Leonhard Paul Euler: his life and his works.Outline 1 Biography Early years St. Petersburgh Berlin...

Leonhard Paul Euler: his life and his works.

Sergey Lapin

MATH 398 // March 20, 2008

S. Lapin () Leonhard Euler 03/20/08 1 / 41

Lisez Euler, lisez Euler, c’est notre maıtre a tous.

– Pierre-Simon Laplace


Outline

1 BiographyEarly yearsSt. PetersburghBerlinReturn to St. Petersburgh

2 Contributions to mathematicsMathematical notationAnalysisNumber theoryGraph theoryApplied mathematicsPhysics and astronomy

3 Selected bibliography


Biography Early years

Outline






Early years

Euler was born in Basel on April 15, 1707.Father: Paul Euler, a pastor of the Reformed Church.Mother: Marguerite Brucker.He had two younger sisters named Anna Maria and Maria Magdalena.Soon after the birth of Leonhard, the Eulers moved to the town of Riehen,where Euler spent most of his childhood.Paul Euler was a friend of the Bernoulli family and Johann Bernoulli, whowas then regarded as Europe’s foremost mathematician.



Early years

Euler’s early formal education started in Basel, where he lived with hismaternal grandmother.Euler’s father wanted his son to follow him into the church and sent himto the University of Basel to prepare for the ministry. He entered theUniversity in 1720, at the age of 13, first to obtain a general educationbefore going on to more advanced studies.Euler was studying theology, Greek, and Hebrew in order to become apastor.



Early years

Johann Bernoulli convinced Paul Euler that Leonhard was destined tobecome a great mathematician.Euler’s own account given in his unpublished autobiographical writings:

... I soon found an opportunity to be introduced to a famousprofessor Johann Bernoulli. ... True, he was very busy and sorefused flatly to give me private lessons; but he gave me muchmore valuable advice to start reading more difficult mathematicalbooks on my own and to study them as diligently as I could; if Icame across some obstacle or difficulty, I was given permission tovisit him freely every Sunday afternoon and he kindly explainedto me everything I could not understand ...



In 1723 Euler completed his Master’s degree in philosophy havingcompared and contrasted the philosophical ideas of Descartes and Newton.Euler completed his PhD at the University of Basel in 1726. He hadstudied many mathematical works during his time in Basel. They includeworks by Varignon, Descartes, Newton, Galileo, van Schooten, JacobBernoulli, Hermann, Taylor and Wallis.By 1726 Euler had already a paper in print, a short article on isochronouscurves in a resisting medium.In 1727 he published another article on reciprocal trajectories andsubmitted an entry for the 1727 Grand Prize of the Paris Academy on thebest arrangement of masts on a ship. He won second place, losing only toPierre Bouguer.Euler subsequently won this annual prize twelve times in his career.


Biography St. Petersburgh

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St. Petersburgh

Around this time Johann Bernoulli’s two sons, Daniel and Nicolas, wereworking at the Imperial Russian Academy of Sciences in St Petersburg.In July 1726, Nicolas died of appendicitis and when Daniel assumed hisbrother’s position in the mathematics/physics division, he recommendedthat his post in physiology be filled by his friend Euler.In November 1726 Euler accepted the offer, but delayed the trip to StPetersburg until spring of 1727.Euler arrived in St. Petersburgh on May 17, 1727. He was promoted fromhis junior post in the medical department of the academy to a position inthe mathematics department.Euler mastered Russian and settled into life in St. Petersburg. He alsotook on an additional job as a medic in the Russian Navy.



St. Petersburgh

The Academy at St. Petersburg, established by Peter the Great, wasintended to improve education in Russia and to close the scientific gapwith Western Europe.The academy possessed ample financial resources and a comprehensivelibrary drawn from the private libraries of Peter himself and of the nobility.The academy emphasized research and offered to its faculty both the timeand the freedom to pursue scientific questions.Euler rose quickly through the ranks in the academy and was madeprofessor of physics in 1731.



St. Petersburgh

When Daniel Bernoulli left St Petersburg to return to Basel in 1733 Eulerwas appointed to be the senior chair of mathematics.On January 7, 1734, he married Katharina Gsell, daughter of a painterfrom the Academy Gymnasium. They had 13 children, but only fivesurvived their infancy.Euler claimed that he made some of his greatest mathematical discoverieswhile holding a baby in his arms with other children playing round his feet.The publication of many articles and his book Mechanica (1736-37),which extensively presented Newtonian dynamics in the form ofmathematical analysis for the first time, started Euler on the way to majormathematical work.


Biography Berlin

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Biography Berlin

Berlin

By 1740 Euler had a very high reputation, having won the Grand Prize ofthe Paris Academy in 1738 and 1740. Political turmoil in Russia forcedEuler to accept an invitation of Frederick the Great of Prussia to take apost in Berlin Academy of Science. He left St. Petersburg on June 19th1741, arriving in Berlin on July 25th.


Biography Berlin

Berlin

Euler’s 25 years in Berlin were very busy and productive. Besides the greatmathematical success he also

served on the Library and Scientific Publications Committee of theBerlin Academy and

was a government advisor on state lotteries, insurance, annuities andpensions, and artillery.

Euler wrote nearly 380 articles during his Berlin period.He also wrote many scientific and popular science books, including famousLetters to a Princess of Germany, which was translated into manylanguages and published almost 40 times. Euler led the Berlin Academy ofScience after the death of Maupertuis in 1759, although he never held theformal title of President.


Biography Berlin

Euler’s health problems began in 1735 when he had a severe fever andalmost lost his life. In his autobiographical writings Euler says that hiseyesight problems began in 1738 with overstrain due to his cartographicwork and that by 1740 he had

... lost an eye and [the other] currently may be in the samedanger.

Euler’s sight in that eye worsened throughout his stay in Germany, somuch so that Frederick referred to him as ”Cyclops”.


Biography Return to St. Petersburgh

Outline






Return to St. Petersburgh

In 1762, the politics in Russia changed. Empress Catherine II, later namedCatherine the Great, came to the throne. The atmosphere in Russiansociety improved dramatically.




Catherine II aimed to create in Russia a regime of Educated Absolutism.She invited many progressive people to Russia and increased the budget ofthe St. Petersburgh Academy to 60000 rubles per year, which was muchmotre than the budget of the Berlin Academy.Catherine II offered Euler an important post in the mathematicsdepartment, conference-secretary of the Academy, with a big salary. Sheinstructed her representative in Berlin to agree to Euler’s terms if he doesnot like her first offer.In 1766 Euler returned to St. Petersburg.




In 1771 Euler’s home was destroyed by fire and he was able to save onlyhimself and his mathematical manuscripts.In September 1771, Euler had surgery to remove his cataract. The surgerywas very successful - the mathematicians vision was restored.Unfortunately, Euler didnt take care of his eyes; he continued to work andafter a few days lost his vision again, this time without any hope ofrecovery.However, because of his remarkable memory he was able to continue withhis work on optics, algebra, and lunar motion.Amazingly after his return to St Petersburg he produced almost half histotal works despite the total blindness!




Euler achieved this remarkable level of output with help of his sons,

Johann Albrecht Euler who was appointed to the chair of physics atthe Academy in St Petersburg in 1766 and

Christoph Euler.

Euler was also helped by two other members of the Academy

W. L. Krafft,

A. J. Lexell,

and Nikolaus von Fuss who was invited to the Academy from Switzerlandin 1772 and became Euler’s assistant in 1776.




On September 18, 1783, Euler passed away in St. Petersburg aftersuffering a stroke, and was buried with his wife in the Smolensk LutheranCemetery on Vasilievsky Island. His eulogy was written for the FrenchAcademy by the French mathematician and philosopher Marquis deCondorcet, and an account of his life, with a list of his works, by Nikolausvon Fuss, Euler’s grandson-in-law and the secretary of the ImperialAcademy of St. Petersburg.

...il cessa de calculer et de vivre -

... he ceased to calculate and to live.

– Marquis de Condorcet


Contributions to mathematics

Contributions to mathematics

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 one of the most prolific of all time; his publication list of 886 papersand books exceeded only by Paul Erdos.After Euler’s death in 1783 the St. Petersburg Academy continued topublish Euler’s unpublished work for nearly 50 more years.


Contributions to mathematics Mathematical notation

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Contributions to mathematics Mathematical notation

Mathematical notation

Euler introduced and popularized several notational conventions.He introduced the concept of a function and was the first to write f (x) todenote 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,

the Greek letter Σ for summations,

the letter i to denote the imaginary unit.

The use of the Greek letter π to denote the ratio of a circle’scircumference to its diameter was also popularized by Euler, although itdid not originate with him.


Contributions to mathematics Analysis

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Analysis

The development of calculus was at the forefront of 18th centurymathematical research. Thanks to the influence of Bernoulli family,studying calculus became the major focus of Euler’s work.He is well known in analysis for his frequent use and development of powerseries, such as

ex =

∞∑

n=0

xn

n!= lim

n→∞

(

1

0!+

x

1!+

x2

2!+ · · · +

xn

n!

)

.

Euler discovered the power series expansions for e and the inverse tangentfunction. Use of power series enabled him to solve the famous Baselproblem in 1735

limn→∞

(

1

12+

1

22+

1

32+ · · · +

1

n2

)

=π

2

6.



Analysis

Euler introduced the use of the exponential function and logarithms inanalytic proofs.

He discovered ways to express various logarithmic functions usingpower series,

and he successfully defined logarithms for negative and complexnumbers, thus greatly expanding the scope of mathematicalapplications of logarithms.

He also defined the exponential function for complex numbers, anddiscovered its relation to the trigonometric functions. For any real numberϕ, Euler’s formula states that

e iϕ = cos ϕ + i sinϕ.

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

e iπ + 1 = 0

voted ”the Most Beautiful Mathematical Formula Ever”.S. Lapin () Leonhard Euler 03/20/08 28 / 41

Contributions to mathematics Number theory

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Number theory

A lot of Euler’s early work on number theory was based on the works ofPierre de Fermat. Euler developed some of Fermat’s ideas, and disprovedsome of his conjectures.Euler linked the nature of prime distribution with ideas in analysis. Heproved that the sum of the reciprocals of the primes diverges. In doing so,he discovered the connection between the Riemann zeta function and theprime numbers; this is known as the Euler product formula for theRiemann 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.



Number theory

Generalized Fermat’s little theorem to what is now known as Euler’stheorem.

Contributed significantly to the understanding of perfect numbers,which has fascinated mathematicians since Euclid.

Made progress toward the prime number theorem, and he conjecturedthe law of quadratic reciprocity.

The two concepts are regarded as fundamental theorems of number theory.By 1772 Euler had proved that 231

− 1 = 2, 147, 483, 647 is a Mersenneprime. It have remained the largest known prime until 1867.


Contributions to mathematics Graph theory

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Contributions to mathematics Graph theory

Graph theory

In 1736, Euler solved the problem known as the Seven Bridges ofKonigsberg.Problem:Is it possible to follow a path that crosses each bridge exactlyonce and returns to the starting point?Answer: It is not.This solution is considered to be the first theorem of graph theory andplanar graph theory. Euler also introduced the notion now known as theEuler characteristic of a space and a formula relating the number of edges,vertices, and faces of a convex polyhedron with this constant.


Contributions to mathematics Applied mathematics

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Applied mathematics

Some of Euler’s greatest successes were in solving real-world problemsanalytically. He

integrated Leibniz’s differential calculus with Newton’s Method ofFluxions,

developed tools that made it easier to apply calculus to physicalproblems,

invented what are now known as the Euler approximations whichleaded to:

Euler’s method and the Euler-Maclaurin formula.

Euler also facilitated the use of differential equations, in particularintroducing the Euler-Mascheroni constant:

γ = limn→∞

(

1 +1

2+

1

3+

1

4+ · · · +

1

n− ln(n)

)

.



Applied mathematics

One of Euler’s more unusual interests was the application of mathematicalideas in music.In 1739 he wrote the Tentamen novae theoriae musicae, hoping toincorporate musical theory as part of mathematics. However, this work didnot receive wide attention and was described as too mathematical formusicians and too musical for mathematicians.Also, Euler became involved in cartography when he was appointeddirector of the St Petersburg Academy’s geography section in 1735. Asthe result the Russian Atlas appeared in 1745, consisting of 20 maps.Euler, in Berlin by the time of its publication, proudly remarked that thiswork put the Russians well ahead of the Germans in the art of cartography.


Contributions to mathematics Physics and astronomy

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Contributions to mathematics Physics and astronomy

Physics and astronomy

Euler helped develop the Euler-Bernoulli beam equation, whichbecame a cornerstone of engineering.

He applied his analytic tools to problems in classical mechanics,

and to celestial problems. His work in astronomy was recognized by anumber of Paris Academy Prizes.

His accomplishments in astronomy include

determining the orbits of comets and other celestial bodies,

understanding the nature of comets, and

calculating the parallax of the sun.

In addition, Euler made important contributions in optics.


Selected bibliography

Euler’s bibliography

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

Elements of Algebra. This elementary algebra text starts with adiscussion of the nature of numbers and gives a comprehensiveintroduction to algebra, including formulae for solutions of polynomialequations.

Introductio in analysin infinitorum (1748). English translationIntroduction to Analysis of the Infinite by John Blanton (Book I,ISBN 0-387-96824-5, Springer-Verlag 1988; Book II, ISBN0-387-97132-7, Springer-Verlag 1989).

Two influential textbooks on calculus: Institutiones calculidifferentialis (1755) and Institutiones calculi integralis (1768-1770).



Lettres a une Princesse d’Allemagne (Letters to a German Princess)(1768-1772). English translation, with notes, and a life of Euler,available online from Google Books: Volume 1, Volume 2

Methodus inveniendi lineas curvas maximi minimive proprietategaudentes, sive solutio problematis isoperimetrici latissimo sensuaccepti (1744). (Method for finding curved lines enjoying propertiesof maximum or minimum, or solution of isoperimetric problems in thebroadest accepted sense.)

A definitive collection of Euler’s works, entitled Opera Omnia, has beenpublished since 1911 by the Euler Commission of the Swiss Academy ofSciences.



The End.

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Leonhard Paul Euler: his life and his works.Outline 1 Biography Early years St. Petersburgh Berlin...

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