# Leonhard Euler: Wikis

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# Encyclopedia

Leonhard Euler

Portrait by Emanuel Handmann 1756(?)
Born 15 April 1707
Basel, Switzerland
Died 18 September 1783 (aged 76)
[OS: 7 September 1783]
St. Petersburg, Russia
Residence Prussia, Russia
Switzerland
Nationality Swiss
Fields Mathematician and Physicist
Institutions Imperial Russian Academy of Sciences
Alma mater University of Basel
Doctoral students Joseph Louis Lagrange
Known for See full list
Signature
Notes
He is the father of the mathematician Johann Euler
He is listed by academic genealogy authorities as the equivalent to the doctoral advisor of Joseph Louis Lagrange.

Leonhard Paul Euler[citation needed] (15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist who spent most of his life in Russia and Germany. His surname is pronounced /ˈɔɪlər/ OY-lər in English and [ˈɔʏlɐ] in German; the pronunciation /ˈjuːlər/ EW-lər is incorrect.[1][2][3][4]

Euler made important discoveries in fields as diverse as infinitesimal 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.[5] He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy.

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.[6] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."[7]

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 24 May – he was a devout Christian (and believer in biblical inerrancy) who wrote apologetics and argued forcefully against the prominent atheists of his time.[8]

## Life

### Early years

Old Swiss 10 Franc banknote honoring Euler

Euler was born in Basel to Paul Euler, a pastor of the Reformed Church, and Marguerite Brucker, a pastor's daughter. 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. 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. Euler's early formal education started in Basel, where he was sent to live with his maternal grandmother. At the age of thirteen he enrolled at the University of Basel, and in 1723, received his Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics.[9] Euler was at this point studying theology, Greek, and Hebrew at his father's urging, in order to become a pastor, but Bernoulli 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[10] 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.[11]

### 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.[12]

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, academician Leonhard Euler.

Euler arrived in the Russian capital on 17 May 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.[13]

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.[11]

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.

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.[14]

On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell, a painter from the Academy Gymnasium.[15] The young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood.[16]

### 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 V + FE = 2.

Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia. He lived for twenty-five years in Berlin, where he wrote over 380 articles. 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,[17] published in 1755 on differential calculus.[18] In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences.

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.[18]

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.[18] 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. 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![19] ”
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.[20]

### Eyesight deterioration

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 in 1766. 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.[6]

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. 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 Katharina of 40 years, daughter of Swiss painter Georg Gsell. Three years after his wife's death Euler married her half sister, Salome Abigail Gsell (1723–1794).[21] This marriage would last until his death.

On 18 September 1783, Euler died 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,

 “ …il cessa de calculer et de vivre — … he ceased to calculate and to live.[22] ”
Euler's grave at the Alexander Nevsky Lavra

## Contributions to mathematics and physics

 Part of a series of articles on The mathematical constant e 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, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes.[6] Euler's name is associated with a large number of topics.

### 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[5] 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.[23] 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.[24]

### Analysis

The development of infinitesimal calculus was at the forefront of 18th century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler's work. While some of Euler's proofs are not acceptable by modern standards of mathematical rigour,[25] his ideas led to many great advances. Euler is well-known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as

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

Notably, Euler directly proved the power series expansions for e and the inverse tangent function. (Indirect proof via the inverse power series technique was given by Newton and Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the famous Basel problem in 1735 (he provided a more elaborate argument in 1741):[25]

$\sum_{n=1}^\infty {1 \over n^2} = \lim_{n \to \infty}\left(\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots + \frac{1}{n^2}\right) = \frac{\pi ^2}{6}.$
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.[23] 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.[26] In 1988, readers of the Mathematical Intelligencer voted it "the Most Beautiful Mathematical Formula Ever".[27] In total, Euler was responsible for three of the top five formulae in that poll.[27]

De Moivre's formula is a direct consequence of Euler's formula.

In addition, Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. He also found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis, and invented the calculus of variations including its best-known result, the Euler–Lagrange equation.

Euler also pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions. For example, he proved the infinitude of primes using the divergence of the harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler's work in this area led to the development of the prime number theorem.[28]

### 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.

Euler linked the nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and the prime numbers; this is known as the Euler product formula for the Riemann zeta function.

Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and he made distinct contributions to Lagrange's four-square theorem. He also invented the totient function φ(n) which is the number of positive integers less than or equal to 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 theory of perfect numbers, which had 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.[29]

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.[30]

### 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.[31] 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.[31]

Euler also discovered the formula V − E + F = 2 relating the number of vertices, edges, and faces of a convex polyhedron,[32] and hence of a planar graph (for a planar graph, V − E + F = 1). 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.[33] The study and generalization of this formula, specifically by Cauchy[34] and L'Huillier,[35] is at the origin of topology.

### Applied mathematics

Some of Euler's greatest successes were in solving real-world problems analytically, and in describing numerous applications of the Bernoulli 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:

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

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.[36]

### Physics and astronomy

Classical mechanics
$\mathbf{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \mathbf{v})$
Newton's Second Law
History of ...
Scientists
Isaac Newton · Jeremiah Horrocks · Leonhard Euler · Jean le Rond d'Alembert · Alexis Clairaut
Joseph Louis Lagrange · Pierre-Simon Laplace · William Rowan Hamilton · Siméon-Denis Poisson

Euler helped develop the Euler-Bernoulli beam equation, which became a cornerstone of engineering. Aside from successfully applying his analytic tools to problems in classical mechanics, Euler also applied these techniques to celestial problems. His work in astronomy was recognized by a number of Paris Academy Prizes over the course of his career. His accomplishments include determining with great accuracy the orbits of comets and other celestial bodies, understanding the nature of comets, and calculating the parallax of the sun. His calculations also contributed to the development of accurate longitude tables.[37]

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.[38]

### Logic

He is also credited with using closed curves to illustrate syllogistic reasoning (1768). These diagrams have become known as Euler diagrams.[39]

## Personal philosophy and religious beliefs

Euler and his friend Daniel Bernoulli were opponents of Leibniz's monadism and the philosophy of Christian Wolff. Euler insisted that knowledge is founded in part on the basis of precise quantitative laws, something that monadism and Wolffian science were unable to provide. Euler's religious leanings might also have had a bearing on his dislike of the doctrine; he went so far as to label Wolff's ideas as "heathen and atheistic".[40]

Much of what is known of Euler's religious beliefs can be deduced from his Letters to a German Princess and an earlier work, Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister (Defense of the Divine Revelation against the Objections of the Freethinkers). These works show that Euler was a devout Christian who believed the Bible to be inspired; the Rettung was primarily an argument for the divine inspiration of scripture.[8]

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, $\frac{a+b^n}{z}=x$, hence God exists—reply!". Diderot, to whom (says the story) all mathematics was gibberish, stood dumbstruck as peals of laughter erupted from the court. The phrase's scary nature could be more understood if taken into account that the current mathematical notation was not invented at the time and these equations were expressed in words, most likely in Latin. Embarrassed, Diderot 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.[41]

## Selected bibliography

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

Euler has an extensive bibliography. His best known books include:

• 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 Institutionum 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.[42]

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.

## References and notes

1. ^ "Euler", Oxford English Dictionary, second edition, Oxford University Press, 1989.
2. ^ "Euler", Merriam–Webster's Online Dictionary, 2009.
3. ^ "Euler, Leonhard", The American Heritage Dictionary of the English Language, fourth edition, Houghton Mifflin Company, Boston, 2000.
4. ^ Peter M. Higgins (2007). Nets, Puzzles, and Postmen: An Exploration of Mathematical Connections. Oxford University Press. p. 43.
5. ^ a b Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America. p. 17.
6. ^ a b c Finkel, B.F. (1897). "Biography- Leonard Euler". The American Mathematical Monthly 4 (12): 300. doi:10.2307/2968971.
7. ^ 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."
8. ^ a b Euler, Leonhard (1960). Orell-Fussli. ed. "Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister". Leonhardi Euleri Opera Omnia (series 3) 12.
9. ^ James, Ioan (2002). Remarkable Mathematicians: From Euler to von Neumann. Cambridge. p. 2. ISBN 0-521-52094-0.
10. ^ Translation of Euler's Ph.D in English by Ian Bruce
11. ^ 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.
12. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 125. doi:10.1006/hmat.1996.0015.
13. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 127. doi:10.1006/hmat.1996.0015.
14. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 128–129. doi:10.1006/hmat.1996.0015.
15. ^ Gekker, I.R.; Euler, A.A. (2007). "Leonhard Euler's family and descendants". in Bogoliubov, N.N.; Mikhaĭlov, G.K.; Yushkevich, A.P.. Euler and modern science. Mathematical Association of America. ISBN 088385564X. , p. 402.
16. ^ Fuss, Nicolas. "Eulogy of Euler by Fuss". Retrieved 30 August 2006.
17. ^
18. ^ a b c Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America. xxiv–xxv.
19. ^ Frederick II of Prussia (1927). Letters of Voltaire and Frederick the Great, Letter H 7434, 25 January 1778. New York: Brentano's.
20. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 154–155. doi:10.1006/hmat.1996.0015.
21. ^ Gekker, I.R.; Euler, A.A. (2007). "Leonhard Euler's family and descendants". in Bogoliubov, N.N.; Mikhaĭlov, G.K.; Yushkevich, A.P.. Euler and modern science. Mathematical Association of America. ISBN 088385564X. , p. 405.
22. ^ Marquis de Condorcet. "Eulogy of Euler - Condorcet". Retrieved 30 August 2006.
23. ^ a b Boyer, Carl B.; Uta C. Merzbach (1991). A History of Mathematics. John Wiley & Sons. pp. 439–445. ISBN 0-471-54397-7.
24. ^ Wolfram, Stephen. "Mathematical Notation: Past and Future". Retrieved August 2006.
25. ^ a b Wanner, Gerhard; Harrier, Ernst (March 2005). Analysis by its history (1st ed.). Springer. p. 62.
26. ^ Feynman, Richard (1970). "Chapter 22: Algebra". The Feynman Lectures on Physics: Volume I. p. 10.
27. ^ a b Wells, David (1990). "Are these the most beautiful?". Mathematical Intelligencer 12 (3): 37–41. doi:10.1007/BF03024015.
Wells, David (1988). "Which is the most beautiful?". Mathematical Intelligencer 10 (4): 30–31. doi:10.1007/BF03023741.
28. ^ Dunham, William (1999). "3,4". Euler: The Master of Us All. The Mathematical Association of America.
29. ^ Dunham, William (1999). "1,4". Euler: The Master of Us All. The Mathematical Association of America.
30. ^ Caldwell, Chris. The largest known prime by year
31. ^ 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.
32. ^ Peter R. Cromwell (1997). Polyhedra. Cambridge: Cambridge University Press. pp. 189–190.
33. ^ Alan Gibbons (1985). Algorithmic Graph Theory. Cambridge: Cambridge University Press. p. 72.
34. ^ Cauchy, A.L. (1813). "Recherche sur les polyèdres—premier mémoire". Journal de l'Ecole Polytechnique 9 (Cahier 16): 66–86.
35. ^ L'Huillier, S.-A.-J. (1861). "Mémoire sur la polyèdrométrie". Annales de Mathématiques 3: 169–189.
36. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 144–145. doi:10.1006/hmat.1996.0015.
37. ^ Youschkevitch, A P; Biography in Dictionary of Scientific Biography (New York 1970–1990).
38. ^ Home, R.W. (1988). "Leonhard Euler's 'Anti-Newtonian' Theory of Light". Annals of Science 45 (5): 521–533. doi:10.1080/00033798800200371.
39. ^ 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.
40. ^ Calinger, Ronald (1996). "Leonhard Euler: The First St. Petersburg Years (1727–1741)". Historia Mathematica 23 (2): 153–154. doi:10.1006/hmat.1996.0015.
41. ^ 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.
42. ^ 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-Guinness, 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 0883853280
• Fraser, Craig G., 2005, "Leonhard Euler's 1744 book on the calculus of variations" in Grattan-Guinness, 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-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 181–90.
• Richeson, David S. (2008) Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
• Sandifer, Edward C. (2007), The Early Mathematics of Leonhard Euler, Mathematical Association of America. IBSN 0883855593
• 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". Mathematics Magazine 56 (5). November 1983.

# Quotes

Up to date as of January 14, 2010

### From Wikiquote

Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.

Leonhard Euler () Swiss mathematician and physicist, considered to be one of the greatest mathematicians of all time.

## Sourced

To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.
• Madam, I have come from a country where people are hanged if they talk.
• In Berlin, to the Queen Mother of Prussia, on his lack of conversation in his meeting with her, on his return from Russia; as quoted in Science in Russian Culture : A History to 1860 (1963) Alexander Vucinich
• Variant: Madame... I have come from a country where one can be hanged for what one says.
• Now I will have less distraction.
• Upon losing the use of his right eye; as quoted in In Mathematical Circles (1969) by H. Eves
• Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.
• As quoted in Calculus Gems (1992) by G. Simmons
• Since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear ... there is absolutely no doubt that every affect in the universe can be explained satisfactorily from final causes, by the aid of the method of maxima and minima, as it can be from the effective causes themselves ... Of course, when the effective causes are too obscure, but the final causes are readily ascertained, the problem is commonly solved by the indirect method...
• As quoted in The Anthropic Cosmological Principle (1986) by John D. Barrow and Frank J. Tipler, p. 150
• To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.
• As quoted in Fundamentals of Teaching Mathematics at University Level (2000) by Benjamin Baumslag, p. 214

### Introductio in analysin infinitorum (1748)

Translated as Introduction to Analysis of the Infinite (1988-89) by John Blanton (Book I ISBN 0387968245; Book II ISBN 0387971327).
• Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.

He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air. ~ François Arago
Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence. ~ Keith Devlin
Alphabetized by author
• He calculated without any apparent effort, just as men breathe, as eagles sustain themselves in the air.
• François Arago; Variant: Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind.
• The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum.
• Carl B. Boyer on Euler's Introduction to the Analysis of the Infinite in "The Foremost Textbook of Modern Times" (1950)
• The Introductio does not boast an impressive number of editions, yet its influence was pervasive. In originality and in the richness of its scope it ranks among the greatest of textbooks; but it is outstanding also for clarity of exposition. Published two hundred and two years ago, it nevertheless possesses a remarkable modernity of terminology and notation, as well as of viewpoint. Imitation is indeed the sincerest form of flattery.
• Carl B. Boyer in "The Foremost Textbook of Modern Times" (1950)
• Euler calculated the force of the wheels necessary to raise the water in a reservoir … My mill was carried out geometrically and could not raise a drop of water fifty yards from the reservoir. Vanity of vanities! Vanity of geometry!
• Read Euler: he is our master in everything.
• Pierre Simon de Laplace, as quoted in Calculus Gems (1992) variant: Read Euler, read Euler. He is the master of us all.
• He was later to write that he had made some of his best discoveries while holding a baby in his arms surrounded by playing children.
• Richard Mankiewicz, in The Story of Mathematics (2000), p. 142
• Euler and Ramanujan are mathematicians of the greatest importance in the history of constants (and of course in the history of Mathematics ...)
• E. W. Middlemast
• It is the invaluable merit of the great Basle mathematician Leonard Euler, to have freed the analytical calculus from all geometric bounds, and thus to have established analysis as an independent science, which from his time on has maintained an unchallenged leadership in the field of mathematics.
• Thomas Reid, as quoted in Mathematical Maxims and Minims (1988) by N. Rose
• Perhaps the most surprising thing about mathematics is that it is so surprising. The rules which we make up at the beginning seem ordinary and inevitable, but it is impossible to foresee their consequences. These have only been found out by long study, extending over many centuries. Much of our knowledge is due to a comparatively few great mathematicians such as Newton, Euler, Gauss, or Riemann; few careers can have been more satisfying than theirs. They have contributed something to human thought even more lasting than great literature, since it is independent of language.

### On "Euler's identity"

In mathematical analysis, Euler's identity is the equation "$e^{i \pi} + 1 = 0. \,\!$"
Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth. ~ Benjamin Peirce
• One of the most frequently mentioned equations was Euler's equation, $e^{i \pi} + 1 = 0. \,\!$ Respondents called it "the most profound mathematical statement ever written"; "uncanny and sublime"; "filled with cosmic beauty"; and "mind-blowing". Another asked: "What could be more mystical than an imaginary number interacting with real numbers to produce nothing?" The equation contains nine basic concepts of mathematics — once and only once — in a single expression. These are: e (the base of natural logarithms); the exponent operation; π; plus (or minus, depending on how you write it); multiplication; imaginary numbers; equals; one; and zero.
• Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence.
• Keith Devlin, as quoted in Dr. Euler's Fabulous Formula : Cures Many Mathematical Ills (2006) ISBN 978-0691118222
• Our jewel ... one of the most remarkable, almost astounding, formulas in all of mathematics.
• There is a famous formula, perhaps the most compact and famous of all formulas — developed by Euler from a discovery of de Moivre: $e^{i \pi} + 1 = 0. \,\!$ It appeals equally to the mystic, the scientist, the philosopher, the mathematician.
• Edward Kasner and James Newman in Mathematics and the Imagination (1940).
• Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don't know what it means. But we have proved it, and therefore we know it must be the truth.
• Benjamin Peirce, as quoted in notes by W. E. Byerly, published in Benjamin Peirce, 1809-1880 : Biographical Sketch and Bibliography (1925) by R. C. Archibald; also in Mathematics and the Imagination (1940) by Edward Kasner and James Newman

# 1911 encyclopedia

Up to date as of January 14, 2010

### From LoveToKnow 1911

LEONHARD EULER (1707-1783), Swiss mathematician, was born at Basel on the 15th of April 5707, his father Paul Euler, who had considerable attainments as a mathematician, being Calvinistic pastor of the neighbouring village of Riechen. After receiving preliminary instructions in mathematics from his father, he was sent to the university of Basel, where geometry soon became his favourite study. His mathematical genius gained for him a high place in the 'esteem of Jean Bernoulli, who was at that time one of the first mathematicians in Europe, as well as of his sons Daniel and Nicolas Bernoulli. Having taken his degree as master of arts in 1723, Euler applied himself, at his father's desire, to the study of theology and the Oriental languages with the view of entering the church, but, with his father's consent, he soon returned to geometry as his principal pursuit. At the same time, by the advice of the younger Bernoullis, who had removed to St Petersburg in 1725, he applied himself to the study of physiology, to which he made a happy application of his mathematical knowledge; and he also attended the medical lectures at Basel. While he was engaged in physiological researches, he composed a dissertation on the nature and propagation of sound, and an answer to a prize question concerning the masting of ships, to which the French Academy of Sciences adjudged the second rank in the year 5727.

In 1727, on the invitation of Catherine I., Euler took up his residence in St Petersburg, and was made an associate of the Academy of Sciences. In 1730 he became professor of physics, and in 1733 he succeeded Daniel Bernoulli in the chair of mathematics. At the commencement of his new career he enriched the academical collection with many memoirs, which excited a noble emulation between him and the Bernoullis, though this did not in any way affect their friendship. It was at this time that he carried the integral calculus to a higher degree of perfection, invented the calculation of sines, reduced analytical operations to a greater simplicity, and threw new light on nearly all parts of pure mathematics. In 1735 a problem proposed by the academy, for the solution of which several eminent mathematicians had demanded the space of some months, was solvecdby Euler in three days,but the effort threw him into a fever which endangered his life and deprived him of the use of his right eye. The Academy of Sciences at Paris in 1738 adjudged the prize to his memoir on the nature and properties of fire, and in 1740 his treatise on the tides shared the prize with those of Colin Maclaurin and Daniel Bernoulli - a higher honour than if he had carried it away from inferior rivals.

In 1741 Euler accepted the invitation of Frederick the Great to Berlin, where he was made a member of the Academy of Sciences and professor of mathematics. He enriched the last volume of the Mélanges or Miscellanies of Berlin with five memoirs, and these were followed, with an astonishing rapidity, by a great number of important researches, which are scattered throughout the annual memoirs of the Prussian Academy. At the same time he continued his philosophical contributions to the Academy of St Petersburg, which granted him a pension in 1742. The respect in which he was held by the Russians was strikingly shown in 1760, when a farm he occupied near Charlottenburg happened to be pillaged by the invading Russian army. On its being ascertained that the farm belonged to Euler, the general immediately ordered compensation to be paid, and the empress Elizabeth sent an additional sum of four thousand crowns.

In 1766 Euler with difficulty obtained permission from the king of Prussia to return to St Petersburg, to which he had been originally invited by Catherine II. Soon after his return to St Petersburg a cataract formed in his left eye, which ultimately deprived him almost entirely of sight. It was in these circumstances that he dictated to his servant, a tailor's apprentice, who was absolutely devoid of mathematical knowledge, his Anleitung zur Algebra (1770), a work which, though purely elementary, displays the mathematical genius of its author, and is still reckoned one of the best works of its class. Another task to which he set himself immediately after his return to St Petersburg was the preparation of his Lettres a une princesse d'Allemagne sur quelques sujets de physique et de philosophie (3 vols., 1768-1772). They were written at the request of the princess of Anhalt-Dessau, and contain an admirably clear exposition of the principal facts of mechanics, optics, acoustics and physical astronomy. Theory, however, is frequently unsoundly applied in it, and it is to be observed generally that Euler's strength lay rather in pure than in applied mathematics.

In 1755 Euler had been elected a foreign member of the Academy of Sciences at Paris, and some time afterwards the academical prize was adjudged to three of his memoirs Concerning the Inequalities in the Motions of the Planets. The two prizequestions proposed by the same academy for 1770 and 1772 were designed to obtain a more perfect theory of the moon's motion. Euler, assisted by his eldest son Johann Albert, was a competitor for these prizes, and obtained both. In the second memoir he reserved for further consideration several ine q ualities of the moon's motion, which he could not determine in his first theory on account of the complicated calculations in which the method he then employed had engaged him. ' He afterwards reviewed his whole theory with the assistance of his son and W. L. Krafft and A. J. Lexell, and pursued his researches until he had constructed the new tables, which appeared in his Theoria motuum lunae (1772). Instead of confining himself, as before, to the fruitless integration of three differential equations of the second degree, which are furnished by mathematical principles, he reduced them to the three co-ordinates which determine the place of the moon; and he divided into classes all the inequalities of that planet, as far as they depend either on the elongation of the sun and moon, or upon the eccentricity, or the parallax, or the inclination of the lunar orbit. The inherent difficulties of this task were immensely enhanced by the fact that Euler was virtually blind, and had to carry all the elaborate computations it involved in his memory. A further difficulty arose from the burning of his house and the destruction of the greater part of his property in 1771. His manuscripts were fortunately preserved. His own life was only saved by the courage of a native of Basel, Peter Grimmon, who carried him out of the burning house.

Some time after this an operation restored Euler's sight; but a too harsh use of the recovered faculty, along with some carelessness on the part of the surgeons, brought about a relapse. With the assistance of his sons, and of Krafft and Lexell, however, he continued his labours, neither the loss of his sight nor the infirmities of an advanced age being sufficient to check his activity. Having engaged to furnish the Academy of St Petersburg with as many memoirs as would be sufficient to complete its Acta for twenty years after his death, he in seven years transmitted to the academy above seventy memoirs, and left above two hundred more, which were revised and completed by another hand.

Euler's knowledge was more general than might have been expected in one who had pursued with such unremitting ardour mathematics and astronomy as his favourite studies. He had made very considerable progress in medical, botanical and chemical science, and he was an excellent classical scholar, and extensively read in general literature. He was much indebted to an uncommon memory, which seemed to retain every idea that was conveyed to it, either from reading or meditation. He could repeat the Aeneid of Virgil from the beginning to the end without hesitation, and indicate the first and last line of every page of the edition which he used. Euler's constitution was uncommonly vigorous, and his general health was always good. He was enabled to continue his labours;to the very close of his life. His last subject of investigation was the motion of balloons, and the last subject on which he conversed was the newly discovered planet Herschel (Uranus). He died of apoplexy on the ,8th of September 1783, whilst he was amusing himself at tea with one of his grandchildren.

Euler's genius was great and his industry still greater. His works, if printed in their completeness, would occupy from 60 to 80 quarto volumes. He was simple and upright in his character, and had a strong religious faith. He was twice married, his second wife being a half-sister of his first, and he had a numerous family, several of whom attained to distinction. His eloge was written for the French Academy by the marquis de Condorcet, and an account of his life, with a list of his works, was written by Von Fuss, the secretary to the Imperial Academy of St Petersburg.

The works which Euler published separately are: Dissertatio physica de sono (Basel, 1727, in 4to); Mechanica, sive motus scientia analytice exposita (St Petersburg, 1736, in 2 vols. 4to); Einleitung in die Arithmetik (ibid., 1738, in 2 vols. 8vo), in German and Russian; Tentamen novae theoriae musicae (ibid. 1739, in 4to); Methodus inveniendi lineas curvas, maximi minimive proprietate gaudentes (Lausanne, 1744, in 4to); Theoria motuum planetarum et cometarum (Berlin, 1744, in 4to); Beantwortung, &c., or Answers to Different Questions respecting Comets (ibid., 1744, in 8vo); Neue Grundsatze, &c., or New Principles of Artillery, translated from the English of Benjamin Robins, with notes and illustrations (ibid., 1745, in 8vo); Opuscula varii argumenti (ibid., 1746-1751, in 3 vols. 4to); Novae et correctae tabulae ad loca lunae computanda (ibid., 1746, in 4to); Tabulae astronomicae solis et lunae (ibid., 4to); Gedanken, &c., or Thoughts on the Elements of Bodies (ibid. 4to); Rettung der gottlichen Offenbarung, &c., Defence of Divine Revelation against Freethinkers (ibid., 1747, in 4to); Introductio in analysin infinitorum (Lausanne, 1748, in 2 vols. 4to); Scientia navalis, tractatus de construendis ac dirigendis navibus (St Petersburg, 1749, in 2 vols. 4to); Theoria motus lunae (Berlin, 1753, in 4to); Dissertatio de principio minimae actionis, una cum examine objectionum cl. prof. Koenigii (ibid., 1753, in 8vo); Institutiones calculi diferentialis, cum ejus usu in analysi Infinitorum ac doctrina serierum (ibid., 1755, in 4to); Constructio lentium objectivarum, &c. (St Petersburg, 1762, in 4to); Theoria motus corporum solidorum seu rigidorum (Rostock, 1765, in 4to); Institutiones calculi integralis (St Petersburg, 1768-1770, in 3 vols. 4to); Lettres a sine Princesse d'Allemagne sur quelques sujets de physique et de philosophie (St Petersburg, 1768-1772, in 3 vols. 8vo); Anleitung zur Algebra, or Introduction to Algebra (ibid., 1770, in 8vo); Dioptrica (ibid., 1767-1771, in 3 vols. 4to); Theoria motuum lunae nova methodo pertractata (ibid., 1772, in 4to); Novae tabulae lunares (ibid., in 8vo); Theorie complete de la construction et de la manoeuvre des vaisseaux (ibid., 1773, in 8vo); Eclaircissements sur etablissements en faveur tant des veuves que des morts, without a date; Opuscula analytica (St Petersburg, 1783-1785, in 2 vols. 4to). See Rudio, Leonhard Euler (Basel, 1884); M. Cantor, Geschichte der Mathematik.

# Simple English

Leonhard Euler

Leonhard Euler (pronounced Oiler) (April 15, 1707September 7, 1783) was a Swiss mathematician and physicist. He spent most of his life in Russia and Germany.

Euler made important discoveries in fields like calculus and topology. He also made many of the words used in math today. He came up with the idea of a mathematical function.[1] He is also known for his work in mechanics, optics, and astronomy.

Euler is considered to be the most important mathematician of the 18th century, one of the greatest mathematicians of all time, and one of the mathematicians who wrote the most. His collected works fill 60–80 volumes.[2] Another mathematician, Pierre-Simon Laplace said, "Read Euler, read Euler, he is a master for us all".[3]

Euler was featured on the sixth series of the Swiss 10-franc bill[4] and on numerous Swiss, German, and Russian postage stamps. The asteroid 2002 Euler was named in his honor. He is also celebrated by the Lutheran Church on their Calendar of Saints on May 24.

## = Early years

=

File:Euler-10 Swiss Franc banknote (front).jpg
Old Swiss 10 Franc banknote honoring Euler

Euler was born in Basel to Paul Euler. He was a pastor of the Reformed Church. His mother was Marguerite Brucker, a pastor's daughter. 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. Euler's started school in Basel, where he was sent to live with his grandmother. At the age of thirteen he went to the University of Basel. In 1723, he received his Master of Philosophy. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics.[5]

## References

1. Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America. pp. 17.
2. Finkel, B.F. (1897). [Expression error: Unexpected < operator "Biography- Leonard Euler"]. The American Mathematical Monthly 4 (12): 300.
3. Dunham, William (1999). Euler: The Master of Us All. The Mathematical Association of America. pp. xiii. "Lisez Euler, lisez Euler, c'est notre maître à tous."
4.
5. James, Ioan (2002). Remarkable Mathematicians: From Euler to von Neumann. Cambridge. p. 2. ISBN 0-521-52094-0.