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### Identifying interesting or difficult problems

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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 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.

In , almost 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. 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. 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 s 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. In he published an important set of equations for inviscid flow , that are now known as the Euler equations. 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: [51].

Euler is also credited with using closed curves to illustrate syllogistic reasoning These diagrams have become known as Euler diagrams. 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.

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

A first point of Euler's musical theory is the definition of "genres", i. Euler describes 18 such genres, with the general definition 2 m A, where A is the "exponent" of the genre i. Genres 12 2 m. Genre 18 2 m. 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.

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".

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. There is a famous legend [65] inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. 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. Embarrassed, he asked to leave Russia, a request that was graciously granted by the Empress. However amusing the anecdote may be, it is apocryphal , given that Diderot himself did research in mathematics. Euler was featured on the sixth series of the Swiss franc banknote and on numerous Swiss, German, and Russian postage stamps.

The asteroid 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. Euler has an extensive bibliography. His best-known books include:. The first collection of Euler's work was made by Paul Heinrich von Fuss in Illustration from Solutio problematis From Wikipedia, the free encyclopedia.

Swiss mathematician, physicist, and engineer. For other uses, see Euler disambiguation.

- In mathematical circles; a selection of mathematical stories.
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Portrait by Jakob Emanuel Handmann Basel , Switzerland. Saint Petersburg , Russian Empire. 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. Second law of motion. History Timeline. Newton's laws of motion.

Analytical mechanics Lagrangian mechanics Hamiltonian mechanics Routhian mechanics Hamilton—Jacobi equation Appell's equation of motion Udwadia—Kalaba equation Koopman—von Neumann mechanics. Core topics. Circular motion Rotating reference frame Centripetal force Centrifugal force reactive Coriolis force Pendulum Tangential speed Rotational speed. The title page of Euler's Methodus inveniendi lineas curvas. Mathematics portal Switzerland portal Biography portal. Peter M. Higgins Oxford University Press.

From p. Leonhard Euler: Mathematical Genius in the Enlightenment. Princeton University Press. Remarkable Mathematicians: From Euler to von Neumann. Retrieved 14 September Petersburg Years — ".

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Historia Mathematica. MacTutor History of Mathematics archive. University of St Andrews. Retrieved 24 January Retrieved 30 August Internet Archive, Digitzed by Google. Retrieved 15 April Richard Aldington. New York: Brentano's. Richeson Quoted from Howard W. Eves The American Mathematical Monthly. Leonhard Euler mathematical genius in the Enlightenment. Yakovlev Leonhard Euler. Par M. Washington, D.