Retellings of the Eulenspiegel tradition have been published in modern literature, since the later 19th century. The Ulenspiegel modern Dutch: Tijl Uilenspiegel from this novel became a symbol of Flemish independence.

According to the chapbook, Eulenspiegel was born in Kneitlingen near Brunswick around The first known chapbook on Eulenspiegel was printed in c.

It is reasonable to place the folkloristic origins of the tradition in the 15th century, although, in spite of often-repeated [ by whom?

There has been a debate surrounding the possible existence of an older Low German edition, now lost. This version has been attributed to Hans Dorn , the only known printer active in Brunswick at the beginning of the 16th century active from at least Sodmann in support of this hypothesis adduced the woodcut of a fool on horseback holding a hand mirror used by Dorn in later prints, as the title illustration of his Liber vagatorum and Grobianus , which may originally have served as the title illustration of the lost Eulenspiegel edition.

There are several suggestions as to the author of the chapbook, none of which has gained mainstream acceptance.

He gives the year as the date when he originally began to collect the tales, stressing the difficulty of the project and how he wanted to abandon it several times, saying that he is now publishing it after all to "lighten the mood in hard times".

The preface also announces the inclusion of material from Pfaff Amis and Pfaff vom Kahlenberg , but no such material is present in the edition.

The literal translation of the High German name "Eulenspiegel" is "owl mirror" hence owle-glasse. It is both innocuous and indicative of his character and has been explained as a garbled form of an expression for "wipe-the-arse": Despite of the name's High German meaning, the book's hero acts out the Plattdeutsch version of his name: Many of Till's pranks are scatological in nature, and involve tricking people into touching, smelling, or even eating Till's excrement.

In modern scholarship, there have been some attempts to find evidence for the historicity of Till Eulenspiegel's person.

In Mölln , the reported site of Eulenspiegel's death and burial in the plague year of , a memorial stone was commissioned by the town council in , now on display in an alcove on the outside wall of the tower of St.

The stone is inscribed in Low German, as follows:. The inscription including the date of was allegedly copied from an older tombstone, now lost.

This older tombstone is referred to in the chapbook of , and it is mentioned as still being extant in Moryson also reports that in his time, the citizen of Mölln held a yearly festival in Eulenspiegel's honour, on which occasion they would present the clothes worn by Eulenspiegel when he died.

The first edition was unknown before sixteen folia of printing proofs were discovered in in the binding of a Latin edition of Reynard.

Only a single specimen of the first edition has been preserved, discovered in A previous owner has replaced the missing pages by pages torn from an Eulenspiegel edition of c.

It was most likely published in The sixteen folia discovered by Honegger are likely printing proofs for this edition.

It consists of folia with 66 woodcuts of high quality. The text is divided into 95 chapters numbered to 96 as chapter number 42 is missing.

The edition is decidedly inferior, missing many of the illustrations of the older edition, and showing signs of careless copying of the text.

The initials of the final six chapters form the "acrostic" ERMANB , which has been taken as a hidden reference to the book's author.

The first chapters are dedicated to Till's childhood. In chapter nine, Till leaves his mother and begins his life as a vagrant.

He takes up diverse occupations, but each chapter ends with his moving on to another place. The final seven chapters are dedicated to his death and burial.

In the chapbooks, Eulenspiegel is presented as a trickster who plays practical jokes on his contemporaries, exposing vices at every turn, greed and folly, hypocrisy and foolishness.

As Peter Carels notes, "The fulcrum of his wit in a large number of the tales is his literal interpretation of figurative language.

The "Antwerp group" of Eulenspiegel editions comprises a number of Flemish, French and English publications. It is only slightly younger than the Strasbourg editions, these books were probably printed from before , first in Antwerp by Jan van Doesborch.

It is possible that the first English translation was printed by van Doesborch as early as In this edition the name Ulenspiegel is rendered Howleglas as it were "owl-glass".

Later English editions, derived from the Antwerp group, were printed by William Copland in London, in and The first modern edition of the chapbook of is by Lappenberg Lappenberg was not yet aware of the existence of the edition.

The and editions were published in facsimile by Insel-Ferlag in and , respectively. An English translation by Paul Oppenheimer was published in Editions of Eulenspiegel in German, Dutch, Flemish, French and English remained popular throughout the early modern period.

In the eighteenth century, German satirists adopted episodes for social satire, and in the nineteenth and early twentieth-century versions of the tales are bowdlerized , to render them fit for children, who had come to be considered their chief natural audience, by expurgating their many scatological references.

The Legend of Thyl Ulenspiegel and Lamme Goedzak , an novel by Belgian author Charles De Coster , has been translated, often in mutilated versions, into many languages.

It was De Coster who first transferred Ulenspiegel from his original late medieval surroundings to the Protestant Reformation. The author gives him a father, Claes, and mother, Soetkin, as well as a girlfriend, Nele, and a best friend, Lamme Goedzak.

In the course of the story Claes is taken prisoner by the Spanish oppressors and burned at the stake, while Soetkin goes insane as a result.

This tempts Thyl to start resistance against the Spanish oppressors. Thanks to the novel, Ulenspiegel has become a symbol of Flemish nationalism , with a statue dedicated to him in Damme.

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 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. 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.

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.

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".

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 divine inspiration of scripture.

There is a famous legend [64] 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: Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced this non-sequitur: 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:. A definitive collection of Euler's works, entitled Opera Omnia , has been published since by the Euler Commission of the Swiss Academy of Sciences.

A complete chronological list of Euler's works is available at the following page: From Wikipedia, the free encyclopedia. For other uses, see Euler disambiguation.

Portrait by Jakob Emanuel Handmann 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.

Map of Königsberg in Euler's time showing the actual layout of the seven bridges , highlighting the river Pregel and the bridges.

Second law of motion. 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.

Circular motion Rotating reference frame Centripetal force Centrifugal force reactive Coriolis force Pendulum Tangential speed Rotational speed.

Mathematics portal Switzerland portal Biography portal. Nets, Puzzles, and Postmen: An Exploration of Mathematical Connections.

The American Mathematical Monthly. Gugliemo Libri January , Book review: Mathematical Genius in the Enlightenment. From Euler to von Neumann.

Retrieved 14 September Petersburg Years — ". MacTutor History of Mathematics archive. University of St Andrews. Retrieved 30 August Internet Archive, Digitzed by Google.

Retrieved 15 April The Polyhedron Formula and the Birth of Topology. Quoted from Howard W. A Selection of Mathematical Stories and Anecdotes.

Leonhard Euler mathematical genius in the Enlightenment. American Academy of Arts and Sciences. Retrieved 28 July A History of Mathematics.

Retrieved 23 September Analysis by its history 1st ed. The Feynman Lectures on Physics. Bulletin of the American Mathematical Society.

Dictionary of Scientific Biography. Archived from the original on 29 April Noll eds, Springer, , pp.

Leonhardi Euleri Opera Omnia series 3. A Concise History of Mathematics 3rd revised ed. Retrieved on 14 September Lexikon der Naturwissenschaftler , , Heidelberg: Euler and Modern Science.

Translated by Robert Burns. Mathematical Association of America. Ronald Calinger, Leonhard Euler: Landmark Writings in Western Mathematics — The Master of Us All.

The Genius of Euler: Reflections on his Life and Work.

His calculations also contributed to the development of accurate longitude tables. Analysis by its history 1st ed. Navigation Hauptseite Themenportale Zufälliger Artikel. Durch die Intensivierung der Landwirtschaft stehen auch casino rama or fallsview Kleinsäuger als Nahrungsquelle Beste Spielothek in Oberrothenbach finden Verfügung. Euler diagrams consist of simple closed curves usually circles in the plane that depict sets. Grupo EULEN is a member of the Global Compact and shows a clear commitment to society through socially responsible policies, cultural patronage and environmental protection. Zu diesem Ruf haben sicherlich der starre, ruhige Blick ihrer Internet casino vergleich beigetragen. His life is set club social casino de badajoz the first half of the 14th century, and the final chapters of the chapbook describe his death from the Beste Spielothek in Ueterlande finden of Dies gilt insbesondere für den Backgammon casinospel - Prova spelet gratis på nätet nu, bei dem Waldkauz und Waldohreule einen nicht unerheblichen Beutebestandteil ausmachen. However, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and exceptional memory. A Selection of Mathematical Stories and Anecdotes. Merkur casino app Eberhard Junkersdorf adapted the story into a feature-length**Aarteet odottavat Xing Guardian -slotissa Casumolla**film. Euler mastered Russian and settled into life in Saint Petersburg.

## Eulen wiki -

Eulen wurden vor dem Versteck des Jägers aufgebaumt und die so angelockten Vögel abgeschossen oder mit Netzen eingefangen. Als auf die nächtliche Jagd spezialisierte Vögel unterscheiden sich Eulen von anderen Vögeln durch spezifische anatomische Merkmale. Der Uhu wurde früher auch häufig als König oder Herrscher der Nacht bezeichnet. Im Alter von 8 Wochen beginnen die Junguhus mit dem Fliegen. Die Brutplätze finden sich vor allem in Felswänden und Steilhängen und in alten Greifvogelhorsten, seltener an Gebäuden oder auf dem Boden. Traditionell gebaute Scheunen und Ställe haben deshalb in vielen Regionen sogenannte Eulentüren oder Eulenlöcher Uhlenloch oder Uhlenflucht , die den Vögeln Zugang zu geeigneten Brutplätzen bieten. Danach führt das Männchen Balzflüge aus, bei denen es mitunter so langsam fliegt, bis es nach unten durchsackt.In Walter van der Kamp directed Uilenspiegel , a Dutch film. Michael Rosen adapted the story into a children's novel, illustrated by Fritz Wegner: In Eberhard Junkersdorf adapted the story into a feature-length animated film.

In Christian Theede directed the film Till Eulenspiegel. Clive Barker incorporated elements of the Till Eulenspiegel story in his play Crazyface. There are three museums in Germany featuring Till Eulenspiegel.

One is located in the town of Schöppenstedt in Lower Saxony, which is nearby his assumed birthplace Kneitlingen.

The second is located in the supposed place of his death, the city of Mölln in Schleswig-Holstein, and the third in Bernburg Saale , Sachsen-Anhalt.

In the town of Damme, Belgium, there is also a museum honoring him, and there is a fountain and statue featuring Till Eulenspiegel in the Marktplatz of Magdeburg, capital city of Sachsen-Anhalt.

From Wikipedia, the free encyclopedia. For other uses, see Till Eulenspiegel disambiguation. Paul Oppenheimer, "Introduction" in: See also Swabian salute.

Two German Folk Heroes", Folklore In support of Bote being the author: Sichtermann, Die Wandlungen des Till Eulenspiegel Schulz-Grobert argues against Bote's authorship, assuming that the acrostic, if genuine, more likely refers to Buschius.

Als man zalt von Crist geburt. Eulenspiegel "stands" rather than lies in his grave due to a mishap during his burial described in the final story of the book.

Archived from the original on 14 November Retrieved 13 November Beiträge zur Forschung und Katalog der Ausstellung vom 6.

Oktober bis By the 17th century it was noted as "often renewed". His Adventures" , Introduction, p. Lexicon of Modern Hebrew Literature in Hebrew.

Retrieved 1 January Jester Wise fool Clowns. Retrieved from " https: Till Eulenspiegel s books Chapbooks German books Fictional tricksters Humor and wit characters German folklore German legends European folklore characters.

Wikipedia articles needing page number citations from March All articles lacking reliable references Articles lacking reliable references from March All articles with unsourced statements Articles with unsourced statements from March Articles needing the year an event occurred from March CS1 Hebrew-language sources he Articles with specifically marked weasel-worded phrases from March Commons category with local link different than on Wikidata Wikipedia articles incorporating a citation from the Encyclopaedia Britannica with Wikisource reference Wikipedia articles with GND identifiers Wikipedia articles with ISNI identifiers Wikipedia articles with VIAF identifiers.

Views Read Edit View history. In other projects Wikimedia Commons. This page was last edited on 7 November , at By using this site, you agree to the Terms of Use and Privacy Policy.

Wikimedia Commons has media related to Ein kurtzweilig lesen von Dyl Vlenspiegel. Gemeinsam mit den Federohren dient der Gesichtsschleier im Feind- und Sozialkontakt auch dazu, Stimmungen auszudrücken, und ist aus diesem Grunde häufig auffällig gefärbt.

Bewegliche Ohrläppchen vor und hinter der Ohröffnung sind mit kurzen, harten Federn ausgestattet und unterstützen die Geräuschortung. Ebenfalls die Geräuschortung unterstützend ist der im Vergleich zu anderen Vogelarten breitere Schädel.

Ein seitliches Geräusch wird dadurch von einem Ohr den Bruchteil einer Sekunde früher wahrgenommen. Der Teil des Gehirns, in dem sich das Gehörzentrum befindet, ist sehr gut entwickelt.

Bei der Schleiereule z. Die Eulen sind jedoch weniger empfindlich für Geräusche mit niedriger Frequenz, hingegen ist die Empfindlichkeit gegenüber hohen Frequenzen sehr gut entwickelt.

Dies ermöglicht Eulen einen geräuscharmen Flug. Dieser wird auch dadurch unterstützt, dass die Flugfedern der meisten Gattungen einen weichen und kammförmigen Rand haben.

Die Ausnahme davon stellen die Fischeulen und Fischuhus dar, die sich auf Fische als Nahrungstiere spezialisiert haben.

Bei den Eigentlichen Eulen ist die nach hinten weisende Innenzehe etwas verkürzt. Die Normalstellung ausgewachsener Eulen ist dabei zygodactyl , also mit zwei nach vorn und zwei nach hinten weisenden Zehen.

Eulenarten sind weltweit mit Ausnahme der Antarktis sowie einzelner Inseln verbreitet. Sie besiedeln fast alle Arten von Lebensräumen, von den trockenen und feuchten Urwäldern über Savannen , Sumpfgebieten und Wäldern bis hin zur Tundra.

Dabei leben die meisten Arten in den tropischen und subtropischen Lebensräumen Südamerikas und Asiens. Das nördlichste Verbreitungsgebiet weist die Schnee-Eule auf, die in der Tundra Nord sibiriens , Nord kanadas und sogar an den Küsten Grönlands anzutreffen ist.

Die meisten Eulen sind nachtaktiv. Sie jagen in der Nacht und schlafen am Tag. Ausnahmen sind beispielsweise die tagaktive Schnee-Eule , die Sperbereule , die Sumpfohreule oder der oft auch am Tag aktive Sperlingskauz.

Eulen sind vor allem auf nachtaktive Beutetiere spezialisiert. Die von den Eulen praktizierte Jagdtechnik ist dabei artspezifisch, von den jeweiligen örtlichen Gegebenheiten geprägt und auch beuteabhängig.

Generell praktizieren die Arten, die im Wald leben, eher eine Ansitzjagd, bei der sie von einer Warte Ausschauplatz aus auf Beute lauern.

Eulen, die offenere Landschaften bewohnen, jagen durch Pirschflüge, bei der sie aus dem Flug heraus ihre Beute erspähen oder hören.

Letzteres gilt beispielsweise für die Schleiereule. Diese greift aber auf die Ansitzjagd zurück, wenn schlechtes Wetter diese Pirschflüge einschränkt.

Für die meisten Eulenarten sind Kleinsäuger wie Mäuse die bevorzugte Beute. Dies gilt insbesondere für den Uhu, bei dem Waldkauz und Waldohreule einen nicht unerheblichen Beutebestandteil ausmachen.

Die meisten Eulenarten besitzen Erkennungsmerkmale, die sie eindeutig als Eule charakterisieren. Andere Vogelarten erkennen darin ihren Fressfeind und reagieren, wenn sie während des Tages Eulen in ihrem Versteck entdecken, mit aggressivem Verhalten.

Eulen wurden vor dem Versteck des Jägers aufgebaumt und die so angelockten Vögel abgeschossen oder mit Netzen eingefangen. Heute ist diese sogenannte Hüttenjagd mit lebenden Lockvögeln verboten.

Fast alle Eulenarten gelten in Deutschland als gefährdet. 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 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. 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.

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.

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".

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