[ The study and generalization of this formula, specifically by Augustin-Louis Cauchy, Cauchy] and Simon Antoine Jean L'Huilier, L'Huilier, is at the origin of topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

.
Physics, astronomy, and engineering
Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series, Euler numbers, the constants and pi, , continued fractions and integrals. He integrated Gottfried Leibniz, Leibniz's differential calculus
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
with Newton's Method of Fluxions, and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. The most notable of these approximations are Euler's method[ and the Euler–Maclaurin formula.][
Euler helped develop the Euler–Bernoulli beam equation, which became a cornerstone of engineering.][ Besides successfully applying his analytic tools to problems in classical mechanics, Euler applied these techniques to celestial problems. His work in astronomy was recognized by multiple Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the solar parallax, parallax of the Sun. His calculations contributed to the development of accurate History of longitude, longitude tables.][
Euler made important contributions in ]optics
Optics is the branch of that studies the behaviour and properties of , including its interactions with and the construction of that use or it. Optics usually describes the behaviour of , , and light. Because light is an , other forms of s ...

.[ He disagreed with Newton's corpuscular theory of light, which was then the prevailing theory. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christiaan Huygens would become the dominant mode of thought, at least until the development of the wave-particle duality, quantum theory of light.]
In fluid dynamics
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...
, Euler was the first to predict the phenomenon of cavitation, in 1754, long before its first observation in the late 19th century, and the Euler number (physics), Euler number used in fluid flow calculations comes from his related work on the efficiency of turbines. In 1757 he published an important set of equations for inviscid flow in fluid dynamics
In physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. " ...
, that are now known as the Euler equations (fluid dynamics), Euler equations.[
Euler is well known in structural engineering for his formula giving Euler's critical load, the critical buckling load of an ideal strut, which depends only on its length and flexural stiffness.
]
Logic
Euler is credited with using closed curves to illustrate syllogism, syllogistic reasoning (1768). These diagrams have become known as Euler diagrams.[
]
An Euler diagram is a diagrammatic means of representing Set (mathematics), sets and their relationships. Euler diagrams consist of simple closed curves (usually circles) in the plane that depict Set (mathematics), sets. Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the element (mathematics), elements of the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important; the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection (set theory), intersection, subset and Disjoint sets, disjointness). Curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection (set theory), intersection of the sets). A curve that is contained completely within the interior zone of another represents a subset of it.
Euler diagrams (and their refinement to Venn diagrams) were incorporated as part of instruction in set theory as part of the new math movement in the 1960s.[ Since then, they have come into wide use as a way of visualizing combinations of characteristics.][
]
Music
One of Euler's more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the ''Tentamen novae theoriae musicae'' (''Attempt at a New Theory of Music''), hoping to eventually incorporate musical theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians. Even when dealing with music, Euler's approach is mainly mathematical,[ including for instance the introduction of binary logarithms as a way of describing numerically the subdivision of octaves into fractional parts.][ His writings on music are not particularly numerous (a few hundred pages, in his total production of about thirty thousand pages), but they reflect an early preoccupation and one that did not leave him throughout his life.][
A first point of Euler's musical theory is the definition of "genres", i.e. of possible divisions of the octave using the prime numbers 3 and 5. Euler describes 18 such genres, with the general definition 2mA, where A is the "exponent" of the genre (i.e. the sum of the exponents of 3 and 5) and 2m (where "m is an indefinite number, small or large, so long as the sounds are perceptible"][), expresses that the relation holds independently of the number of octaves concerned. The first genre, with A = 1, is the octave itself (or its duplicates); the second genre, 2m.3, is the octave divided by the fifth (fifth + fourth, C–G–C); the third genre is 2m.5, major third + minor sixth (C–E–C); the fourth is 2m.32, two-fourths and a tone (C–F–B–C); the fifth is 2m.3.5 (C–E–G–B–C); etc. Genres 12 (2m.33.5), 13 (2m.32.52) and 14 (2m.3.53) are corrected versions of the Genus (music), diatonic, chromatic and enharmonic, respectively, of the Ancients. Genre 18 (2m.33.52) is the "diatonico-chromatic", "used generally in all compositions",][ and which turns out to be identical with the system described by Johann Mattheson.][ Euler later envisaged the possibility of describing genres including the prime number 7.][
Euler devised a specific graph, the ''Speculum musicum'',][ to illustrate the diatonico-chromatic genre, and discussed paths in this graph for specific intervals, recalling his interest in the Seven Bridges of Königsberg (see #Graph theory, above). The device drew renewed interest as the Tonnetz in neo-Riemannian theory (see also Lattice (music)).][
Euler further used the principle of the "exponent" to propose a derivation of the ''gradus suavitatis'' (degree of suavity, of agreeableness) of intervals and chords from their prime factors – one must keep in mind that he considered just intonation, i.e. 1 and the prime numbers 3 and 5 only.][ Formulas have been proposed extending this system to any number of prime numbers, e.g. in the form
where ''p''''i'' are prime numbers and ''k''''i'' their exponents.][
]
Personal philosophy and religious beliefs
Euler opposed the concepts of Gottfried Leibniz, Leibniz's monadism and the philosophy of Christian Wolff (philosopher), Christian Wolff. Euler insisted that knowledge is founded in part on the basis of precise quantitative laws, something that monadism and Wolffian science were unable to provide. Euler's religious leanings might also have had a bearing on his dislike of the doctrine; he went so far as to label Wolff's ideas as "heathen and atheistic".
Euler stayed a religious person throughout his life. Much of what is known of Euler's religious beliefs can be deduced from his ''Letters to a German Princess'' and an earlier work, ''Rettung der Göttlichen Offenbahrung gegen die Einwürfe der Freygeister'' (''Defense of the Divine Revelation against the Objections of the Freethinkers''). These works show that Euler was a devout Christian who believed the Bible to be inspired; the ''Rettung'' was primarily an argument for the Biblical inspiration, divine inspiration of scripture.
There is a famous legend inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg Academy. The French philosopher Denis Diderot was visiting Russia on Catherine the Great's invitation. However, the Empress was alarmed that the philosopher's arguments for atheism were influencing members of her court, and so Euler was asked to confront the Frenchman. Diderot was informed that a learned mathematician had produced a proof of the existence of God: he agreed to view the proof as it was presented in court. Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced this Non sequitur (literary device), non-sequitur: "Sir, =''x'', hence God exists—reply!"
Diderot, to whom (says the story) all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. Embarrassed, he asked to leave Russia, a request that was graciously granted by the Empress. However amusing the anecdote may be, it is wikt:apocryphal, apocryphal, given that Diderot himself did research in mathematics.[
The legend was apparently first told by
Dieudonné Thiébault with embellishment by Augustus De Morgan.]
Commemorations
Euler was featured on both the sixth[ and seventh][ series of the Swiss 10-Swiss franc, franc banknote and on numerous Swiss, German, and Russian postage stamps. In 1782 he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences.] The asteroid 2002 Euler was named in his honour.[
]
Selected bibliography
Euler has an Contributions of Leonhard Euler to mathematics#Works, extensive bibliography. His books include:
* ''Mechanica'' (1736).
''Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti'' (1744)
[ Latin translation: ''a method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense''.][
* '']Introductio in analysin infinitorum
''Introductio in analysin infinitorum'' (Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Throug ...
'' (1748).[ English translation: ''Introduction to Analysis of the Infinite''][
* '']Institutiones calculi differentialis
''Institutiones calculi differentialis'' (''Foundations of differential calculus'') is a mathematical work written in 1748 by Leonhard Euler and published in 1755 that lays the groundwork for the differential calculus. It consists of a single volum ...
'' (1755).[
* ''Vollständige Anleitung zur Algebra'' (1765).
* ''Institutiones calculi integralis'' (1768–1770).][
* ''Letters to a German Princess'' (1768–1772).
* ''Dioptrica'', published in three volumes beginning in 1769.][
It took until 1830 for the bulk of Euler's posthumous works to be individually published,][ with an additional batch of 61 unpublished works discovered by Paul Heinrich von Fuss, Euler's great-grandson and Nicolas Fuss's son, and published as a collection in 1862.][ After several delays in the 19th century,][ a definitive collection of Euler's works, entitled ''Opera Omnia Leonhard Euler, Opera Omnia'', has been published since 1911 by the Euler Commission of the Swiss Academies of Arts and Sciences, Swiss Academy of Sciences. A chronological catalog of Euler's works was compiled by Swedish mathematician Gustaf Eneström and published from 1910 to 1913, and Euler's works are often cited by their number in the Eneström index, from E1 to E866.][ The Euler Archive was started at Dartmouth College][ before moving to the Mathematical Association of America][ and, most recently, to University of the Pacific (United States), University of the Pacific in 2017.Euler Archive]
, University of the Pacific
File:Acta Eruditorum - II geometria, 1744 – BEIC 13411238.jpg, Illustration from ''Solutio problematis... a. 1743 propositi'' published in Acta Eruditorum, 1744
File:Methodus inveniendi - Leonhard Euler - 1744.jpg, The title page of Euler's ''Methodus inveniendi lineas curvas''.
Notes
References
Sources
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Further reading
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External links
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Euler Tercentenary 2007
The Euler Society
Euler's Correspondence with Frederick the Great, King of Prussia
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