Born Leonhard Euler Portrait by Johann Georg Brucker April 15, 1707Basel, Switzerland September 18 [O.S. September 7] 1783St Petersburg, Russia PrussiaRussiaSwitzerland Swiss Mathematician and physicist Imperial Russian Academy of SciencesBerlin Academy University of Basel Johann Bernoulli Johann HennertJoseph Lagrange Euler's number Calvinist[1][2] Signature

Euler made important discoveries in fields as diverse as calculus and graph theory. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. Analysis has its beginnings in the rigorous formulation of Calculus. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function [3] He is also renowned for his work in mechanics, optics, and astronomy. Mechanics ( Greek) is the branch of Physics concerned with the behaviour of physical bodies when subjected to Forces or displacements Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study

Euler is considered to be the preeminent mathematician of the 18th century and one of the greatest of all time. The 18th century lasted from 1701 to 1800 in the Gregorian calendar, in accordance with the Anno Domini / Common Era numbering system He is also one of the most prolific; his collected works fill 60–80 quarto volumes. Bookbinding is the process of physically assembling a Book from a number of folded or unfolded sheets of Paper or other material [4] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master [i. e. , teacher] of us all. "[5]

Euler was featured on the sixth series of the Swiss 10-franc banknote and on numerous Swiss, German, and Russian postage stamps. The franc ( German: Franken, French and Romansh: franc, Italian: franco; code: CHF A postage stamp is an adhesive paper evidence of pre-paying a fee for postal services The asteroid 2002 Euler was named in his honor. Asteroids, sometimes called Minor planets or planetoids', are bodies—primarily of the inner Solar System —that are smaller than planets but 2002 Euler is an Asteroid named after the Swiss mathematician and physicist Leonhard Euler, who was considered to be one of the greatest mathematicians of all time He is also commemorated by the Lutheran Church on their Calendar of Saints on May 24th. Lutheranism is a major branch of Western Christianity that identifies with the teachings of the sixteenth-century German reformer Martin Luther The Lutheran Calendar of Saints is a listing which details the primary annual festivals and events that are celebrated liturgically by the Lutheran Church Events 1218 - The Fifth Crusade leaves Acre for Egypt. 1276 - Magnus Ladulås is crowned

Life

Early years

Old Swiss 10 Franc banknote honoring Euler, the most successful Swiss mathematician in history.

Euler was born in Basel to Paul Euler, a pastor of the Reformed Church, and Marguerite Brucker, a pastor's daughter. "Basilia" redirects here For the Fly Genus, see Basilia (fly. A pastor is an official person within a Protestant group of people and related to the positions of Priest or Bishop within the Anglican, Roman Catholic The Reformed churches are a group of Christian Protestant Denominations formally characterized by a similar Calvinist system of doctrine historically He had two younger sisters named Anna Maria and Maria Magdalena. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Riehen is a municipality in the canton of Basel-City in Switzerland. Paul Euler was a friend of the Bernoulli familyJohann Bernoulli, who was then regarded as Europe's foremost mathematician, would eventually be the most important influence on young Leonhard. The Bernoullis were a family of traders and scholars from Basel, Switzerland. Johann Bernoulli ( Basel, 27 July 1667 - 1 January 1748 was a Swiss Mathematician. A mathematician is a person whose primary area of study and research is the field of Mathematics. Euler's early formal education started in Basel, where he was sent to live with his maternal grandmother. At the age of thirteen he matriculated at the University of Basel, and in 1723, received his M.Phil with a dissertation that compared the philosophies of Descartes and Newton. The University of Basel (German Universität Basel) is located at Basel, Switzerland. In the usage of India, United Kingdom, United States, Australia, New Zealand, Hong Kong and some other countries the Master Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. [6]

Euler was at this point studying theology, Greek, and Hebrew at his father's urging, in order to become a pastor. Theology is the study of a god or the gods from a religious perspective Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Johann Bernoulli intervened, and convinced Paul Euler that Leonhard was destined to become a great mathematician. In 1726, Euler completed his Ph. D. dissertation on the propagation of sound with the title De Sono[7] and in 1727, he entered the Paris Academy Prize Problem competition, where the problem that year was to find the best way to place the masts on a ship. Sound is a vibration that travels through an elastic medium as a Wave. The French Academy of Sciences ( French: Académie des sciences) is a Learned society, founded in 1666 by Louis XIV at the The mast of a sailing ship is a tall vertical or near vertical Spar, or arrangement of Spars which supports the Sails Large ships have several masts He won second place, losing only to Pierre Bouguer—who is now known as "the father of naval architecture". Pierre Bouguer ( February 16, 1698 &ndash August 15, 1758) was a French Mathematician and astronomer Euler subsequently won this coveted annual prize twelve times in his career. [8]

St. Petersburg

Around this time Johann Bernoulli's two sons, Daniel and Nicolas, were working at the Imperial Russian Academy of Sciences in St Petersburg. Daniel Bernoulli ( Groningen, 29 January 1700 &ndash 27 July 1782 was a Dutch - Swiss Mathematician, who is particularly remembered for his applications Nicolaus II Bernoulli, aka Niklaus Bernoulli, Nikolaus Bernoulli, ( February 6 1695, Basel, Switzerland – July The Russian Academy of Sciences (Российская Академия Наук Rossi'iskaya Akade'miya Nau'k, shortened to PAH RAN) consists of the National Saint Petersburg ( tr: Sankt-Peterburg,) is a city and a federal subject of Russia located on the Neva River In July 1726, Nicolas died of appendicitis after spending a year in Russia, and when Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. Appendicitis (or epityphlitis) is a condition characterized by Inflammation of the appendix. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. [9]

1957 stamp of the former Soviet Union commemorating the 250th birthday of Euler. The Union of Soviet Socialist Republics (USSR was a constitutionally Socialist state that existed in Eurasia from 1922 to 1991 The text says: 250 years from the birth of the great mathematician and academician, Leonhard Euler.

Euler arrived in the Russian capital on May 17, 1727. Events 1521 - Edward Stafford 3rd Duke of Buckingham, is executed for Treason. Year 1727 ( MDCCXXVII) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or a Common He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he often worked in close collaboration. Euler mastered Russian and settled into life in St Petersburg. Russian ( transliteration:,) is the most geographically widespread language of Eurasia, the most widely spoken of the Slavic languages He also took on an additional job as a medic in the Russian Navy. The Russian Navy or VMF ( Russian: Военно-Морской Флот (ВМФ России- Voyenno-Morskoy Flot Rossii (VMF or literally Military Maritime [10]

The Academy at St. Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy so as to lessen the faculty's teaching burden, and the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions. [8]

The Academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. Yekaterina (Catherine I Alexeyevna (In Russian: Екатерина I Алексеевна (born Martha Elena Scowronska, Marta Elena Skavronska later The Russian nobility then gained power upon the ascension of the twelve-year-old Peter II. Pyotr (Peter II Alekseyevich ( Russian: Пётр II Алексеевич or Pyotr II Alekseyevich) ( October 23, 1715 &ndash January The nobility were suspicious of the academy's foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues.

Conditions improved slightly upon the death of Peter II, and Euler swiftly rose through the ranks in the academy and was made professor of physics in 1731. Two years later, Daniel Bernoulli, who was fed up with the censorship and hostility he faced at St. Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. [11]

On January 7, 1734, he married Katharina Gsell, daughter of a painter from the Academy Gymnasium. Events 1325 - Alfonso IV becomes King of Portugal. 1558 - France takes Calais, the last continental Year 1734 ( MDCCXXXIV) was a Common year starting on Friday (link will display the full calendar of the Gregorian calendar (or a The young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood. [12]

Berlin

Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. The German Democratic Republic ( GDR; Deutsche Demokratische Republik DDR; commonly known in English as East Germany) was a Socialist state In the middle, it shows his polyhedral formula VE + F = 2. Kuratowski's and Wagner's theorems The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of Forbidden

Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on June 19, 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia. Events 1179 - The Norwegian Battle of Kalvskinnet outside Nidaros. Year 1741 ( MDCCXLI) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar (or a Common year The Prussian Academy of Sciences (Preußische Akademie der Wissenschaften was an Academy established in Berlin on July 11 1700. Frederick II (Friedrich II January 24 1712 August 17 1786) was a King of Prussia (1740&ndash1786 from the He lived for twenty-five years in Berlin, where he wrote over 380 articles. Berlin is the capital city and one of sixteen states of Germany. In Berlin, he published the two works which he would be most renowned for: the Introductio in analysin infinitorum, a text on functions published in 1748, and the Institutiones calculi differentialis,[13] published in 1755 on differential calculus. Algebra Theory of equations Hisab Algebra Theory of equations Hisab Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change [14]

In addition, Euler was asked to tutor the Princess of Anhalt-Dessau, Frederick's niece. Euler wrote over 200 letters to her, 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 work contained Euler's exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler's personality and religious beliefs. This book became more widely read than any of his mathematical works, and it 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. [14]

Despite Euler's immense contribution to the Academy's prestige, he was eventually forced to leave Berlin. This was partly because of a conflict of personality with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the Academy. Voltaire was among those in Frederick's employ, and the Frenchman enjoyed a prominent position in the king's social circle. François-Marie Arouet ( 21 November 1694 30 May 1778) better known by the Pen name Voltaire, was a French Euler, a simple religious man and a hard worker, was very conventional in his beliefs and tastes. He was in many ways the direct opposite of Voltaire. Euler had limited training in rhetoric, and tended to debate matters that he knew little about, making him a frequent target of Voltaire's wit. Rhetoric has had many definitions no simple definition can do it justice [14] Frederick also expressed disappointment with Euler's practical engineering abilities:

 “ 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. Frederick II (Friedrich II January 24 1712 August 17 1786) was a King of Prussia (1740&ndash1786 from the Sanssouci is the former summer palace of Frederick the Great, King of Prussia, at Potsdam, near My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry![15] ”

Eyesight deterioration

A 1753 portrait by Emanuel Handmann. Emanuel Handmann (born 1718 in Basel, died 1781 in Bern) was a Swiss painter. This portrayal suggests problems of the right eyelid, and possible strabismus. Strabismus (from Greek: στραβισμός strabismos, from στραβίζειν strabizein "to squint" from στραβός strabos The left eye appears healthy; it was later affected by a cataract. [16]

Euler's eyesight worsened throughout his mathematical career. In Psychology, visual perception is the ability to interpret information from Visible light reaching the Eyes The resulting Perception is also Three years after suffering a near-fatal fever in 1735 he became nearly blind in his right eye, but Euler rather blamed his condition on the painstaking work on cartography he performed for the St. Fever (also known as pyrexia, from the Greek pyretos meaning fire or a febrile response, from the Latin word Febris Petersburg Academy. Euler's sight in that eye worsened throughout his stay in Germany, so much so that Frederick referred to him as "Cyclops". In Greek mythology and later Roman mythology, a cyclops (ˈsaɪklɒps or kyklops ( Greek) is a member of a primordial race of Euler later suffered a cataract in his good left eye, rendering him almost totally blind a few weeks after its discovery. A cataract is a clouding that develops in the crystalline lens of the Eye or in its envelope varying in degree from slight to complete opacity Even so, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and photographic memory. Eidetic memory, photographic memory, or total recall is the ability to recall Images Sounds, or objects in 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. For the group of nine Ancient Egyptian deities see Ennead. The Aeneid (əˈniːɪd in Publius Vergilius Maro ( October 15, 70 BCE &ndash September 21, 19 BCE later called Virgilius, and known in English as Virgil or 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 1775. [4]

Euler's grave at the Alexander Nevsky Lavra. Alexander Nevsky Lavra or Alexander Nevsky Monastery was founded by Peter the Great in 1710 at the eastern end of the Nevsky Prospekt in

The situation in Russia had improved greatly since the accession to the throne of Catherine the Great, and in 1766 Euler accepted an invitation to return to the St. Catherine II, called Catherine the Great (Екатерина II Великая Yekaterina II Velikaya;) reigned as Empress of Russia for 34 years Petersburg Academy and spent the rest of his life in Russia. His second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife of 40 years. Three years after his wife's death Euler married her half sister. This marriage would last until his death.

On September 18, 1783, Euler passed away in St. Events 96 - Nerva is proclaimed Roman Emperor after Domitian is assassinated Year 1783 ( MDCCLXXXIII) was a Common year starting on Wednesday (link will display the full calendar of the Gregorian calendar (or Petersburg after suffering a brain hemorrhage, and was buried with his wife in the Smolensk Lutheran Cemetery on Vasilievsky Island (the Soviets destroyed the cemetery after transferring Euler's remains to the Orthodox Alexander Nevsky Lavra). A stroke is the rapidly developing loss of brain functions due to a disturbance in the blood vessels supplying blood to the brain Vasilievsky Island is an island in Saint Petersburg, bordered by the rivers Bolshaya Neva and Malaya Neva (in the delta of Neva) from Alexander Nevsky Lavra or Alexander Nevsky Monastery was founded by Peter the Great in 1710 at the eastern end of the Nevsky Prospekt in His eulogy was written for the French Academy by the French mathematician and philosopher Marquis de Condorcet, and an account of his life, with a list of his works, by Nikolaus von Fuss, Euler's son-in-law and the secretary of the Imperial Academy of St. Petersburg. The Russian Academy of Sciences (Российская Академия Наук Rossi'iskaya Akade'miya Nau'k, shortened to PAH RAN) consists of the National Condorcet commented,

 “ …il cessa de calculer et de vivre — … he ceased to calculate and to live. [17] ”

Contributions to mathematics

 Part of a series of articles onThe mathematical constant, e Natural logarithm Applications in Compound interest · Euler's identity & Euler's formula  · Half-lives & Exponential growth/decay People John Napier  · Leonhard Euler

Euler worked in almost all areas of mathematics: geometry, calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational Compound interest is the concept of adding accumulated Interest back to the principal so that interest is earned on interest from that moment on In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic Half-Life (computer-game page here It's already listed in the disambiguation page Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value In Mathematics, the series representation of Euler's number e e = \sum_{n = 0}^{\infty} \frac{1}{n!}\! can be used to prove The Mathematical constant ''e'' can be represented in a variety of ways as a Real number. In Mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers For other people with the same name see John Napier (disambiguation. In Mathematics, specifically Transcendence theory, Schanuel's conjecture is the following statement Given any n Complex numbers Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes The Lunar theory is an explanation by mathematical reasoning of perturbations in the movements of the Moon founded on the Law of gravitation. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. 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. The size of a specific Book is measured from the head to tail of the spine and from edge to edge across the covers [4] Euler's name is associated with a large number of topics. In Mathematics and Physics, there are a large number of topics named in honour of Leonhard Euler ( pronounced '''''Oiler''''')

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[3] and was the first to write f(x) to denote the function f applied to the argument x. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function He also introduced the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm (now also known as Euler's number), the Greek letter Σ for summations and the letter i to denote the imaginary unit. The natural logarithm, formerly known as the Hyperbolic logarithm is the Logarithm to the base e, where e is an irrational The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line Sigma (upper case Σ, lower case σ; Greek Σιγμα lower case in word-final position ς) is the eighteenth letter of the Greek Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation [18] The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it did not originate with him. Pi (uppercase &Pi, lower case &pi) is the sixteenth letter of the Greek alphabet. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems [19]

Analysis

The development of 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. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives 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,[20] his ideas led to many great advances. Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse

He is well known in analysis for his frequent use and development of power series: that is, the expression of functions as sums of infinitely many terms, such as

$e^x = \sum_{n=0}^\infty {x^n \over n!} = \lim_{n \to \infty}\left(\frac{1}{0!} + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!}\right).$

Notably, Euler discovered the power series expansions for e and the inverse tangent function. In Mathematics, a power series (in one variable is an Infinite series of the form f(x = \sum_{n=0}^\infty a_n \left( x-c \right^n = a_0 + His daring (and, by modern standards, technically incorrect) use of power series enabled him to solve the famous Basel problem in 1735:[20]

$\lim_{n \to \infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}\right) = \frac{\pi ^2}{6}.$
A geometric interpretation of Euler's formula

Euler introduced the use of the exponential function and logarithms in analytic proofs. The Basel problem is a famous problem in Number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735 The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce 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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted [18] He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions. For any real number φ, Euler's formula states that the complex exponential function satisfies

$e^{i\varphi} = \cos \varphi + i\sin \varphi.\,$

A special case of the above formula is known as Euler's identity,

$e^{i \pi} +1 = 0 \,$

called "the most remarkable formula in mathematics" by Richard Feynman, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, e, i and π[21]. In Mathematics, the real numbers may be described informally in several different ways Phi (uppercase Φ, lowercase φ or ϕ) pronounced in modern Greek and as in English is the 21st letter of the Greek alphabet This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic In Mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation e^{i \pi} + 1 = 0 \\! where Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum In 1988, the readers of the Mathematical Intelligencer voted it "the Most Beautiful Mathematical Formula Ever"[22]. The Mathematical Intelligencer is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone rather than the technical and In total, Euler was responsible for three of the top five formulae in that poll[22].

In addition, Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. A transcendental function is a function that does not satisfy a Polynomial equation whose Coefficients are themselves polynomials in contrast to an In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function In Mathematics, a quartic equation is one which can be expressed as a Quartic function equalling zero He also found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis, and invented the calculus of variations including its best-known result, the Euler–Lagrange equation. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Calculus of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions. In Calculus of variations, the Euler–Lagrange equation, or Lagrange's equation is a Differential equation whose solutions are the functions

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 Mathematics, analytic number theory is a branch of Number theory that uses methods from Mathematical analysis to solve number-theoretical problems 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. In Mathematics, a hypergeometric series is a Power series in which the ratios of successive Coefficients k is a Rational function In Mathematics, in the area of Combinatorics, a q-Pochhammer symbol, also called a q-shifted factorial, is a Q-analog of the common Pochhammer In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular functions In analysis, a generalized continued fraction is a generalization of regular continued fractions in canonical form in which the partial numerators and the 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. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 See Harmonic series (music for the (related musical concept In Mathematics, the harmonic series is the Infinite series In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 Euler's work in this area led to the development of the prime number theorem. [23]

Number theory

Euler's interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Christian Goldbach ( March 18, 1690 &ndash November 20, 1764) was a Prussian Mathematician who also studied Law Petersburg Academy. A lot of Euler's early work on number theory was based on the works of Pierre de Fermat. Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the 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 the third century BC Euclid proved the existence of infinitely many Prime numbers In the 18th century Leonhard Euler proved a stronger statement the sum 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. We will prove that the following formula holds \begin{align} \\\zeta(s & = 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\frac{1}{5^s}+ \cdots \\ & = \prod_{p} \frac{1}{1-p^{-s}}

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. In Mathematics, Newton's identities, also known as the Newton–Girard formulae give relations between two types of Symmetric polynomials namely between power Fermat's little theorem (not to be confused with Fermat's last theorem) states that if p is a Prime number, then for any Integer a In Number theory, Pierre de Fermat 's theorem on sums of two squares states that an odd prime p is expressible as p = x^2 Lagrange's four-square theorem, also known as Bachet's conjecture, was proven in 1770 by Joseph Louis Lagrange. He also invented the totient function φ(n) which is the number of positive integers less than the integer n that are coprime to n. In Number theory, the totient \varphi(n of a Positive integer n is defined to be the number of positive integers less than or equal to In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem. In Number theory, Euler's theorem (also known as the Fermat-Euler theorem or Euler's totient theorem) states that if n is a positive Integer He contributed significantly to the understanding of perfect numbers, which has fascinated mathematicians since Euclid. In mathematics a perfect number is defined as a positive integer which is the sum of its proper positive Divisors that is the sum of the positive divisors excluding Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euler also made progress toward the prime number theorem, and he conjectured the law of quadratic reciprocity. The law of quadratic reciprocity is a theorem from Modular arithmetic, a branch of Number theory, which shows a remarkable relationship between the solvability The two concepts are regarded as fundamental theorems of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German [24]

By 1772 Euler had proved that 231 − 1 = 2,147,483,647 is a Mersenne prime. In Mathematics, a Mersenne number is a positive integer that is one less than a Power of two: M_n=2^n-1 It may have remained the largest known prime until 1867. [25]

Graph theory

Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges. In Geometry, the Euler line, named after Leonhard Euler, is a line determined from any Triangle that is not equilateral; it passes In Geometry, the nine-point circle is a Circle that can be constructed for any given triangle. The Seven Bridges of Königsberg is a famous historical problem in mathematics

In 1736, Euler solved the problem known as the Seven Bridges of Königsberg. The Seven Bridges of Königsberg is a famous historical problem in mathematics [26] The city of Königsberg, Prussia was set on the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. Königsberg (Karaliaučius Low German: Königsbarg; Królewiec see also other names) was until 1946 the name of Kaliningrad. The Kingdom of Prussia (Königreich Preußen was a German kingdom from 1701 to 1918 and from 1871 was the leading state of the German Empire, comprising The Pregolya or Pregola (Преголя Pregel Prieglius is a River in the Russian Kaliningrad Oblast exclave 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: there is no Eulerian circuit. In Graph theory, an Eulerian path is a path in a graph which visits each edge exactly once This solution is considered to be the first theorem of graph theory, specifically of planar graph theory. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects Kuratowski's and Wagner's theorems The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of Forbidden [26]

Euler also discovered the formula VE + F = 2 relating the number of edges, vertices, and faces of a convex polyhedron[27], and hence of a planar graph. Kuratowski's and Wagner's theorems The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of Forbidden What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions Kuratowski's and Wagner's theorems The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of Forbidden 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. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant In Mathematics, genus has a few different but closely related meanings Topology Orientable surface [28] The study and generalization of this formula, specifically by Cauchy[29] and L'Huillier,[30] is at the origin of topology. Simon Antoine Jean L'Huilier (or L'Huillier) ( Geneva, 24 April 1750 - Geneva, 28 March 1840) was a Swiss Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of

Applied mathematics

Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of Bernoulli's numbers, Fourier series, Venn diagrams, Euler numbers, the constants e and π, continued fractions and integrals. In Mathematics, the Bernoulli numbers are a Sequence of Rational numbers with deep connections to Number theory. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions Venn diagrams or set diagrams are Diagrams that show all hypothetically possible Logical relations between a finite collection of sets (groups For other uses see Euler number (topology and Eulerian number. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems 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. Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change Method of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736. He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) In Mathematics and Computational science, the Euler method, named after Leonhard Euler, is a first order numerical procedure for solving The most notable of these approximations are Euler's method and the Euler–Maclaurin formula. In Mathematics and Computational science, the Euler method, named after Leonhard Euler, is a first order numerical procedure for solving In Mathematics, the Euler–Maclaurin formula provides a powerful connection between Integrals (see Calculus) and sums He also facilitated the use of differential equations, in particular introducing the Euler-Mascheroni constant:

$\gamma = \lim_{n \rightarrow \infty } \left( 1+ \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{n} - \ln(n) \right).$

One of Euler's more unusual interests was the application of mathematical ideas in music. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the The Euler–Mascheroni constant (also called the Euler constant) is a Mathematical constant recurring in analysis and Number theory, usually Music is an Art form in which the medium is Sound organized in Time. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually incorporate musical theory as part of mathematics. Music theory is the field of study that deals with the Mechanics of music and how Music works 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. [31]

Physics and astronomy

Classical mechanics
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Euler helped develop the Euler-Bernoulli beam equation, which became a cornerstone of engineering. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Early Ideas on Motion The Greek philosophers, and Aristotle in particular were the first to propose that there are abstract principles governing nature Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher Johannes Kepler (ˈkɛplɚ ( December 27 1571 &ndash November 15 1630) was a German Mathematician, Astronomer Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Euler-Bernoulli beam theory, or just beam theory, is a simplification of the linear Theory of elasticity which provides a means of calculating the load-carrying Aside from successfully applying his analytic tools to problems in classical mechanics, Euler also applied these techniques to celestial problems. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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. Parallax is an apparent displacement or difference of orientation of an object viewed along two different lines of sight and is measured by the angle or semi-angle of inclination between His calculations also contributed to the development of accurate longitude tables. In Celestial navigation, lunar distance is the angle between the Moon and another Celestial body. [32]

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. Opticks is a book written by English physicist Isaac Newton that was released to the public in 1704. His 1740s papers on optics helped ensure that the wave theory of light proposed by Christian Huygens would become the dominant mode of thought, at least until the development of the quantum theory of light. In Physics and Chemistry, wave–particle duality is the concept that all Matter and Energy exhibits both Wave -like and Christiaan Huygens (ˈhaɪgənz in English ˈhœyɣəns in Dutch) ( April 14, 1629 &ndash July 8, 1695) was a Dutch In Physics and Chemistry, wave–particle duality is the concept that all Matter and Energy exhibits both Wave -like and [33]

Logic

He is also credited with using closed curves to illustrate syllogistic reasoning (1768). In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object A syllogism, or logical appeal, (συλλογισμός &mdash "conclusion" "inference" (usually the categorical syllogism) is a kind of These diagrams have become known as Euler diagrams. Syllogism-Set-Diagramsjpg|thumb|Examples of small Venn diagrams with shaded regions representing Empty sets that are easily transformed into Euler diagrams [34]

Personal philosophy and religious beliefs

Euler and his friend Daniel Bernoulli were opponents of Leibniz's monism and the philosophy of Christian Wolff. Monism is the metaphysical and Theological view that all is one that all reality is subsumed under the most fundamental category of being or existence Christian Wolff (less correctly Wolf; also known as Wolfius) baron ( 24 January 1679 - 9 April 1754) was a German 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". [35]

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 present Euler as a staunch Christian and a biblical literalist (for example, the Rettung was primarily an argument for the divine inspiration of scripture). A Christian is a person who adheres to Christianity, a monotheistic Religion centered on the life and teachings of Jesus of Nazareth Biblical literalism (also called Biblicism) is a primarily pejorative term referring to the adherence to an explicit and literal sense of the Bible. Biblical inspiration is the doctrine in Christian theology concerned with the divine origin of the Bible and what the Bible teaches about itself [36]

There is a famous anecdote 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. Denis Diderot ( October 5, 1713 – July 31, 1784) was a French Philosopher and writer 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. Atheism Diderot was later 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. Arguments for and against the existence of God have been proposed by philosophers theologians and others Euler appeared, advanced toward Diderot, and in a tone of perfect conviction announced, "Sir, (a + bn)z = 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 apocryphal, given that Diderot was a capable mathematician who had published mathematical treatises. [37]

Selected bibliography

The cover page of Euler's Methodus inveniendi lineas curvas.

Euler has an extensive bibliography but his best known books include:

• Elements of Algebra. The 18th-century Swiss mathematician Leonhard Euler (1707&ndash1783 is among the most prolific and successful mathematicians in the history of the field. This elementary algebra text starts with a discussion of the nature of numbers and gives a comprehensive introduction to algebra, including formulae for solutions of polynomial equations.
• Introductio in analysin infinitorum (1748). English translation Introduction to Analysis of the Infinite by John Blanton (Book I, ISBN 0-387-96824-5, Springer-Verlag 1988; Book II, ISBN 0-387-97132-7, Springer-Verlag 1989).
• Two influential textbooks on calculus: Institutiones calculi differentialis (1755) and Institutiones calculi integralis (1768–1770).
• Lettres à une Princesse d'Allemagne (Letters to a German Princess) (1768–1772). Available online (in French). English translation, with notes, and a life of Euler, available online from Google Books: Volume 1, Volume 2
• Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti (1744). Google Book Search is a tool from Google that searches the full text of books that Google scans OCRs, and stores in its digital database The Latin title translates as a method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense. [38]

A definitive collection of Euler's works, entitled Opera Omnia, has been published since 1911 by the Euler Commission of the Swiss Academy of Sciences. The Swiss Academy of Sciences is a Swiss organization that supports and networks the sciences at a regional national and international level

Notes

1. ^ Dan Graves (1996). In Mathematics and Physics, there are a large number of topics named in honour of Leonhard Euler ( pronounced '''''Oiler''''') Scientists of Faith. Grand Rapids, MI: Kregel Resources, 85–86.
2. ^ E. T. Bell (1953). Men of Mathematics, Vol. 1. London: Penguin, 155.
3. ^ a b Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America, 17.
4. ^ a b c Finkel, B. F. (1897). "Biography- Leonard Euler". The American Mathematical Monthly 4 (12): 300.
5. ^ Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America, xiii.  “Lisez Euler, lisez Euler, c'est notre maître à tous. ”
6. ^ James, Ioan (2002). Remarkable Mathematicians: From Euler to von Neumann. Cambridge, 2. ISBN 0-521-52094-0.
7. ^ Translation of Euler's Ph.D in English by Ian BrucePDF (232 KiB)
8. ^ a b Calinger, Ronald (1996). A kibibyte (a contraction of ki lo bi nary byte) is a unit of Information or Computer storage, established by the International "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 156. doi:10.1006/hmat.1996.0015. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
9. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 125. doi:10.1006/hmat.1996.0015. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
10. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 127. doi:10.1006/hmat.1996.0015. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
11. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 128–129. doi:10.1006/hmat.1996.0015. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
12. ^ Fuss, Nicolas. Eulogy of Euler by Fuss. Retrieved on August 30, 2006.
13. ^ [http://www.math.dartmouth.edu/~euler/pages/E212.html .
14. ^ a b c Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America, xxiv–xxv.
15. ^ Frederick II of Prussia (1927). Frederick II (Friedrich II January 24 1712 August 17 1786) was a King of Prussia (1740&ndash1786 from the Letters of Voltaire and Frederick the Great, Letter H 7434, 25 January 1778. New York: Brentano's.
16. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 154–155. doi:10.1006/hmat.1996.0015. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
17. ^ Marquis de Condorcet. Eulogy of Euler - Condorcet. Retrieved on August 30, 2006.
18. ^ a b Boyer, Carl B. ; Uta C. Merzbach. A History of Mathematics. John Wiley & Sons, 439–445. John Wiley & Sons Inc, also referred to as Wiley, is a global Publishing company that markets its products to professionals and consumers students and instructors ISBN 0-471-54397-7.
19. ^ Wolfram, Stephen. Mathematical Notation: Past and Future.
20. ^ a b Wanner, Gerhard; Harrier, Ernst (March 2005). Analysis by its history, 1st, Springer, 62.
21. ^ Feynman, Richard [June 1970]. "Chapter 22: Algebra", The Feynman Lectures on Physics: Volume I, p. 10.
22. ^ a b Wells, David (1990). "Are these the most beautiful?". Mathematical Intelligencer 12 (3): 37–41.
Wells, David (1988). "Which is the most beautiful?". Mathematical Intelligencer 10 (4): 30–31.
23. ^ Dunham, William (1999). "3,4", Euler: The Master of Us All. The Mathematical Association of America.
24. ^ Dunham, William (1999). "1,4", Euler: The Master of Us All. The Mathematical Association of America.
25. ^ Caldwell, Chris. The largest known prime by year
26. ^ a b Alexanderson, Gerald (July 2006). "Euler and Königsberg's bridges: a historical view". Bulletin of the American Mathematical Society.
27. ^ Peter R. Cromwell (1997). Polyhedra. Cambridge: Cambridge University Press, 189-190.
28. ^ Alan Gibbons (1985). Algorithmic Graph Theory. Cambridge: Cambridge University Press, 72.
29. ^ Cauchy, A. L. (1813). "Recherche sur les polyèdres—premier mémoire". Journal de l'Ecole Polytechnique 9 (Cahier 16): 66–86.
30. ^ L'Huillier, S. -A. -J. (1861). "Mémoire sur la polyèdrométrie". Annales de Mathématiques 3: 169–189.
31. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 144–145. doi:10.1006/hmat.1996.0015. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
32. ^ Youschkevitch, A P; Biography in Dictionary of Scientific Biography (New York 1970–1990).
33. ^ Home, R. W. (1988). "Leonhard Euler's 'Anti-Newtonian' Theory of Light". Annals of Science 45 (5): 521–533. doi:10.1080/00033798800200371. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
34. ^ Baron, M. E. ; A Note on The Historical Development of Logic Diagrams. The Mathematical Gazette: The Journal of the Mathematical Association. Vol LIII, no. 383 May 1969.
35. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 153–154. doi:10.1006/hmat.1996.0015. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
36. ^ Euler, Leonhard (1960). "Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister". Leonhardi Euleri Opera Omnia (series 3) 12.
37. ^ Brown, B. H. (May 1942). "The Euler-Diderot Anecdote". The American Mathematical Monthly 49 (5): 302–303.  ; Gillings, R. J. (February 1954). "The So-Called Euler-Diderot Incident". The American Mathematical Monthly 61 (2): 77–80.
38. ^ E65 — Methodus… entry at Euler Archives

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• Gladyshev, Georgi, P. (2007) “Leonhard Euler’s methods and ideas live on in the thermodynamic hierarchical theory of biological evolution,International Journal of Applied Mathematics & Statistics (IJAMAS) 11 (N07), Special Issue on Leonhard Paul Euler’s: Mathematical Topics and Applications (M. T. A. ).
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• Heimpell, Hermann, Theodor Heuss, Benno Reifenberg (editors). 1956. Die großen Deutschen, volume 2, Berlin: Ullstein Verlag.
• Krus, D. J. (2001) "Is the normal distribution due to Gauss? Euler, his family of gamma functions, and their place in the history of statistics," Quality and Quantity: International Journal of Methodology, 35: 445-46.
• Nahin, Paul (2006) Dr. Euler's Fabulous Formula, New Jersey: Princeton, ISBN 978-06-9111-822-2
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• Sandifer, Edward C. (2007), The Early Mathematics of Leonhard Euler, Washington: Mathematical Association of America. IBSN 10: 0-88385-559-3
• Simmons, J. (1996) The giant book of scientists: The 100 greatest minds of all time, Sydney: The Book Company.
• Singh, Simon. (1997). Fermat's last theorem, Fourth Estate: New York, ISBN 1-85702-669-1
• Thiele, Rüdiger. (2005). The mathematics and science of Leonhard Euler, in Mathematics and the Historian's Craft: The Kenneth O. May Lectures, G. Van Brummelen and M. Kinyon (eds. ), CMS Books in Mathematics, Springer Verlag. ISBN 0-387-25284-3.
• "A Tribute to Leohnard Euler 1707-1783" (November 1983). Mathematics Magazine 56 (5). Mathematics Magazine is a refereed bimonthly publication of the Mathematical Association of America.