier, S.-A.-J. (1861). "Mémoire sur la polyèdrométrie".
*Annales de Mathématiques*.**3**: 169–89. - ↑ Calinger 1996, pp. 144–45
- ↑ Youschkevitch, A P (1970–1990).
*Dictionary of Scientific Biography*. New York. - ↑ Home, R.W. (1988). "Leonhard Euler's 'Anti-Newtonian' Theory of Light".
*Annals of Science*.**45**(5): 521–33. doi:10.1080/00033798800200371. - ↑ Euler, Leonhard (1757). "Principes généraux de l'état d'équilibre d'un fluide" [General principles of the state of equilibrium of a fluid](PDF).
*Académie Royale des Sciences et des Belles-Lettres de Berlin, Mémoires*.**11**: 217–73. - ↑ Gautschi, Walter (2008). "Leonhard Euler: His Life, the Man, and His Work" (PDF).
*SIAM Review*.**50**(1): 3–33. Bibcode:2008SIAMR..50....3G. CiteSeerX 10.1.1.177.8766 . doi:10.1137/070702710. - ↑ Baron, M.E. (May 1969). "A Note on The Historical Development of Logic Diagrams".
*The Mathematical Gazette*.**LIII**(383): 113–25. JSTOR 3614533. - ↑ "Strategies for Reading Comprehension Venn Diagrams". Archived from the original on 29 April 2009.
- ↑ Peter Pesic,
*Music and the Making of Modern Science*, p. 133. - ↑ Leonhard Euler,
*Tentamen novae theoriae musicae*, St Petersburg, 1739, p. 115 - ↑ Eric Emery,
*Temps et musique*, Lausanne, L'Âge d'homme, 2000, pp. 344–45. - ↑ Johannes Mattheson,
*Grosse General-Baß-Schule*, Hamburg, 1731, Vol. I, pp. 104–06, mentioned by Euler; and*Exemplarische Organisten-Probe*, Hamburg, 1719, pp. 57–59. - ↑ Wilfrid Perret,
*Some Questions of Musical Theory*, Cambridge, 1926, pp. 60–62; "What is an Euler-Fokker genus?", http://www.huygens-fokker.org/microtonality/efg.html, retrieved 12-6-2015. - ↑ Leonhard Euler,
*Tentamen novae theoriae musicae*, St Petersburg, 1739, p. 147;*De harmoniae veris principiis*, St Petersburg, 1774, p. 350. - ↑ Edward Gollin, "Combinatorial and Transformational Aspects of Euler's
*Speculum Musicum*",*Mathematics and Computation in Music*, T. Klouche and Th. Noll eds, Springer, 2009, pp. 406–11. - ↑ Mark Lindley and Ronald Turner-Smith,
*Mathematical Models of Musical Scales*, Bonn, Verlag für systematische Musikwissenschaft, 1993, pp. 234–39. See also Catherine Nolan, "Music Theory and Mathematics",*The Cambridge History of Western Music Theory*, Th. Christensen ed., New York, CUP, 2002, pp. 278–79. - ↑ Patrice Bailhache, "La Musique traduite en Mathématiques: Leonhard Euler", http://patrice.bailhache.free.fr/thmusique/euler.html, retrieved 12-6-2015.
- ↑ Calinger 1996, pp. 153–54
- 1 2 Euler, Leonhard (1960). Orell-Fussli, ed. "Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister".
*Leonhardi Euleri Opera Omnia (series 3)*.**12**. - ↑ Brown, B.H. (May 1942). "The Euler–Diderot Anecdote".
*The American Mathematical Monthly*.**49**(5): 302–03. doi:10.2307/2303096. JSTOR 2303096.; Gillings, R.J. (February 1954). "The So-Called Euler–Diderot Incident".*The American Mathematical Monthly*.**61**(2): 77–80. doi:10.2307/2307789. JSTOR 2307789. - ↑ Marty, Jacques (1988). "Quelques aspects des travaux de Diderot en Mathematiques Mixtes".
*Recherches Sur Diderot et Sur l'Encyclopédie*.**4**(1): 145–147. - ↑ Brown, B.H. (May 1942). "The Euler–Diderot Anecdote".
*American Mathematical Monthly*.**49**(5): 302–03. doi:10.2307/2303096. JSTOR 2303096. - ↑ Struik, Dirk J. (1967).
*A Concise History of Mathematics*(3rd revised ed.). Dover Books. p. 129. ISBN 978-0-486-60255-4. - ↑ Gillings, R.J. (Feb 1954). "The So-Called Euler-Diderot Anecdote".
*American Mathematical Monthly*.**61**(2): 77–80. doi:10.2307/2307789. JSTOR 2307789. - ↑ E65 – Methodus... entry at Euler Archives. Math.dartmouth.edu. Retrieved on 14 September 2011.

- Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)".
*Historia Mathematica*.**23**(2): 121–66. doi:10.1006/hmat.1996.0015. - Dunham, William (1999).
*Euler: The Master of Us All*. Mathematical Association of America. ISBN 978-0-88385-328-3. - Gekker, I.R.; Euler, A.A. (2007). "Leonhard Euler's family and descendants". In Bogolyubov, Nikolaĭ Nikolaevich; Mikhaĭlov, G.K.; Yushkevich, Adolph Pavlovich.
*Euler and Modern Science*. Translated by Robert Burns. Mathematical Association of America. ISBN 978-0-88385-564-5.

*Lexikon der Naturwissenschaftler*, (2000), Heidelberg: Spektrum Akademischer Verlag.- Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward (2007).
*Euler at 300: An Appreciation*. Mathematical Association of America. ISBN 978-0-88385-565-2. - Demidov, S.S. (2005). "Treatise on the differential calculus". In Grattan-Guinness, Ivor.
*Landmark Writings in Western Mathematics 1640–1940*. Elsevier. pp. 191–98. ISBN 978-0-08-045744-4. - Dunham, William (2007).
*The Genius of Euler: Reflections on his Life and Work*. Mathematical Association of America. ISBN 978-0-88385-558-4. - Fraser, Craig G. (2005-02-11).
*Leonhard Euler's 1744 book on the calculus of variations*. ISBN 978-0-08-045744-4. In Grattan-Guinness 2005, pp. 168–80 - Gautschi, Walter (2008). "Leonhard Euler: his life, the man, and his works" (PDF).
*SIAM Review*.**50**(1): 3–33. Bibcode:2008SIAMR..50....3G. CiteSeerX 10.1.1.177.8766 . doi:10.1137/070702710. - Hascher, Xavier and Papadopoulos, Athanase (editors). 2015.
*Leonhard Euler : Mathématicien, physicien et théoricien de la musique*, Paris, CNRS Editions, 2015, 516 p. ( ISBN 978-2-271-08331-9) - Heimpell, Hermann, Theodor Heuss, Benno Reifenberg (editors). 1956.
*Die großen Deutschen*, volume 2, Berlin: Ullstein Verlag. - Krus, D.J. (November 2001). "Is the normal distribution due to Gauss? Euler, his family of gamma functions, and their place in the history of statistics".
*Quality & Quantity*.**35**(4): 445–46. doi:10.1023/A:1012226622613. Archived from the original on 10 February 2006. - Nahin, Paul J. (2006).
*Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills*. Princeton University Press. ISBN 978-0-691-11822-2. - du Pasquier, Louis-Gustave (2008).
*Leonhard Euler And His Friends*. Translated by John S.D. Glaus. CreateSpace. ISBN 978-1-4348-3327-3. - Reich, Karin (2005-02-11).
*'Introduction' to analysis*. ISBN 978-0-08-045744-4. In Grattan-Guinness 2005, pp. 181–90 - Richeson, David S. (2011).
*Euler's Gem: The Polyhedron Formula and the Birth of Topology*. Princeton University Press. ISBN 978-0-691-12677-7. - Sandifer, C. Edward (2007).
*The Early Mathematics of Leonhard Euler*. Mathematical Association of America. ISBN 978-0-88385-559-1. - Sandifer, C. Edward (2007).
*How Euler Did It*. Mathematical Association of America. ISBN 978-0-88385-563-8. - Simmons, J. (1996).
*The giant book of scientists: The 100 greatest minds of all time*. Sydney: The Book Company. ISBN 978-1-86309-647-8. - Singh, Simon (1997).
*Fermat's Last Theorem*. New York: Fourth Estate. ISBN 978-1-85702-669-6. - Thiele, Rüdiger (2005). "The mathematics and science of Leonhard Euler". In Kinyon, Michael; van Brummelen, Glen.
*Mathematics and the Historian's Craft: The Kenneth O. May Lectures*. Springer. pp. 81–140. ISBN 978-0-387-25284-1. - "A Tribute to Leohnard Euler 1707–1783".
*Mathematics Magazine*.**56**(5). November 1983. - Derbyshire, John (2003).
*Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics*. Washington, DC: John Henry Press. ISBN 978-0-309-08549-6..

Wikisource has the text of the 1911 Encyclopædia Britannica article . Euler, Leonhard |

- LeonhardEuler.com
- Weisstein, Eric Wolfgang (ed.). "Euler, Leonhard (1707–1783)".
*ScienceWorld*. - Encyclopædia Britannica article
- Leonhard Euler at the Mathematics Genealogy Project
- How Euler did it contains columns explaining how Euler solved various problems
- Euler Archive
- Leonhard Euler – Œuvres complètes Gallica-Math
- Euler Committee of the Swiss Academy of Sciences
- References for Leonhard Euler
- Euler Tercentenary 2007
- The Euler Society
- Euler Family Tree
- Euler's Correspondence with Frederick the Great, King of Prussia
- O'Connor, John J.; Robertson, Edmund F., "Leonhard Euler",
*MacTutor History of Mathematics archive*, University of St Andrews . - Euler Quartic Conjecture
- Portrait of Leonhard Euler from the Lick Observatory Records Digital Archive, UC Santa Cruz Library's Digital Collections
- Euler's (1769–1771)
*Dioptricae, 3 vols.*– digital facsimile from the Linda Hall Library - Works by Leonhard Euler at LibriVox (public domain audiobooks)

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