Editing Leonhard Euler (section)

==Contributions to mathematics and physics==
{{main|Contributions of Leonhard Euler to mathematics}}
{{E (mathematical constant)}}
Euler worked in almost all areas of mathematics, including [[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 (text)|quarto]] volumes.<ref name="volumes"/> Euler's name is associated with a [[List of topics named after Leonhard Euler|large number of topics]]. Euler's work averages 800 pages a year from 1725 to 1783. He also wrote over 4500 letters and hundreds of manuscripts. It has been estimated that Leonhard Euler was the author of a quarter of the combined output in mathematics, physics, mechanics, astronomy, and navigation in the 18th century, while other researchers credit Euler for a third of the output in mathematics in that century.<ref name="assad"/>

===Mathematical notation===
Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a [[function (mathematics)|function]]{{sfn|Dunham|1999|p=17}} 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 {{math|''e''}} for the base of the [[natural logarithm]] (now also known as [[Euler's number]]), the Greek letter [[Sigma|Σ]] for summations and the letter {{math|''i''}} to denote the [[imaginary unit]].<ref name=Boyer/> The use of the Greek letter ''[[pi (letter)|π]]'' to denote the [[pi|ratio of a circle's circumference to its diameter]] was also popularized by Euler, although it originated with [[Welsh people|Welsh]] mathematician [[William Jones (mathematician)|William Jones]].<ref name=arndt/>

===Analysis===
The development of [[infinitesimal calculus]] was at the forefront of 18th-century mathematical research, and the [[Bernoulli family|Bernoullis]]—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of [[mathematical rigor|mathematical rigour]]<ref name = "Basel"/> (in particular his reliance on the principle of the [[generality of algebra]]), his ideas led to many great advances.
Euler is well known in [[Mathematical analysis|analysis]] for his frequent use and development of [[power series]], the expression of functions as sums of infinitely many terms,{{sfn|Ferraro|2008|p=155}} such as
<math display=block>e^x = \sum_{n=0}^\infty {x^n \over n!} = \lim_{n \to \infty} \left(\frac{1}{0!} + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}\right).</math>

Euler's use of power series enabled him to solve the [[Basel problem]], finding the sum of the reciprocals of squares of every natural number, in 1735 (he provided a more elaborate argument in 1741). The Basel problem was originally posed by [[Pietro Mengoli]] in 1644, and by the 1730s was a famous open problem, popularized by [[Jacob Bernoulli]] and unsuccessfully attacked by many of the leading mathematicians of the time. Euler found that:<ref name ="Morris PhD thesis ">{{cite thesis |last=Morris |first=Imogen I. |title=Mechanising Euler's use of Infinitesimals in the Proof of the Basel Problem |degree=PhD |publisher=University of Edinburgh | date=24 October 2023 | doi=10.7488/ERA/3835 }}</ref>{{sfn|Dunham|1999}}<ref name="Basel"/>

<math display=block>\sum_{n=1}^\infty {1 \over n^2} = \lim_{n \to \infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}\right) = \frac{\pi ^2}{6}.</math>

Euler introduced the constant
<math display=block>\gamma = \lim_{n \rightarrow \infty } \left( 1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} - \ln(n) \right) \approx 0.5772,</math>
now known as [[Euler's constant]] or the Euler–Mascheroni constant, and studied its relationship with the [[harmonic series (mathematics)|harmonic series]], the [[gamma function]], and values of the [[Riemann zeta function]].<ref name=lagarias/>

[[File:Euler's formula.svg|thumb|A geometric interpretation of [[Euler's formula]]]]

Euler introduced the use of the [[exponential function]] and [[logarithms]] in [[analytic proof]]s. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and [[complex number]]s, thus greatly expanding the scope of mathematical applications of logarithms.<ref name=Boyer/> He also defined the exponential function for complex numbers and discovered its relation to the [[trigonometric function]]s. For any [[real number]] {{math|[[φ]]}} (taken to be radians), [[Euler's formula]] states that the [[Exponential function#On the complex plane|complex exponential]] function satisfies
<math display=block>e^{i\varphi} = \cos \varphi + i\sin \varphi</math>

which was called "the most remarkable formula in mathematics" by [[Richard Feynman]].<ref name="Feynman"/>

A special case of the above formula is known as [[Euler's identity]],
<math display=block>e^{i \pi} +1 = 0 </math>

Euler elaborated the theory of higher [[transcendental function]]s by introducing the [[gamma function]]{{sfn|Ferraro|2008|p=159}}<ref name=davis/> and introduced a new method for solving [[quartic equation]]s.<ref name=nickalls/> He found a way to calculate integrals with complex limits, foreshadowing the development of modern [[complex analysis]]. He invented the [[calculus of variations]] and formulated the [[Euler–Lagrange equation]] for reducing [[Mathematical optimization|optimization problems]] in this area to the solution of [[differential equation]]s.

Euler pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, [[analytic number theory]]. In breaking ground for this new field, Euler created the theory of [[Generalized hypergeometric series|hypergeometric series]], [[q-series]], [[hyperbolic functions|hyperbolic trigonometric functions]], and the analytic theory of [[generalized continued fraction|continued fractions]]. For example, he proved the [[infinitude of primes]] using the divergence of the [[harmonic series (mathematics)|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]].{{sfn|Dunham|1999|loc=Ch. 3, Ch. 4}}

===Number theory===
Euler's interest in number theory can be traced to the influence of [[Christian Goldbach]],{{sfn|Calinger|1996|p=130}} his friend in the St. Petersburg Academy.{{sfn|Gautschi|2008|p=6}} Much of Euler's early work on number theory was based on the work of [[Pierre de Fermat]]. Euler developed some of Fermat's ideas and disproved some of his conjectures, such as his conjecture that all numbers of the form <math display="inline">2^{2^n}+1</math> ([[Fermat numbers]]) are prime.{{sfn|Dunham|1999|p=7}}

Euler linked the nature of prime distribution with ideas in analysis. He proved that [[Proof that the sum of the reciprocals of the primes diverges|the sum of the reciprocals of the primes diverges]]. In doing so, he discovered the connection between the [[Riemann zeta function]] and prime numbers; this is known as the [[Proof of the Euler product formula for the Riemann zeta function|Euler product formula for the Riemann zeta function]].<ref name=patterson/>

Euler invented the [[totient function]] φ(''n''), the number of positive integers less than or equal to the integer ''n'' that are [[coprime]] to ''n''. Using properties of this function, he generalized [[Fermat's little theorem]] to what is now known as [[Euler's theorem]].<ref name=shiu/> He contributed significantly to the theory of [[perfect number]]s, which had fascinated mathematicians since [[Euclid]]. He proved that the relationship shown between even perfect numbers and [[Mersenne prime]]s (which he had earlier proved) was one-to-one, a result otherwise known as the [[Euclid–Euler theorem]].<ref name=stillwell/> Euler also conjectured the law of [[quadratic reciprocity]]. The concept is regarded as a fundamental theorem within number theory, and his ideas paved the way for the work of [[Carl Friedrich Gauss]], particularly ''[[Disquisitiones Arithmeticae]]''.{{sfn|Dunham|1999|loc=Ch. 1, Ch. 4}} By 1772 Euler had proved that 2<sup>31</sup>&nbsp;−&nbsp;1 = [[2147483647|2,147,483,647]] is a Mersenne prime. It may have remained the [[largest known prime]] until 1867.<ref name=caldwell/>

Euler also contributed major developments to the theory of [[partitions of an integer]].<ref name=hopwil/>

===Graph theory===
[[File:Konigsberg bridges.png|frame|right|Map of [[Königsberg]] in Euler's time showing the actual layout of the [[Seven Bridges of Königsberg|seven bridges]], highlighting the river [[Pregolya|Pregel]] and the bridges]]
In 1735, Euler presented a solution to the problem known as the [[Seven Bridges of Königsberg]].<ref name="bridge"/> The city of [[Königsberg]], [[Kingdom of Prussia|Prussia]] was set on the [[Pregolya|Pregel]] River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once. Euler showed that it is not possible: there is no [[Eulerian path]]. This solution is considered to be the first theorem of [[graph theory]].<ref name="bridge"/>

Euler also discovered the [[Planar graph#Euler's formula|formula]] <math>V - E + F = 2</math> relating the number of vertices, edges, and faces of a [[Convex polytope|convex polyhedron]],{{sfn|Richeson|2012}} 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 (mathematics)|genus]] of the object.<ref name=gibbons/> The study and generalization of this formula, specifically by [[Augustin-Louis Cauchy|Cauchy]]<ref name="Cauchy"/> and [[Simon Antoine Jean L'Huilier|L'Huilier]],<ref name="Lhuillier"/> is at the origin of [[topology]].{{sfn|Richeson|2012}}

===Physics, astronomy, and engineering===
{{Classical mechanics|cTopic=Scientists}}
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 number]]s, the constants {{math|[[E (mathematical constant)|e]]}} and [[pi|{{pi}}]], continued fractions, and integrals. He integrated [[Gottfried Leibniz|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]]<ref name=butcher/> and the [[Euler–Maclaurin formula]].{{sfn|Calinger|2016|pp=96, 137}}{{sfn|Ferraro|2008|loc=Chapter 14: Euler's derivation of the Euler–Maclaurin summation formula|pages=171–180}}<ref name=mills/>

Euler helped develop the [[Euler–Bernoulli beam equation]], which became a cornerstone of engineering.<ref name=ojalvo/> Besides successfully applying his analytic tools to problems in [[classical mechanics]], Euler applied these techniques to celestial problems. His work in astronomy was recognized by multiple [[French Academy of Sciences|Paris Academy]] Prizes over the course of his career. His accomplishments include determining with great accuracy the [[orbit]]s of [[comet]]s and other celestial bodies, understanding the nature of comets, and calculating the [[solar parallax|parallax]] of the Sun. His calculations contributed to the development of accurate [[History of longitude|longitude tables]].<ref name=yousch/>

Euler made important contributions in [[optics]].<ref name=davidson/> He disagreed with Newton's [[corpuscular theory of light]],{{sfn|Calinger|1996|pp=152–153}} which was the prevailing theory of the time. His 1740s papers on optics helped ensure that the [[wave theory of light]] proposed by [[Christiaan Huygens]] would become the dominant mode of thought, at least until the development of the [[wave-particle duality|quantum theory of light]].<ref name="optics"/>

In [[fluid dynamics]], Euler was the first to predict the phenomenon of [[cavitation]], in 1754, long before its first observation in the late 19th century, and the [[Euler number (physics)|Euler number]] used in fluid flow calculations comes from his related work on the efficiency of [[turbine]]s.{{r|li}} In 1757 he published an important set of equations for [[inviscid flow]] in [[fluid dynamics]], that are now known as the [[Euler equations (fluid dynamics)|Euler equations]].<ref name=euler2/>

Euler is well known in [[structural engineering]] for his formula giving [[Euler's critical load]], the critical [[buckling]] load of an ideal strut, which depends only on its length and [[Flexural rigidity|flexural stiffness]].{{sfn|Gautschi|2008|p=22}}

===Logic===
Euler is credited with using [[closed curve]]s to illustrate [[syllogism|syllogistic]] reasoning (1768). These diagrams have become known as [[Euler diagram]]s.<ref name=logic/>
[[File:Euler Diagram.svg|thumb|upright|An Euler diagram]]

An Euler diagram is a [[diagram]]matic means of representing [[Set (mathematics)|sets]] and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict [[Set (mathematics)|sets]]. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the [[element (mathematics)|elements]] of the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important; the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships ([[intersection (set theory)|intersection]], [[subset]], and [[Disjoint sets|disjointness]]). Curves whose interior zones do not intersect represent [[disjoint sets]]. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the [[intersection (set theory)|intersection]] of the sets). A curve that is contained completely within the interior zone of another represents a [[subset]] of it.

Euler diagrams (and their refinement to [[Venn diagram]]s) were incorporated as part of instruction in [[set theory]] as part of the [[new math]] movement in the 1960s.<ref name=lemanski/> Since then, they have come into wide use as a way of visualizing combinations of characteristics.<ref name=rodgers/>

===Music===
One of Euler's more unusual interests was the application of [[Music and mathematics|mathematical ideas in music]]. In 1739 he wrote the ''Tentamen novae theoriae musicae'' (''Attempt at a New Theory of Music''), hoping to eventually incorporate [[musical theory]] as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians.{{sfn|Calinger|1996|pp=144–145}} Even when dealing with music, Euler's approach is mainly mathematical,<ref name=pesic/> for instance, his introduction of [[binary logarithm]]s as a way of numerically describing the subdivision of [[octave]]s into fractional parts.<ref name=tegg/> 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 remained with him throughout his life.<ref name=pesic/>

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<sup>m</sup>A, where A is the "exponent" of the genre (i.e. the sum of the exponents of 3 and 5) and 2<sup>m</sup> (where "m is an indefinite number, small or large, so long as the sounds are perceptible"{{sfn|Euler|1739|p=115}}), 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<sup>m</sup>.3, is the octave divided by the fifth (fifth + fourth, C–G–C); the third genre is 2<sup>m</sup>.5, major third + minor sixth (C–E–C); the fourth is 2<sup>m</sup>.3<sup>2</sup>, two-fourths and a tone (C–F–B{{music|b}}–C); the fifth is 2<sup>m</sup>.3.5 (C–E–G–B–C); etc. Genres 12 (2<sup>m</sup>.3<sup>3</sup>.5), 13 (2<sup>m</sup>.3<sup>2</sup>.5<sup>2</sup>) and 14 (2<sup>m</sup>.3.5<sup>3</sup>) are corrected versions of the [[Genus (music)|diatonic, chromatic and enharmonic]], respectively, of the Ancients. Genre 18 (2<sup>m</sup>.3<sup>3</sup>.5<sup>2</sup>) is the "diatonico-chromatic", "used generally in all compositions",<ref name=emery/> and which turns out to be identical with the system described by [[Johann Mattheson]].<ref name=mattheson/> Euler later envisaged the possibility of describing genres including the prime number 7.<ref name=perret/>

Euler devised a specific graph, the ''Speculum musicum'',{{sfn|Euler|1739|p=147}}<ref name="de harmoniae"/> to illustrate the diatonico-chromatic genre, and discussed paths in this graph for specific intervals, recalling his interest in the Seven Bridges of Königsberg (see [[#Graph theory|above]]). The device drew renewed interest as the [[Tonnetz]] in [[Neo-Riemannian theory]] (see also [[Lattice (music)]]).<ref name=gollin/>

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.<ref name=lindley/> Formulas have been proposed extending this system to any number of prime numbers, e.g. in the form
<math display=block>ds=\sum_i(k_ip_i-k_i)+1,</math>
where ''p''<sub>''i''</sub> are prime numbers and ''k''<sub>''i''</sub> their exponents.<ref name=bailhache/>