Leonhard Euler

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"Euler" redirects here. Leonhard Euler_sentence_0

For other uses, see Euler (disambiguation). Leonhard Euler_sentence_1

Leonhard Euler_table_infobox_0

Leonhard EulerLeonhard Euler_header_cell_0_0_0
BornLeonhard Euler_header_cell_0_1_0 (1707-04-15)15 April 1707

Basel, SwitzerlandLeonhard Euler_cell_0_1_1

DiedLeonhard Euler_header_cell_0_2_0 18 September 1783(1783-09-18) (aged 76)

OS: 7 September 1783] Saint Petersburg, Russian EmpireLeonhard Euler_cell_0_2_1

Alma materLeonhard Euler_header_cell_0_3_0 University of Basel (MPhil)Leonhard Euler_cell_0_3_1
Known forLeonhard Euler_header_cell_0_4_0 See full listLeonhard Euler_cell_0_4_1
Spouse(s)Leonhard Euler_header_cell_0_5_0 Katharina Gsell (1734–1773)

Salome Abigail Gsell (1776–1783)Leonhard Euler_cell_0_5_1

FieldsLeonhard Euler_header_cell_0_6_0 Mathematics and physicsLeonhard Euler_cell_0_6_1
InstitutionsLeonhard Euler_header_cell_0_7_0 Imperial Russian Academy of Sciences

Berlin AcademyLeonhard Euler_cell_0_7_1

ThesisLeonhard Euler_header_cell_0_8_0 (1726)Leonhard Euler_cell_0_8_1
Doctoral advisorLeonhard Euler_header_cell_0_9_0 Johann BernoulliLeonhard Euler_cell_0_9_1
Doctoral studentsLeonhard Euler_header_cell_0_10_0 Johann HennertLeonhard Euler_cell_0_10_1
Other notable studentsLeonhard Euler_header_cell_0_11_0 Nicolas Fuss

Stepan Rumovsky Joseph-Louis Lagrange (epistolary correspondent)Leonhard Euler_cell_0_11_1

SignatureLeonhard Euler_header_cell_0_12_0
NotesLeonhard Euler_header_cell_0_13_0

Leonhard Euler (/ˈɔɪlər/ OY-lər; German: [ˈɔʏlɐ (listen); 15 April 1707 – 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory. Leonhard Euler_sentence_2

He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. Leonhard Euler_sentence_3

He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Leonhard Euler_sentence_4

Euler was one of the most eminent mathematicians of the 18th century and is held to be one of the greatest in history. Leonhard Euler_sentence_5

He is also widely considered to be the most prolific, as his collected works fill 92 volumes, more than anyone else in the field. Leonhard Euler_sentence_6

He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. Leonhard Euler_sentence_7

Amongst his many discoveries and developments, Euler is credited for introducing the Greek letter pi to denominate the Archimedes constant (the ratio of a circle's circumference to its diameter), and for developing a new mathematical constant, the "e" (also known as Euler's Number), which is equivalent to a logarithm's natural base, and has several applications such as to calculate compound interest. Leonhard Euler_sentence_8

A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Leonhard Euler_sentence_9

Early life Leonhard Euler_section_0

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. Leonhard Euler_sentence_10

He had two younger sisters, Anna Maria and Maria Magdalena, and a younger brother, Johann Heinrich. Leonhard Euler_sentence_11

Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Switzerland, where Leonhard spent most of his childhood. Leonhard Euler_sentence_12

Paul was a friend of the Bernoulli family; Johann Bernoulli, then regarded as Europe's foremost mathematician, would eventually be the most important influence on young Leonhard. Leonhard Euler_sentence_13

Euler's formal education started in Basel, where he was sent to live with his maternal grandmother. Leonhard Euler_sentence_14

In 1720, at age thirteen, he enrolled at the University of Basel. Leonhard Euler_sentence_15

In 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. Leonhard Euler_sentence_16

During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. Leonhard Euler_sentence_17

At that time Euler's main studies included theology, Greek and Hebrew at his father's urging to become a pastor, but Bernoulli convinced his father that Leonhard was destined to become a great mathematician. Leonhard Euler_sentence_18

In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. Leonhard Euler_sentence_19

At that time, he was unsuccessfully attempting to obtain a position at the University of Basel. Leonhard Euler_sentence_20

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. Leonhard Euler_sentence_21

Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place. Leonhard Euler_sentence_22

Euler later won this annual prize twelve times. Leonhard Euler_sentence_23

Career Leonhard Euler_section_1

Saint Petersburg Leonhard Euler_section_2

Around this time Johann Bernoulli's two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. Leonhard Euler_sentence_24

On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia. Leonhard Euler_sentence_25

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. Leonhard Euler_sentence_26

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. Leonhard Euler_sentence_27

Euler arrived in Saint Petersburg on 17 May 1727. Leonhard Euler_sentence_28

He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. Leonhard Euler_sentence_29

He lodged with Daniel Bernoulli with whom he often worked in close collaboration. Leonhard Euler_sentence_30

Euler mastered Russian and settled into life in Saint Petersburg. Leonhard Euler_sentence_31

He also took on an additional job as a medic in the Russian Navy. Leonhard Euler_sentence_32

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. Leonhard Euler_sentence_33

As a result, it was made especially attractive to foreign scholars like Euler. Leonhard Euler_sentence_34

The academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Leonhard Euler_sentence_35

Very few students were enrolled in the academy to lessen the faculty's teaching burden. Leonhard Euler_sentence_36

The academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions. Leonhard Euler_sentence_37

The Academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. Leonhard Euler_sentence_38

The Russian nobility then gained power upon the ascension of the twelve-year-old Peter II. Leonhard Euler_sentence_39

The nobility, suspicious of the academy's foreign scientists, cut funding and caused other difficulties for Euler and his colleagues. Leonhard Euler_sentence_40

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. Leonhard Euler_sentence_41

Two years later, Daniel Bernoulli, who was fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Leonhard Euler_sentence_42

Euler succeeded him as the head of the mathematics department. Leonhard Euler_sentence_43

On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell, a painter from the Academy Gymnasium. Leonhard Euler_sentence_44

The young couple bought a house by the Neva River. Leonhard Euler_sentence_45

Of their thirteen children, only five survived childhood. Leonhard Euler_sentence_46

Berlin Leonhard Euler_section_3

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. Leonhard Euler_sentence_47

He lived for 25 years in Berlin, where he wrote over 380 articles. Leonhard Euler_sentence_48

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, published in 1755 on differential calculus. Leonhard Euler_sentence_49

In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences. Leonhard Euler_sentence_50

In addition, Euler was asked to tutor Friederike Charlotte of Brandenburg-Schwedt, the Princess of Anhalt-Dessau and Frederick's niece. Leonhard Euler_sentence_51

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. Leonhard Euler_sentence_52

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. Leonhard Euler_sentence_53

This book became more widely read than any of his mathematical works and was published across Europe and in the United States. Leonhard Euler_sentence_54

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. Leonhard Euler_sentence_55

Despite Euler's immense contribution to the Academy's prestige, he eventually incurred the ire of Frederick and ended up having to leave Berlin. Leonhard Euler_sentence_56

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. Leonhard Euler_sentence_57

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. Leonhard Euler_sentence_58

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. Leonhard Euler_sentence_59

Frederick also expressed disappointment with Euler's practical engineering abilities: Leonhard Euler_sentence_60

Personal life Leonhard Euler_section_4

Eyesight deterioration Leonhard Euler_section_5

Euler's eyesight worsened throughout his mathematical career. Leonhard Euler_sentence_61

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. Leonhard Euler_sentence_62

Euler's vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as "Cyclops". Leonhard Euler_sentence_63

Euler remarked on his loss of vision, "Now I will have fewer distractions." Leonhard Euler_sentence_64

He later developed a cataract in his left eye, which was discovered in 1766. Leonhard Euler_sentence_65

Just a few weeks after its discovery, a failed surgical restoration rendered him almost totally blind. Leonhard Euler_sentence_66

He was 59 years old then. Leonhard Euler_sentence_67

However, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and exceptional memory. Leonhard Euler_sentence_68

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. Leonhard Euler_sentence_69

With the aid of his scribes, Euler's productivity on many areas of study actually increased. Leonhard Euler_sentence_70

He produced, on average, one mathematical paper every week in the year 1775. Leonhard Euler_sentence_71

The Eulers bore a double name, Euler-Schölpi, the latter of which derives from schelb and schief, signifying squint-eyed, cross-eyed, or crooked. Leonhard Euler_sentence_72

This suggests that the Eulers had a susceptibility to eye problems. Leonhard Euler_sentence_73

Return to Russia and death Leonhard Euler_section_6

In 1760, with the Seven Years' War raging, Euler's farm in Charlottenburg was sacked by advancing Russian troops. Leonhard Euler_sentence_74

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. Leonhard Euler_sentence_75

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. Leonhard Euler_sentence_76

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. Leonhard Euler_sentence_77

All of these requests were granted. Leonhard Euler_sentence_78

He spent the rest of his life in Russia. Leonhard Euler_sentence_79

However, his second stay in the country was marred by tragedy. Leonhard Euler_sentence_80

A fire in St. Petersburg in 1771 cost him his home, and almost his life. Leonhard Euler_sentence_81

In 1773, he lost his wife Katharina after 40 years of marriage. Leonhard Euler_sentence_82

Three years after his wife's death, Euler married her half-sister, Salome Abigail Gsell (1723–1794). Leonhard Euler_sentence_83

This marriage lasted until his death. Leonhard Euler_sentence_84

In 1782 he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences. Leonhard Euler_sentence_85

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. Leonhard Euler_sentence_86

He died a few hours later. Leonhard Euler_sentence_87

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, which he delivered at a memorial meeting. Leonhard Euler_sentence_88

In his eulogy for the French Academy, French mathematician and philosopher Marquis de Condorcet, wrote: Leonhard Euler_sentence_89

Euler was buried next to Katharina at the Smolensk Lutheran Cemetery on Goloday Island. Leonhard Euler_sentence_90

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. Leonhard Euler_sentence_91

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. Leonhard Euler_sentence_92

Contributions to mathematics and physics Leonhard Euler_section_7

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. Leonhard Euler_sentence_93

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. Leonhard Euler_sentence_94

Euler's name is associated with a large number of topics. Leonhard Euler_sentence_95

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. Leonhard Euler_sentence_96

It is not known whether γ is rational or irrational. Leonhard Euler_sentence_97

Mathematical notation Leonhard Euler_section_8

Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Leonhard Euler_sentence_98

Most notably, he introduced the concept of a function and was the first to write f(x) to denote the function f applied to the argument x. Leonhard Euler_sentence_99

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. Leonhard Euler_sentence_100

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. Leonhard Euler_sentence_101

Analysis Leonhard Euler_section_9

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. Leonhard Euler_sentence_102

Thanks to their influence, studying calculus became the major focus of Euler's work. Leonhard Euler_sentence_103

While some of Euler's proofs are not acceptable by modern standards of mathematical rigour (in particular his reliance on the principle of the generality of algebra), his ideas led to many great advances. Leonhard Euler_sentence_104

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 Leonhard Euler_sentence_105

Euler directly proved the power series expansions for e and the inverse tangent function. Leonhard Euler_sentence_106

(Indirect proof via the inverse power series technique was given by Newton and Leibniz between 1670 and 1680.) Leonhard Euler_sentence_107

His daring use of power series enabled him to solve the famous Basel problem in 1735 (he provided a more elaborate argument in 1741): Leonhard Euler_sentence_108

Euler introduced the use of the exponential function and logarithms in analytic proofs. Leonhard Euler_sentence_109

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. Leonhard Euler_sentence_110

He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions. Leonhard Euler_sentence_111

For any real number φ (taken to be radians), Euler's formula states that the complex exponential function satisfies Leonhard Euler_sentence_112

A special case of the above formula is known as Euler's identity, Leonhard Euler_sentence_113

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. Leonhard Euler_sentence_114

In 1988, readers of the Mathematical Intelligencer voted it "the Most Beautiful Mathematical Formula Ever". Leonhard Euler_sentence_115

In total, Euler was responsible for three of the top five formulae in that poll. Leonhard Euler_sentence_116

De Moivre's formula is a direct consequence of Euler's formula. Leonhard Euler_sentence_117

Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. Leonhard Euler_sentence_118

He found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis. Leonhard Euler_sentence_119

He invented the calculus of variations including its best-known result, the Euler–Lagrange equation. Leonhard Euler_sentence_120

Euler pioneered the use of analytic methods to solve number theory problems. Leonhard Euler_sentence_121

In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. Leonhard Euler_sentence_122

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. Leonhard Euler_sentence_123

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. Leonhard Euler_sentence_124

Euler's work in this area led to the development of the prime number theorem. Leonhard Euler_sentence_125

Number theory Leonhard Euler_section_10

Euler's interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy. Leonhard Euler_sentence_126

A lot of Euler's early work on number theory was based on the works of Pierre de Fermat. Leonhard Euler_sentence_127

Euler developed some of Fermat's ideas and disproved some of his conjectures. Leonhard Euler_sentence_128

Euler linked the nature of prime distribution with ideas in analysis. Leonhard Euler_sentence_129

He proved that the sum of the reciprocals of the primes diverges. Leonhard Euler_sentence_130

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. Leonhard Euler_sentence_131

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. Leonhard Euler_sentence_132

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. Leonhard Euler_sentence_133

He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. Leonhard Euler_sentence_134

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. Leonhard Euler_sentence_135

Euler also conjectured the law of quadratic reciprocity. Leonhard Euler_sentence_136

The concept is regarded as a fundamental theorem of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss. Leonhard Euler_sentence_137

By 1772 Euler had proved that 2 − 1 = 2,147,483,647 is a Mersenne prime. Leonhard Euler_sentence_138

It may have remained the largest known prime until 1867. Leonhard Euler_sentence_139

Graph theory Leonhard Euler_section_11

In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg. Leonhard Euler_sentence_140

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. Leonhard Euler_sentence_141

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. Leonhard Euler_sentence_142

It is not possible: there is no Eulerian circuit. Leonhard Euler_sentence_143

This solution is considered to be the first theorem of graph theory, specifically of planar graph theory. Leonhard Euler_sentence_144

Applied mathematics Leonhard Euler_section_12

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. Leonhard Euler_sentence_145

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. Leonhard Euler_sentence_146

He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. Leonhard Euler_sentence_147

The most notable of these approximations are Euler's method and the Euler–Maclaurin formula. Leonhard Euler_sentence_148

He also facilitated the use of differential equations, in particular introducing the Euler–Mascheroni constant: Leonhard Euler_sentence_149

One of Euler's more unusual interests was the application of mathematical ideas in music. Leonhard Euler_sentence_150

In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually incorporate musical theory as part of mathematics. Leonhard Euler_sentence_151

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. Leonhard Euler_sentence_152

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. Leonhard Euler_sentence_153

Physics and astronomy Leonhard Euler_section_13

Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering. Leonhard Euler_sentence_154

Besides successfully applying his analytic tools to problems in classical mechanics, Euler applied these techniques to celestial problems. Leonhard Euler_sentence_155

His work in astronomy was recognized by multiple Paris Academy Prizes over the course of his career. Leonhard Euler_sentence_156

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. Leonhard Euler_sentence_157

His calculations contributed to the development of accurate longitude tables. Leonhard Euler_sentence_158

Euler made important contributions in optics. Leonhard Euler_sentence_159

He disagreed with Newton's corpuscular theory of light in the Opticks, which was then the prevailing theory. Leonhard Euler_sentence_160

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. Leonhard Euler_sentence_161

In 1757 he published an important set of equations for inviscid flow, that are now known as the Euler equations. Leonhard Euler_sentence_162

In differential form, the equations are: Leonhard Euler_sentence_163

where Leonhard Euler_sentence_164

Leonhard Euler_unordered_list_0

  • ρ is the fluid mass density,Leonhard Euler_item_0_0
  • u is the fluid velocity vector, with components u, v, and w,Leonhard Euler_item_0_1
  • E = ρ e + ½ ρ (u + v + w) is the total energy per unit volume, with e being the internal energy per unit mass for the fluid,Leonhard Euler_item_0_2
  • p is the pressure,Leonhard Euler_item_0_3
  • ⊗ denotes the tensor product, andLeonhard Euler_item_0_4
  • 0 being the zero vector.Leonhard Euler_item_0_5

Euler is 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: Leonhard Euler_sentence_165

where Leonhard Euler_sentence_166

Leonhard Euler_unordered_list_1

  • F = maximum or critical force (vertical load on column),Leonhard Euler_item_1_6
  • E = modulus of elasticity,Leonhard Euler_item_1_7
  • I = area moment of inertia,Leonhard Euler_item_1_8
  • L = unsupported length of column,Leonhard Euler_item_1_9
  • K = column effective length factor, whose value depends on the conditions of end support of the column, as follows.Leonhard Euler_item_1_10

Leonhard Euler_description_list_2

  • Leonhard Euler_item_2_11
    • For both ends pinned (hinged, free to rotate), K = 1.0.Leonhard Euler_item_2_12
    • For both ends fixed, K = 0.50.Leonhard Euler_item_2_13
    • For one end fixed and the other end pinned, K = 0.699…Leonhard Euler_item_2_14
    • For one end fixed and the other end free to move laterally, K = 2.0.Leonhard Euler_item_2_15

Leonhard Euler_unordered_list_3

  • K L is the effective length of the column.Leonhard Euler_item_3_16

Logic Leonhard Euler_section_14

Euler is credited with using closed curves to illustrate syllogistic reasoning (1768). Leonhard Euler_sentence_167

These diagrams have become known as Euler diagrams. Leonhard Euler_sentence_168

An Euler diagram is a diagrammatic means of representing sets and their relationships. Leonhard Euler_sentence_169

Euler diagrams consist of simple closed curves (usually circles) in the plane that depict sets. Leonhard Euler_sentence_170

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. Leonhard Euler_sentence_171

The sizes or shapes of the curves are not important; the significance of the diagram is in how they overlap. Leonhard Euler_sentence_172

The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection, subset and disjointness). Leonhard Euler_sentence_173

Curves whose interior zones do not intersect represent disjoint sets. Leonhard Euler_sentence_174

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). Leonhard Euler_sentence_175

A curve that is contained completely within the interior zone of another represents a subset of it. Leonhard Euler_sentence_176

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. Leonhard Euler_sentence_177

Since then, they have also been adopted by other curriculum fields such as reading. Leonhard Euler_sentence_178

Music Leonhard Euler_section_15

Even when dealing with music, Euler's approach is mainly mathematical. Leonhard Euler_sentence_179

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. Leonhard Euler_sentence_180

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. Leonhard Euler_sentence_181

Euler describes 18 such genres, with the general definition 2A, where A is the "exponent" of the genre (i.e. the sum of the exponents of 3 and 5) and 2 (where "m is an indefinite number, small or large, so long as the sounds are perceptible"), expresses that the relation holds independently of the number of octaves concerned. Leonhard Euler_sentence_182

The first genre, with A = 1, is the octave itself (or its duplicates); the second genre, 2.3, is the octave divided by the fifth (fifth + fourth, C–G–C); the third genre is 2.5, major third + minor sixth (C–E–C); the fourth is 2.3, two-fourths and a tone (C–F–B♭–C); the fifth is 2.3.5 (C–E–G–B–C); etc. Leonhard Euler_sentence_183

Genres 12 (2.3.5), 13 (2.3.5) and 14 (2.3.5) are corrected versions of the diatonic, chromatic and enharmonic, respectively, of the Ancients. Leonhard Euler_sentence_184

Genre 18 (2.3.5) is the "diatonico-chromatic", "used generally in all compositions", and which turns out to be identical with the system described by Johann Mattheson. Leonhard Euler_sentence_185

Euler later envisaged the possibility of describing genres including the prime number 7. Leonhard Euler_sentence_186

Euler devised a specific graph, the Speculum musicum, to illustrate the diatonico-chromatic genre, and discussed paths in this graph for specific intervals, recalling his interest in the Seven Bridges of Königsberg (see above). Leonhard Euler_sentence_187

The device drew renewed interest as the Tonnetz in neo-Riemannian theory (see also Lattice (music)). Leonhard Euler_sentence_188

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. Leonhard Euler_sentence_189

Formulas have been proposed extending this system to any number of prime numbers, e.g. in the form Leonhard Euler_sentence_190

Leonhard Euler_description_list_4

  • ds = Σ (kipi – ki) + 1Leonhard Euler_item_4_17

where pi are prime numbers and ki their exponents. Leonhard Euler_sentence_191

Personal philosophy and religious beliefs Leonhard Euler_section_16

Euler and his friend Daniel Bernoulli were opponents of Leibniz's monadism and the philosophy of Christian Wolff. Leonhard Euler_sentence_192

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. Leonhard Euler_sentence_193

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". Leonhard Euler_sentence_194

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). Leonhard Euler_sentence_195

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. Leonhard Euler_sentence_196

There is a famous legend inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg Academy. Leonhard Euler_sentence_197

The French philosopher Denis Diderot was visiting Russia on Catherine the Great's invitation. Leonhard Euler_sentence_198

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. Leonhard Euler_sentence_199

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. Leonhard Euler_sentence_200

Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced this non-sequitur: "Sir, a+b/n=x, hence God exists—reply!" Leonhard Euler_sentence_201

Diderot, to whom (says the story) all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. Leonhard Euler_sentence_202

Embarrassed, he asked to leave Russia, a request that was graciously granted by the Empress. Leonhard Euler_sentence_203

However amusing the anecdote may be, it is , given that Diderot himself did research in mathematics. Leonhard Euler_sentence_204

The legend was apparently first told by Dieudonné Thiébault with embellishment by Augustus De Morgan. Leonhard Euler_sentence_205

Commemorations Leonhard Euler_section_17

Euler was featured on the sixth series of the Swiss 10-franc banknote and on numerous Swiss, German, and Russian postage stamps. Leonhard Euler_sentence_206

The asteroid 2002 Euler was named in his honor. Leonhard Euler_sentence_207

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. Leonhard Euler_sentence_208

Selected bibliography Leonhard Euler_section_18

Euler has an extensive bibliography. Leonhard Euler_sentence_209

His best-known books include: Leonhard Euler_sentence_210

Leonhard Euler_unordered_list_5

  • Mechanica (1736).Leonhard Euler_item_5_18
  • . 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.Leonhard Euler_item_5_19
  • 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).Leonhard Euler_item_5_20
  • Two influential textbooks on calculus: Institutiones calculi differentialis (1755) and Institutionum calculi integralis (1768–1770).Leonhard Euler_item_5_21
  • 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.)Leonhard Euler_item_5_22
  • Letters to a German Princess (1768–1772).Leonhard Euler_item_5_23

The first collection of Euler's work was made by Paul Heinrich von Fuss in 1862. Leonhard Euler_sentence_211

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. Leonhard Euler_sentence_212

A complete chronological list of Euler's works is available at The Eneström Index. Leonhard Euler_sentence_213

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. Leonhard Euler_sentence_214

The Euler Archive was started at Dartmouth College before moving to the Mathematical Association of America and, most recently, to University of the Pacific in 2017. Leonhard Euler_sentence_215

Leonhard Euler_unordered_list_6

  • Leonhard Euler_item_6_24
  • Leonhard Euler_item_6_25

See also Leonhard Euler_section_19

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Leonhard Euler.