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Oxford English Dictionary 3rd ed. The asteroid Euler was named in his honor.

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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. Of their thirteen children, only five survived childhood. Concerned about the continuing turmoil in Russia, Euler left St.

Petersburg on 19 June 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 articles. In Berlin, he published the two works for which he would become most renowned: Euler wrote over letters to her in the early s, which were later compiled into a best-selling volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess.

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.

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.

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.

Euler's eyesight worsened throughout his mathematical career. In , 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.

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 This suggests that the Eulers may have had a susceptibility to eye problems.

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 roubles—an exorbitant amount at the time. His conditions were quite exorbitant—a 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 cost him his home, and almost his life. In , 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 — Petersburg on 18 September , 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. In his eulogy for the French Academy, French mathematician and philosopher Marquis de Condorcet , wrote:. In , the Russian Academy of Sciences put a marble bust of Leonhard Euler on a pedestal next to the Director's seat and, in , placed a headstone on Euler's grave.

To commemorate the th anniversary of Euler's birth, the headstone was moved in , together with his remains, to the 18th-century necropolis at the Alexander Nevsky Monastery. 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. Euler is the only mathematician to have two numbers named after him: 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.

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 and His daring use of power series enabled him to solve the famous Basel problem in he provided a more elaborate argument in 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.

A special case of the above formula is known as Euler's identity ,. 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. Euler's interest in number theory can be traced to the influence of Christian Goldbach , his friend in the St.

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. 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.

It may have remained the largest known prime until In , Euler presented a solution to the problem known as the Seven Bridges of Königsberg. 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: This solution is considered to be the first theorem of graph theory , specifically of planar graph theory. 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. 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.

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: 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 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 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. 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. 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. Eine solche Überlagerung als Summation für drei verschieden Wellenlängen im Bild, oben ergibt den im unteren Bild dargestellten Intensitätsverlauf als Funktion der Spiegelverschiebung. Dort besitzen alle Wellen die Phasendifferenz Null und überlagern sich deshalb konstruktiv.

Die Intensität ist maximal. Für zunehmende Spiegelwege unterscheiden nehmen die Phasendifferenzen der einzelnen Wellenlängen unterschiedlich stark, sie addieren sich nicht mehr zum Maximalwert. Dieser Mittelwert enthällt keine spektroskopisch relevante Information. Der Mittelwert oder Gleichanteil des Interferogramms wird elektronisch herausgefiltert. Es ergibt sich für die am Detektor einfallende Intensität als Funktion des Spiegelwegs x. Betrachten wir sich die bereits oben angeführte Gleichung zur Berechnung der Interferenzintensität einer polychromatischen Interferenz.

Diese Gleichung entspricht genau der Fouriercosinustransformation. Die inverse Fouriercosinustransformation entspricht bei geraden Funktionen der Fouriercosinustransformation und ist definiert durch. Für gerade Funktionen entspricht die Fouriercosinustransformation genau der komplexen Fouriertransformation, da die nach der EULERschen Formel enthaltenen Sinusanteile nur ungerade Funktionen oder ungerade Anteile einer Funktion repräsentieren, die bei rein geraden Funktioen verschwinden.

Dieses Spektrum entspricht allerdings noch nicht dem Transmissionsspektrum oder Extinktionsspektrum einer Probe. Vielmehr ist dies das Einstrahlspektrum , das die Energieverteilung der Lichtquelle, die Transmissionsfunktion des Spektrometers und die Empfindlichkeit des Detektors beinhaltet. Das Extiktionsspektrum einer Probe berechnet sich durch Aufnahme des Einstrahlspektrums der Referenz auch Background und des Einstrahlspektrums der Probe.

Die Referenzmessung erfolgte an Stickstoff. Obenstehendes Bild zeigt die Interferogramme der Probe und der Referenz. Da das Interferogramm idealerweise eine gerade Funktion ist, sind die Informationen beidseits des Nullpunktes der Spiegelverschiebung identisch.

Es genügt daher, eine Seite des Interferogramms aufzunehmen.