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

Euler made important discoveries in fields as diverse as calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[3] He is also renowned for his work in mechanics, optics, and astronomy. For other uses, see Calculus (disambiguation). ... A drawing of a graph. ... Analysis has its beginnings in the rigorous formulation of calculus. ... This article is about functions in mathematics. ... For other uses, see Mechanic (disambiguation). ... For the book by Sir Isaac Newton, see Opticks. ... For other uses, see Astronomy (disambiguation). ...

Euler is considered to be the preeminent mathematician of the 18th century and one of the greatest of all time. He is also one of the most prolific; his collected works fill 60–80 quarto volumes.[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] (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... Old book binding and cover Bookbinding is the process of physically assembling a book from a number of folded or unfolded sheets of paper or other material. ... Pierre-Simon, marquis de Laplace (March 23, 1749 - March 5, 1827) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy. ...

Euler was featured on the sixth series of the Swiss 10-franc banknote and on numerous Swiss, German, and Russian postage stamps. The asteroid 2002 Euler was named in his honor. He is also commemorated by the Lutheran Church on their Calendar of Saints on May 24th. A selection of Hong Kong postage stamps A postage stamp is evidence of pre-paying a fee for postal services. ... For other uses, see Asteroid (disambiguation). ... An asteroid named after the mathematician, Leonhard Euler Categories: Asteroid stubs ... The Lutheran movement is a group of denominations of Protestant Christianity by the original definition. ... The Lutheran Calendar of Saints is a listing which details the primary annual festivals and events that are celebrated liturgically by the Lutheran Church. ... is the 144th day of the year (145th in leap years) in the Gregorian calendar. ...

### Early years

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

Euler was at this point studying theology, Greek, and Hebrew at his father's urging, in order to become a pastor. 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. He won second place, losing only to Pierre Bouguer—who is now known as "the father of naval architecture". Euler subsequently won this coveted annual prize twelve times in his career.[8] Theology finds its scholars pursuing the understanding of and providing reasoned discourse of religion, spirituality and God or the gods. ... Hebrew redirects here. ... For other uses, see Speed of sound (disambiguation). ... Louis XIV visiting the AcadÃ©mie in 1671 The French Academy of Sciences (AcadÃ©mie des sciences) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. ... mizzen mast, mainmast and foremast Grand Turk The mast of a sailing ship is a tall vertical pole which supports the sails. ... Pierre Bouguer (February 16, 1698 – August 15, 1758) was a French mathematician. ...

### 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. 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. 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] Daniel Bernoulli Daniel Bernoulli (February 8, 1700 â€“ March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland where he died. ... Nicolaus II Bernoulli (February 6, 1695, Basel, Switzerland – July 31, 1726, Saint Petersburg, Russia) was a Swiss mathematician. ... Russian Academy of Sciences: main building Russian Academy of Sciences (Ð Ð¾ÑÑÐ¸ÌÐ¹ÑÐºÐ°Ñ ÐÐºÐ°Ð´ÐµÌÐ¼Ð¸Ñ ÐÐ°ÑƒÌÐº) is the national academy of Russia. ... Saint Petersburg (Russian: Санкт-Петербу́рг, English transliteration: Sankt-Peterburg), colloquially known as Питер (transliterated Piter), formerly known as Leningrad (Ленингра́д, 1924–1991) and Petrograd (Петрогра́д, 1914–1924), is a city located in Northwestern Russia on the delta of the river Neva at the east end of the Gulf of Finland... Appendicitis (or epityphlitis) is a condition characterized by inflammation of the appendix. ...

1957 stamp of the former Soviet Union commemorating the 250th birthday of Euler. 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. 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. He also took on an additional job as a medic in the Russian Navy.[10] Image File history File links Euler-USSR-1957-stamp. ... Image File history File links Euler-USSR-1957-stamp. ... is the 137th day of the year (138th in leap years) in the Gregorian calendar. ... Events 1727 to 1800 - Lt. ... The Russian Navy or VMF (Russian: Ð’Ð¾ÐµÐ½Ð½Ð¾-ÐœÐ¾Ñ€ÑÐºÐ¾Ð¹ Ð¤Ð»Ð¾Ñ‚ (Ð’ÐœÐ¤) - Voyenno- Morskoy Flot (VMF) or Military Maritime Fleet) is the naval arm of the Russian armed forces. ...

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] Peter the Great or Pyotr Alexeyevich Romanov (Russian: ÐŸÑ‘Ñ‚Ñ€ I ÐÐ»ÐµÐºÑÐµÐµÐ²Ð¸Ñ‡ Pyotr I Alekse`yevich, ÐŸÑ‘Ñ‚Ñ€ Ð’ÐµÐ»Ð¸ÐºÐ¸Ð¹ Pyotr Veli`kiy) (9 June 1672 â€“ 8 February 1725 [30 May 1672â€“28 January 1725 O.S.][1]) ruled Russia from 7 May (27 April O.S.) 1682 until his death, jointly ruling before 1696 with his...

The Academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility then gained power upon the ascension of the twelve-year-old Peter II. The nobility were suspicious of the academy's foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues. Yekaterina (Catherine) I Alexeyevna (In Russian: Ð•ÐºÐ°Ñ‚ÐµÑ€Ð¸Ð½Ð° I ÐÐ»ÐµÐºÑÐµÐµÐ²Ð½Ð°) (born Martha Scowronska, Latvian: , later Marfa Samuilovna Skavronskaya) (April 15, 1684 â€“ May 17, 1727) (April 5, 1684â€“May 6, 1727 O.S.), the second wife of Peter the Great, reigned as Empress of Russia from 1725 until her death. ... Pyotr (Peter) II Alekseyevich (Russian: ÐŸÑ‘Ñ‚Ñ€ II ÐÐ»ÐµÐºÑÐµÐµÐ²Ð¸Ñ‡ or Pyotr II Alekseyevich) (October 23, 1715 â€“ January 30, 1730) was Emperor of Russia from 1727 until his death. ...

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. The young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood.[12] is the 7th day of the year in the Gregorian calendar. ... Events January 8 - Premiere of George Frideric Handels opera Ariodante at the Royal Opera House, Covent Garden. ... River Neva (Нева́) is a 74 km long Russian river flowing from the Lake Ladoga (Ладожское Озеро - Ladozhskoye Ozero) through the Carelian Isthmus (Карельский Перешеек - Karelskii Peresheyek) and the city of Saint Petersburg (Санкт-Петербург - Sankt Peterburg) to the Gulf of Finland (Финский Залив - Finskii Zaliv). ...

### Berlin

Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. In the middle, it shows his polyhedral formula VE + F = 2.

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. 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.[14] Frederick also expressed disappointment with Euler's practical engineering abilities: For other uses, see Voltaire (disambiguation). ... Rhetoric (from Greek , rhÃªtÃ´r, orator, teacher) is generally understood to be the art or technique of persuasion through the use of oral, visual, or written language; however, this definition of rhetoric has expanded greatly since rhetoric emerged as a field of study in universities. ... Frederick II (German: ; January 24, 1712 â€“ August 17, 1786) was a King of Prussia (1740â€“1786) from the Hohenzollern dynasty. ...

 “ I wanted to have a water jet in my garden: Euler calculated the force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci. My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry![15] ”

### Eyesight deterioration

A 1753 portrait by Emanuel Handmann. This portrayal suggests problems of the right eyelid, and possible strabismus. The left eye appears healthy; it was later affected by a cataract.[16]

Euler's eyesight worsened throughout his mathematical career. 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. Petersburg Academy. Euler's sight in that eye worsened throughout his stay in Germany, so much so that Frederick referred to him as "Cyclops". Euler later suffered a cataract in his good left eye, rendering him almost totally blind a few weeks after its discovery. 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. For example, Euler could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last. With the aid of his scribes, Euler's productivity on many areas of study actually increased. He produced on average one mathematical paper every week in the year 1775.[4] Image File history File links Picture of Leonhard Euler by Emanuel Handmann. ... Image File history File links Picture of Leonhard Euler by Emanuel Handmann. ... Jakob Emanuel Handmann, born 1718 in Basel, died 1781 in Bern. ... Strabismus (from Greek: ÏƒÏ„ÏÎ±Î²Î¹ÏƒÎ¼ÏŒÏ‚ strabismos, from ÏƒÏ„ÏÎ±Î²Î¯Î¶ÎµÎ¹Î½ strabizein to squint, from ÏƒÏ„ÏÎ±Î²ÏŒÏ‚ strabos squinting, squint-eyed[1]) is a condition in which the eyes are not properly aligned with each other. ... Visual perception is one of the senses, consisting of the ability to detect light and interpret (see) it as the perception known as sight or naked eye vision. ... An analogue medical thermometer showing the temperature of 38. ... Cartography or mapmaking (in Greek chartis = map and graphein = write) is the study and practice of making maps or globes. ... This page is about the mythical creature. ... Human eye cross-sectional view, showing position of human lens. ... Eidetic memory, photographic memory, or total recall is the ability to recall images, sounds, or objects in memory with extreme accuracy and in abundant volume. ... Aeneas flees burning Troy, Federico Barocci, 1598 Galleria Borghese, Rome The Aeneid (IPA English pronunciation: ; in Latin Aeneis, pronounced â€” the title is Greek in form: genitive case Aeneidos) is a Latin epic written by Virgil in the 1st century BC (between 29 and 19 BC) that tells the legendary story... For other uses, see Virgil (disambiguation). ...

Euler's grave at the Alexander Nevsky Lavra.

On September 18, 1783, Euler passed away in St. 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). 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. Condorcet commented, is the 261st day of the year (262nd in leap years) in the Gregorian calendar. ... 1783 was a common year starting on Wednesday (see link for calendar). ... A cerebral hemorrhage is a condition in the brain in which a blood vessel leaks. ... Spit of the Vasilievsky island Vasilievsky Island is a district of Saint Petersburg, bordered by the rivers Bolshaya Neva and Malaya Neva (in the delta of Neva) from South and Northeast, and by the Gulf of Finland from the West. ... View of the monastery in the early 19th century Alexander Nevsky Monastery was founded by Peter the Great in 1710 at the southern end of the Nevsky Prospect in St Petersburg to house the relics of Alexander Nevsky, patron saint of the newly-founded Russian capital. ... â€œCondorcetâ€ redirects here. ... Russian Academy of Sciences: main building Russian Academy of Sciences (Ð Ð¾ÑÑÐ¸ÌÐ¹ÑÐºÐ°Ñ ÐÐºÐ°Ð´ÐµÌÐ¼Ð¸Ñ ÐÐ°ÑƒÌÐº) is the national academy of Russia. ... Condorcet can refer to two separate things. ...

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

## Contributions to mathematics

 Part of a series of articles on The mathematical constant, e e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... Natural logarithm Image File history File links Euler's_formula. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... Applications in: compound interest · Euler's identity & Euler's formula  · half-lives & exponential growth/decay Compound interest refers to the fact that whenever interest is calculated, it is based not only on the original principal, but also on any unpaid interest that has been added to the principal. ... For other uses, see List of topics named after Leonhard Euler. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... Half-Life For a quantity subject to exponential decay, the half-life is the time required for the quantity to fall to half of its initial value. ... In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... Defining e: proof that e is irrational  · representations of e · Lindemann–Weierstrass theorem In mathematics, the series expansion of the number e can be used to prove that e is irrational. ... The mathematical constant e can be represented in a variety of ways as a real number. ... In mathematics, the Lindemannâ€“Weierstrass theorem states that if Î±1,...,Î±n are algebraic numbers which are linearly independent over the rational numbers, then are algebraically independent over the algebraic numbers; in other words the set has transcendence degree n over . ... People John Napier  · Leonhard Euler For other people with the same name, see John Napier (disambiguation). ... Schanuel's conjecture Schanuels conjecture is that given any set of n complex numbers which have linear independence over the rational numbers, the set (up to twice the size) has transcendence degree of at least n over the rationals. ...

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. 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.[4] Euler's name is associated with a large number of topics. For other uses, see Geometry (disambiguation). ... For other uses, see Calculus (disambiguation). ... Wikibooks has a book on the topic of Trigonometry The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. ... This article is about the branch of mathematics. ... 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 of problems that arise from their study. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... The size of a specific book is measured from the head to tail of the spine, and from edge to edge across the covers. ... Leonhard Euler (1707 - 1783) is the eponym of all of the topics listed below. ...

### 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. 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.[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.[19] This article is about functions in mathematics. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... The mathematical constant e (occasionally called Eulers number after the Swiss mathematician Leonhard Euler, or Napiers constant in honor of the Scottish mathematician John Napier who introduced logarithms) is the base of the natural logarithm function. ... For other uses, see Sigma (disambiguation). ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... For other uses, see Pi (disambiguation) Pi (upper case Î , lower case Ï€ or Ï–) is the sixteenth letter of the Greek alphabet. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ...

### 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. 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. For other uses, see Calculus (disambiguation). ... ...

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 In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

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

Notably, Euler discovered the power series expansions for e and the inverse tangent function. His daring (and, by modern standards, technically incorrect) use of power series enabled him to solve the famous Basel problem in 1735:[20] In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... The Basel problem is a famous problem in number theory, first posed by Pietro Mengoli in 1644, and solved by Leonhard Euler in 1735. ...

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

Euler introduced the use of the exponential function and logarithms in analytic proofs. 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.[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 Image File history File links Euler's_formula. ... Image File history File links Euler's_formula. ... The exponential function is one of the most important functions in mathematics. ... In mathematics, if two variables of bn = x are known, the third can be found. ... A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram In mathematics, the complex numbers are the extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:[1] Every complex number can be... Sine redirects here. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... Phi (upper case Î¦, lower case Ï† or ), pronounced fee or fie (depending on context and, often, personal inclination), is the 21st letter of the Greek alphabet. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

$e^{ivarphi} = cos varphi + isin varphi.,$

A special case of the above formula is known as Euler's identity, For other uses, see List of topics named after Leonhard Euler. ...

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

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 1988, the readers of the Mathematical Intelligencer voted it "the Most Beautiful Mathematical Formula Ever"[22]. In total, Euler was responsible for three of the top five formulae in that poll[22]. This article is about the physicist. ... The Mathematical Intelligencer is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common amongst such journals. ...

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, and invented the calculus of variations including its best-known result, the Euler–Lagrange equation. A transcendental function is a function which does not satisfy a polynomial equation whose coefficients are themselves polynomials. ... The Gamma function along part of the real axis In mathematics, the Gamma function (represented by the capitalized Greek letter Î“) is an extension of the factorial function to real and complex numbers. ... In mathematics, a quartic equation is the result of setting a quartic function equal to zero. ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... Calculus of variations is a field of mathematics that deals with functionals, as opposed to ordinary calculus which deals with 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 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.[23] Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ... In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ... In mathematics, a q-series, also sometimes called a q-shifted factorial, is defined as It is usually considered first as a formal power series; it is also an analytic function of q, in the unit disc. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... In mathematics, a generalized continued fraction is a generalization of the concept of continued fraction in which the numerators are allowed to differ from unity. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... See harmonic series (music) for the (related) musical concept. ... In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are 1 and itself. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...

### Number theory

Euler's interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy. 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. Christian Goldbach (March 18, 1690 - November 20, 1764), was a Prussian mathematician, who was born in KÃ¶nigsberg, Prussia, as son of a pastor. ... Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601 â€“ January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ...

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. In the third century BC, Euclid proved the existence of infinitely many prime numbers. ... We will prove that the following formula holds: where Î¶ denotes the Riemann zeta function and the product extends over all prime numbers p. ...

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. 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. 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 understanding of perfect numbers, which has fascinated mathematicians since Euclid. Euler also made progress toward the prime number theorem, and he conjectured the law of quadratic reciprocity. The two concepts are regarded as fundamental theorems of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss.[24] In mathematics, Newtons identities relate two different ways of describing the roots of a polynomial. ... Fermats little theorem (not to be confused with Fermats last theorem) states that if p is a prime number, then for any integer a, This means that if you start with a number, initialized to 1, and repeatedly multiply, for a total of p multiplications, that number by... In mathematics, Pierre de Fermats theorem on sums of two squares states that an odd prime number p is expressible as with x and y integers, if and only if For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and... Lagranges four-square theorem, also known as Bachets conjecture, was proved in 1770 by Joseph Louis Lagrange. ... In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less than or equal to n and coprime to n. ... In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common factor other than 1 and âˆ’1, or equivalently, if their greatest common divisor is 1. ... In number theory, Eulers theorem (also known as the Fermat-Euler theorem or Eulers totient theorem) states that if n is a positive integer and a is coprime to n, then aÏ†(n) â‰¡ 1 (mod n) where Ï†(n) is Eulers totient function and mod denotes the congruence... In mathematics, a perfect number is an integer which is the sum of its proper positive divisors, excluding itself. ... For other uses, see Euclid (disambiguation). ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ... In number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ... Johann Carl Friedrich Gauss (pronounced ,  ; in German usually GauÃŸ, Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ...

By 1772 Euler had proved that 231 − 1 = 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867.[25] In mathematics, a Mersenne number is a number that is one less than a power of two. ... Graph of number of digits in largest known prime by year - electronic era. ...

### Geometry

In geometry, Eulers line (red line in the image), named after Leonhard Euler, is the line passing through the orthocenter (blue), the circumcenter (green), the centroid (yellow), and the center of the nine-point circle (red point) of any triangle. ...

### 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 1736, Euler solved the problem known as the Seven Bridges of Königsberg.[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. 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. This solution is considered to be the first theorem of graph theory, specifically of planar graph theory.[26] Map of KÃ¶nigsberg in Eulers time showing the actual layout of the seven bridges, highlighting the river Pregolya and the bridges. ... Image File history File links The seven bridges of Konigsberg - old map with bridges highlighted Public Domain Image, modified by me, released under GPL. File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Map of KÃ¶nigsberg in Eulers time showing the actual layout of the seven bridges, highlighting the river Pregolya and the bridges. ... Former German name of the city of Kaliningrad. ... Anthem PreuÃŸenlied, Heil dir im Siegerkranz (both unofficial) The Kingdom of Prussia at its greatest extent, at the time of the formation of the German Empire, 1871 Capital Berlin Government Monarchy King  - 1701 â€” 1713 Frederick I (first)  - 1888 â€” 1918 William II (last) Prime minister  - 1848 Adolf Heinrich von Arnim... Pregolya (Преголя), also spelt as Pregola (German: Pregel) is a river in the Russian exclave of Kaliningrad. ... The KÃ¶nigsberg Bridges graph In the mathematical field of graph theory, an Eulerian path is a path in a graph which visits each edge exactly once. ... A drawing of a graph. ... In graph theory, a planar graph is a graph that can be drawn so that no edges intersect (or that can be embedded) in the plane. ...

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. 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.[28] The study and generalization of this formula, specifically by Cauchy[29] and L'Huillier,[30] is at the origin of topology. In graph theory, a planar graph is a graph that can be drawn so that no edges intersect (or that can be embedded) in the plane. ... For the game magazine, see Polyhedron (magazine). ... In graph theory, a planar graph is a graph that can be drawn so that no edges intersect (or that can be embedded) in the plane. ... In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological spaces shape or structure. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... Augustin Louis Cauchy (August 21, 1789 â€“ May 23, 1857) was a French mathematician. ... Simon Antoine Jean LHuillier(Geneva, 24 April 1750 - Geneva, 28 March 1840) was a Swiss mathematician of French Hugenot descent. ... For other uses, see Topology (disambiguation). ...

### 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. 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: In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... Venn diagrams, Euler diagrams (pronounced oiler) and Johnston diagrams are similar-looking illustrations of set, mathematical or logical relationships. ... The Euler numbers are a sequence En of integers defined by the following Taylor series expansion: (Note that e, the base of the natural logarithm, is also occasionally called Eulers number, as is the Euler characteristic. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... Leibniz redirects here. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... Method of Fluxions was a book by Isaac Newton. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ... An Euler approximation is a numerical method of solving differential equations when the solution to a differential equation cannot be found analytically. ... Numerical ordinary differential equations is the part of numerical analysis which studies the numerical solution of ordinary differential equations (ODEs). ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is Î³ â‰ˆ 0. ...

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

One of Euler's more unusual interests was the application of mathematical ideas in music. In 1739 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.[31] For other uses, see Music (disambiguation). ... Music theory is a set of systems for analyzing, classifying, and composing music and the elements of music. ...

### Physics and astronomy

Classical mechanics
$vec{F} = frac{mathrm{d}}{mathrm{d}t}(m vec{v})$
Newton's Second Law
History of ...
Scientists
Galileo · Kepler · Newton
Laplace · Hamilton · d'Alembert
Cauchy · Lagrange · Euler
This box: view  talk  edit

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 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.[33] For the book by Sir Isaac Newton, see Opticks. ... Opticks or a treatise of the reflections, refractions, inflections and colours of light Opticks is a book written by English physicist Isaac Newton that was released to the public in 1704. ... In physics, wave-particle duality holds that light and matter simultaneously exhibit properties of waves and of particles (or photons). ... Christiaan Huygens Christiaan Huygens (approximate pronunciation: HOW-khens; SAMPA /h9yGEns/ or /h@YG@ns/) (April 14, 1629–July 8, 1695), was a Dutch mathematician and physicist; born in The Hague as the son of Constantijn Huygens. ... In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. ...

### Logic

He is also credited with using closed curves to illustrate syllogistic reasoning (1768). These diagrams have become known as Euler diagrams.[34] For other uses, see Curve (disambiguation). ... A syllogism (Greek: â€” conclusion, inference), usually the categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form. ... An Euler diagram does not need to show all possible intersections. ...

## Personal philosophy and religious beliefs

Euler and his friend Daniel Bernoulli were opponents of Leibniz's monism 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".[35] Leibniz redirects here. ... For other uses, see Monist (disambiguation). ... Christian Wolff (less correctly Wolf; also known as Wolfius) (January 24, 1679 - April 9, 1754) was a German philosopher. ...

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).[36] For other uses, see Christian (disambiguation). ... Biblical inerrancy is the view that the Bible is the Word of God and is in every detail infallible and without error. ... Biblical inspiration is the doctrine in Christian theology concerned with the divine origin of the Bible and what the Bible teaches about itself. ...

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. 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 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. 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] Portrait of Diderot by Louis-Michel van Loo, 1767 Denis Diderot (October 5, 1713 â€“ July 31, 1784) was a French philosopher and writer. ... Atheist redirects here. ... Arguments for and against the existence of God have been proposed by philosophers, theologians, and others. ...

## Selected bibliography

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

Euler has an extensive bibliography but his best known books include: The 18th century Swiss mathematician Leonhard Euler (1707-1783) is among the most prolific and successful mathematicians in the history of the field. ...

• Elements of Algebra. 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). 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. // Google offers a variety of services and tools besides its basic web search. ... The Swiss Academy of Sciences (SCNAT) is a Swiss organization that supports and networks the sciences at a regional, national and international level. ...

Leonhard Euler (1707 - 1783) is the eponym of all of the topics listed below. ...

## Notes

1. ^ Dan Graves (1996). 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. doi:10.2307/2968971.
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). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 156. doi:10.1006/hmat.1996.0015.
9. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 125. doi:10.1006/hmat.1996.0015.
10. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 127. doi:10.1006/hmat.1996.0015.
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.
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). 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.
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. 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 43: 567. doi:10.1090/S0273-0979-06-01130-X.
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.
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.
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.
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. doi:10.2307/2303096. ; Gillings, R.J. (February 1954). "The So-Called Euler-Diderot Incident". The American Mathematical Monthly 61 (2): 77–80. doi:10.2307/2307789.
38. ^ E65 — Methodus… entry at Euler Archives

• Lexikon der Naturwissenschaftler, 2000. Heidelberg: Spektrum Akademischer Verlag.
• Demidov, S.S., 2005, "Treatise on the differential calculus" in Grattan-Guiness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 191-98.
• Dunham, William (1999) Euler: The Master of Us All, Washington: Mathematical Association of America. ISBN 0-88385-328-0.
• Fraser, Craig G., 2005, "Book on the calculus of variations" in Grattan-Guiness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 168-80.
• 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.).
• W. Gautschi (2008). "Leonhard Euler: his life, the man, and his works". SIAM Review 50 (1): 3–33. doi:10.1137/070702710.
• 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
• Reich, Karin, 2005, " 'Introduction' to analysis" in Grattan-Guiness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 181-90.
• 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).

Ivor Grattan-Guinness (Born 23 June 1941, in Bakewell, England) is a prolific historian of mathematics and logic, at Middlesex University. ... Ivor Grattan-Guinness (Born 23 June 1941, in Bakewell, England) is a prolific historian of mathematics and logic, at Middlesex University. ... ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Ivor Grattan-Guinness (Born 23 June 1941, in Bakewell, England) is a prolific historian of mathematics and logic, at Middlesex University. ... Mathematics Magazine is a a bimonthly publication of the Mathematical Association of America. ...

Results from FactBites:

 Leonhard Euler (684 words) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced "oiler", not "yooler") was a Swiss mathematician and physicist. He is the physicist who, with Daniel Bernoulli, established the law that the torque on a thin elastic beam is proportional to a measure of the elasticity of the material and the moment of inertia of a cross section, about an axis through the center of mass and perpendicular to the plane of the couple. Euler wrote Tentamen novae theoriae musicae in 1739 which was an attempt to combine mathematics and music; a biography comments that the work was "for musicians too advanced in its mathematics and for mathematicians too musical".
 Leonhard Euler - Wikipedia, the free encyclopedia (3658 words) Leonhard Euler (pronounced [ˈɔʏlɐ], that is, Oiler) (Basel, Switzerland, April 15, 1707 – September 18, 1783 in St Petersburg, Russia) was a Swiss mathematician and physicist. Euler was at this point studying Theology, Greek and Hebrew on his father's urging, in the hopes of becoming a pastor. Euler was a staunch Christian and a biblical literalist (The Rettung was primarily an argument for the divine inspiration of scripture)
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