Leonhard Euler: Difference between revisions

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friend of the ]—], who was then regarded as Europe's foremost ], 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 matriculated at the ], and in 1723, received his ] with a dissertation that compared the philosophies of ] and ]. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics.<ref name="childhood">{{cite book |last= James |first= Ioan |title= Remarkable Mathematicians: From Euler to von Neumann |publisher= Cambridge |year= 2002|pages=2 |isbn= 0-521-52094-0}}</ref> Euler was at this point studying ], ], and ] 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 ] with the title ''De Sono''<ref>{{PDFlink||232 ]}}</ref> and in 1727, he entered the ''] Prize Problem'' competition, where the problem that year was to find the best way to place the ]s on a ship. He won second place, losing only to ]—who is now known as "the father of naval architecture". Euler subsequently won this coveted annual prize twelve times in his career.<ref name="prize">{{cite journal| author = Calinger, Ronald | year = 1996| title = Leonhard Euler: The First St. Petersburg Years (1727–1741)| journal = Historia Mathematica| volume = 23| issue = 2| pages = 156 | doi = 10.1006/hmat.1996.0015}}</ref>

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philosopher's arguments for ] 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 ]: 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, <math>\frac{a+b^n}{z}=x</math>, 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.<ref name="diderot">{{cite journal| last = Brown | first = B.H.| year = 1942| month = May| title = The Euler-Diderot Anecdote| journal =The American Mathematical Monthly| volume = 49| issue = 5| pages = 302–303| doi = 10.2307/2303096}}; {{cite journal| last = Gillings | first = R.J.| year = 1954| month = February| title = The So-Called Euler-Diderot Incident| journal =The American Mathematical Monthly| volume = 61| issue = 2| pages = 77–80| doi = 10.2307/2307789}}</ref>

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him his home, and almost his life. In 1773, he lost his wife of 40 years. Three years after his wife's death Euler married her half sister, Salome Abigail Gsell (1723–1794).<ref>{{Cite book | first1=I.R. | last1=Gekker | first2=A.A. | last2=Euler | contribution=Leonhard Euler's family and descendants | editor1-first=N.N. | editor1-last=Bogoliubov | editor2-first=G.K. | editor2-last=Mikhaĭlov | editor3-first=A.P. |editor3-last=Yushkevich | title=Euler and modern science | publisher=Mathematical Association of America | year=2007 | isbn=088385564X }}, p. 405.</ref> This marriage would last until his death.

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3868: 1041: 1412: 1188:, 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. 1318:. 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. 3641: 868: 2208: 1489: 1426:, 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 of 40 years. Three years after his wife's death Euler married her half sister, Salome Abigail Gsell (1723–1794). This marriage would last until his death. 706: 1349: 71: 501:

he proved the ] using the divergence of the ], and he used analytic methods to gain some understanding of the way ] are distributed. Euler's work in this area led to the development of the ].<ref name="analysis">{{cite book| last = Dunham| first = William| title = Euler: The Master of Us All | year = 1999| publisher =The Mathematical Association of America | chapter = 3,4 }}</ref>

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3859:, 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. 527:

dislike of the doctrine; he went so far as to label Wolff's ideas as "heathen and atheistic".<ref name="wolff">{{cite journal| author = Calinger, Ronald | year = 1996| title = Leonhard Euler: The First St. Petersburg Years (1727–1741)| journal = Historia Mathematica| volume = 23| issue = 2| pages = 153–154 | doi = 10.1006/hmat.1996.0015}}</ref>

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2202: 3693:, 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 3769:. 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". 1993: 1321:

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

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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

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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, ]. In breaking ground for this new field, Euler created the theory of ], ], ] and the analytic theory of ]. For example,

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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

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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

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The situation in Russia had improved greatly since the accession to the throne of ], 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

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Euler was born in ] to Paul Euler, a ] of the ], 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 ], where Euler spent most of his childhood. Paul Euler was a

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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

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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

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Euler and his friend ] were opponents of ] ] and the philosophy of ]. 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

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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 ] was visiting Russia on Catherine the Great's invitation. However, the Empress was alarmed that the

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On 18 September 1783, Euler died in St. Petersburg after suffering a ], and was buried with his wife in the Smolensk Lutheran Cemetery on ] (the Soviets destroyed the cemetery after transferring Euler's remains to the Orthodox ]). His eulogy was written for the French Academy by the French

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The great 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

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In addition, Euler elaborated the theory of higher ]s by introducing the ] and introduced a new method for solving ]s. He also found a way to calculate integrals with complex limits, foreshadowing the development of modern ], and invented the ] including its best-known result, the

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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

2017: 168: 63: 1804: 1786:—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 551:>{{cite journal| last = Euler| first = Leonhard | editor = Orell-Fussli| year = 1960| title = Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister| journal = Leonhardi Euleri Opera Omnia (series 3)| volume = 12 }}< 1389:

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

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function. (Indirect proof via the inverse power series technique was given by Newton and Leibniz between 1670 and 1680.) His daring (and, by modern standards, technically incorrect) use of power series enabled him to solve the famous

1076:, 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 matriculated at the 5699: 2770: 2308: 1104:

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

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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

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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

2197:{\displaystyle \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}}.} 15: 3892:. 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. 1092:. At this time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil's incredible talent for mathematics. Euler was at this point studying 5624: 1060:, 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 1314:

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

2602: 1195:, 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 5689: 3857: 2913:

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.

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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,

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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

5639: 2484: 2227:, thus greatly expanding the scope of mathematical applications of logarithms. He also defined the exponential function for complex numbers, and discovered its relation to the 415:

mathematician and philosopher ], 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 ]. Condorcet commented,

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mathematician and philosopher ], 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 ]. Condorcet commented,

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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

1988:{\displaystyle 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).} 2480: 5143:

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.

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Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a

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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 "

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in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and

5664: 4657: 4266: 225: 151: 3811:: 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, 5669: 3938: 1696:. 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 1651: 5600: 5597: 5594: 4231: 916: 565:

show that Euler was a devout ] who believed the Bible to be inspired; the ''Rettung'' was primarily an argument for the ].<ref name="theology"/>

3880: 2370:, for its single uses of the notions of addition, multiplication, exponentiation, and equality, and the single uses of the important constants 0, 1, 1199:. The nobility were suspicious of the academy's foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues. 3948:

a method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense

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show that Euler was a devout ] who believed the Bible to be inspired; the ''Rettung'' was primarily an argument for the ].<ref name="theology"

2891:{\displaystyle \gamma =\lim _{n\rightarrow \infty }\left(1+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{4}}+\cdots +{\frac {1}{n}}-\ln(n)\right).} 3672: 4538:

Gekker, I.R.; Euler, A.A. (2007). "Leonhard Euler's family and descendants". In Bogoliubov, N.N.; Mikhaĭlov, G.K.; Yushkevich, A.P. (eds.).

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Gekker, I.R.; Euler, A.A. (2007). "Leonhard Euler's family and descendants". In Bogoliubov, N.N.; Mikhaĭlov, G.K.; Yushkevich, A.P. (eds.).

1128:—who is now known as "the father of naval architecture". Euler subsequently won this coveted annual prize twelve times in his career. 2384:

voted it "the Most Beautiful Mathematical Formula Ever". In total, Euler was responsible for three of the top five formulae in that poll.

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commemorating the 250th birthday of Euler. The text says: 250 years from the birth of the great mathematician, academician Leonhard Euler.

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Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti

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

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

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was visiting Russia on Catherine the Great's invitation. However, the Empress was alarmed that the philosopher's arguments for

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Is the normal distribution due to Gauss? Euler, his family of gamma functions, and their place in the history of statistics,

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Euler, Leonhard (1960). Orell-Fussli (ed.). "Rettung der Göttlichen Offenbahrung Gegen die Einwürfe der Freygeister".

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by John Blanton (Book I, ISBN 0-387-96824-5, Springer-Verlag 1988; Book II, ISBN 0-387-97132-7, Springer-Verlag 1989).

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Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the

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in 1735 he became nearly blind in his right eye, but Euler rather blamed his condition on the painstaking work on

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He is listed by academic genealogy authorities as the equivalent to the doctoral advisor of Joseph Louis Lagrange.

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Leonhard Euler’s methods and ideas live on in the thermodynamic hierarchical theory of biological evolution,

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expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all."

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Cite error: The named reference "Boyer" was defined multiple times with different content (see the

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Cite error: The named reference "prize" was defined multiple times with different content (see the

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Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.

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and settled into life in St Petersburg. He also took on an additional job as a medic in the

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Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess

990: 3595: 8: 5566: 5329: 3929:(in French). English translation, with notes, and a life of Euler, available online from 3690: 3375: 3316: 3294: 3039: 3034: 3029: 2929: 2681: 2562: 2314: 1529: 1077: 783: 1040: 5475: 5153: