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Home , Calculus, Euler method, Exponential function, Leonhard Euler, Pierre de Fermat, Tangent

Leonhard Paul : his life and his works.

Sergey Lapin

MATH 398 // March 20, 2008

S. Lapin () 03/20/08 1 / 41 Lisez Euler, lisez Euler, c’est notre maˆıtre a` tous. – Pierre-Simon Laplace

S. Lapin () Leonhard Euler 03/20/08 2 / 41 Outline

1 Biography Early years St. Petersburgh Return to St. Petersburgh

2 Contributions to Mathematical theory and

3 Selected bibliography

S. Lapin () Leonhard Euler 03/20/08 3 / 41 Biography Early years Outline

1 Biography Early years St. Petersburgh Berlin Return to St. Petersburgh

2 Contributions to mathematics Analysis Graph theory Applied mathematics Physics and astronomy

3 Selected bibliography

S. Lapin () Leonhard Euler 03/20/08 4 / 41 Biography Early years Early years

Euler was born in 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 , who was then regarded as Europe’s foremost .

S. Lapin () Leonhard Euler 03/20/08 5 / 41 Biography Early years Early years

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

S. Lapin () Leonhard Euler 03/20/08 6 / 41 Biography Early years Early years

Johann Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician. Euler’s own account given in his unpublished autobiographical writings: ... I soon found an opportunity to be introduced to a famous professor Johann Bernoulli. ... True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand ...

S. Lapin () Leonhard Euler 03/20/08 7 / 41 Biography Early years

In 1723 Euler completed his Master’s in philosophy having compared and contrasted the philosophical ideas of Descartes and . Euler completed his PhD at the University of Basel in 1726. He had studied many mathematical works during his in Basel. They include works by Varignon, Descartes, Newton, Galileo, van Schooten, Jacob Bernoulli, Hermann, Taylor and Wallis. By 1726 Euler had already a paper in print, a short article on isochronous in a resisting medium. In 1727 he published another article on reciprocal trajectories and submitted an entry for the 1727 Grand Prize of the Paris Academy on the best arrangement of masts on a ship. He won second place, losing only to . Euler subsequently won this annual prize twelve in his career.

S. Lapin () Leonhard Euler 03/20/08 8 / 41 Biography St. Petersburgh Outline

1 Biography Early years St. Petersburgh Berlin Return to St. Petersburgh

2 Contributions to mathematics Mathematical notation Analysis Number theory Graph theory Applied mathematics Physics and astronomy

3 Selected bibliography

S. Lapin () Leonhard Euler 03/20/08 9 / 41 Biography St. Petersburgh St. Petersburgh

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

S. Lapin () Leonhard Euler 03/20/08 10 / 41 Biography St. Petersburgh St. Petersburgh

The Academy at St. Petersburg, established by , was intended to improve education in Russia and to close the scientific gap with Western Europe. The academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. The academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions. Euler rose quickly through the ranks in the academy and was made professor of physics in 1731.

S. Lapin () Leonhard Euler 03/20/08 11 / 41 Biography St. Petersburgh St. Petersburgh

When left St Petersburg to return to Basel in 1733 Euler was appointed to be the senior chair of mathematics. On January 7, 1734, he married Katharina Gsell, daughter of a painter from the Academy Gymnasium. They had 13 children, but only five survived their infancy. Euler claimed that he made some of his greatest mathematical discoveries while holding a baby in his arms with other children playing round his feet. The publication of many articles and his book (1736-37), which extensively presented Newtonian in the form of for the first time, started Euler on the way to major mathematical .

S. Lapin () Leonhard Euler 03/20/08 12 / 41 Biography Berlin Outline

1 Biography Early years St. Petersburgh Berlin Return to St. Petersburgh

2 Contributions to mathematics Mathematical notation Analysis Number theory Graph theory Applied mathematics Physics and astronomy

3 Selected bibliography

S. Lapin () Leonhard Euler 03/20/08 13 / 41 Biography Berlin Berlin

By 1740 Euler had a very high reputation, having won the Grand Prize of the Paris Academy in 1738 and 1740. Political turmoil in Russia forced Euler to accept an invitation of of Prussia to take a post in Berlin Academy of . He left St. Petersburg on June 19th 1741, arriving in Berlin on July 25th. S. Lapin () Leonhard Euler 03/20/08 14 / 41 Biography Berlin Berlin

Euler’s 25 years in Berlin were very busy and productive. Besides the great mathematical success he also served on the Library and Scientific Publications Committee of the Berlin Academy and was a government advisor on state lotteries, insurance, annuities and pensions, and artillery. Euler wrote nearly 380 articles during his Berlin period. He also wrote many scientific and popular science books, including famous Letters to a Princess of Germany, which was translated into many languages and published almost 40 times. Euler led the Berlin Academy of Science after the death of Maupertuis in 1759, although he never held the formal title of President.

S. Lapin () Leonhard Euler 03/20/08 15 / 41 Biography Berlin

Euler’s health problems began in 1735 when he had a severe fever and almost lost his life. In his autobiographical writings Euler says that his eyesight problems began in 1738 with overstrain due to his cartographic work and that by 1740 he had ... lost an eye and [the other] currently may be in the same danger. Euler’s sight in that eye worsened throughout his stay in Germany, so much so that Frederick referred to him as ”Cyclops”. S. Lapin () Leonhard Euler 03/20/08 16 / 41 Biography Return to St. Petersburgh Outline

1 Biography Early years St. Petersburgh Berlin Return to St. Petersburgh

2 Contributions to mathematics Mathematical notation Analysis Number theory Graph theory Applied mathematics Physics and astronomy

3 Selected bibliography

S. Lapin () Leonhard Euler 03/20/08 17 / 41 Biography Return to St. Petersburgh Return to St. Petersburgh

In 1762, the politics in Russia changed. Empress Catherine II, later named , came to the throne. The in Russian society improved dramatically.

S. Lapin () Leonhard Euler 03/20/08 18 / 41 Biography Return to St. Petersburgh Return to St. Petersburgh

Catherine II aimed to create in Russia a regime of Educated Absolutism. She invited many progressive people to Russia and increased the budget of the St. Petersburgh Academy to 60000 rubles per year, which was much motre than the budget of the Berlin Academy. Catherine II offered Euler an important post in the mathematics department, conference-secretary of the Academy, with a big salary. She instructed her representative in Berlin to agree to Euler’s terms if he does not like her first offer. In 1766 Euler returned to St. Petersburg.

S. Lapin () Leonhard Euler 03/20/08 19 / 41 Biography Return to St. Petersburgh Return to St. Petersburgh

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

S. Lapin () Leonhard Euler 03/20/08 20 / 41 Biography Return to St. Petersburgh Return to St. Petersburgh

Euler achieved this remarkable level of output with help of his sons, Johann Albrecht Euler who was appointed to the chair of physics at the 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 in 1772 and became Euler’s assistant in 1776.

S. Lapin () Leonhard Euler 03/20/08 21 / 41 Biography Return to St. Petersburgh Return to St. Petersburgh

On September 18, 1783, Euler passed away in St. Petersburg after suffering a stroke, and was buried with his wife in the Smolensk Lutheran Cemetery on Vasilievsky Island. His eulogy was written for the French Academy by the French mathematician and philosopher , and an account of his life, with a list of his works, by Nikolaus von Fuss, Euler’s grandson-in-law and the secretary of the Imperial Academy of St. Petersburg. ...il cessa de calculer et de vivre - ... he ceased to calculate and to live. – Marquis de Condorcet

S. Lapin () Leonhard Euler 03/20/08 22 / 41 Contributions to mathematics Contributions to mathematics

Euler worked in almost all of mathematics: , , , algebra, and number theory, as well as continuum physics, and other areas of physics. He is one of the most prolific of all time; his publication list of 886 papers and books exceeded only by Paul Erd¨os. After Euler’s death in 1783 the St. Petersburg Academy continued to publish Euler’s unpublished work for nearly 50 more years.

S. Lapin () Leonhard Euler 03/20/08 23 / 41 Contributions to mathematics Mathematical notation Outline

1 Biography Early years St. Petersburgh Berlin Return to St. Petersburgh

2 Contributions to mathematics Mathematical notation Analysis Number theory Graph theory Applied mathematics Physics and astronomy

3 Selected bibliography

S. Lapin () Leonhard Euler 03/20/08 24 / 41 Contributions to mathematics Mathematical notation Mathematical notation

Euler introduced and popularized several notational conventions. He introduced the concept of a 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 , the letter e for the base of the natural , the Greek letter Σ for , the letter i to denote the . The use of the Greek letter π to denote the ratio of a ’s to its was also popularized by Euler, although it did not originate with him.

S. Lapin () Leonhard Euler 03/20/08 25 / 41 Contributions to mathematics Analysis Outline

1 Biography Early years St. Petersburgh Berlin Return to St. Petersburgh

2 Contributions to mathematics Mathematical notation Analysis Number theory Graph theory Applied mathematics Physics and astronomy

3 Selected bibliography

S. Lapin () Leonhard Euler 03/20/08 26 / 41 Contributions to mathematics Analysis Analysis

The development of calculus was at the forefront of 18th century mathematical 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 , such as ∞ n 2 n x x 1 x x x e = = lim + + + · · · + . n! n→∞ 0! 1! 2! n!  Xn=0 Euler discovered the expansions for e and the inverse function. Use of power series enabled him to solve the famous in 1735

1 1 1 1 π2 lim + + + · · · + = . n→∞ 12 22 32 n2  6

S. Lapin () Leonhard Euler 03/20/08 27 / 41 Contributions to mathematics Analysis Analysis

Euler introduced the use of the and in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex , 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 trigonometric functions. For any ϕ, Euler’s formula states that i e ϕ = cos ϕ + i sin ϕ. A special case of the above formula is known as Euler’s , i e π + 1 = 0 voted ”the Most Beautiful Mathematical Formula Ever”. S. Lapin () Leonhard Euler 03/20/08 28 / 41 Contributions to mathematics Number theory Outline

1 Biography Early years St. Petersburgh Berlin Return to St. Petersburgh

2 Contributions to mathematics Mathematical notation Analysis Number theory Graph theory Applied mathematics Physics and astronomy

3 Selected bibliography

S. Lapin () Leonhard Euler 03/20/08 29 / 41 Contributions to mathematics Number theory Number theory

A lot of Euler’s early work on number theory was based on the works of . Euler developed some of Fermat’s ideas, and disproved some of his . Euler linked the nature of 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 and the prime numbers; this is known as the 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.

S. Lapin () Leonhard Euler 03/20/08 30 / 41 Contributions to mathematics Number theory Number theory

Generalized Fermat’s little theorem to what is now known as Euler’s theorem. Contributed significantly to the understanding of perfect numbers, which has fascinated mathematicians since . Made progress toward the theorem, and he conjectured the law of . 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 . It have remained the largest known prime until 1867.

S. Lapin () Leonhard Euler 03/20/08 31 / 41 Contributions to mathematics Graph theory Outline

1 Biography Early years St. Petersburgh Berlin Return to St. Petersburgh

2 Contributions to mathematics Mathematical notation Analysis Number theory Graph theory Applied mathematics Physics and astronomy

3 Selected bibliography

S. Lapin () Leonhard Euler 03/20/08 32 / 41 Contributions to mathematics Graph theory Graph theory

In 1736, Euler solved the problem known as the Seven Bridges of K¨onigsberg. Problem:Is it possible to follow a path that crosses each bridge exactly once and returns to the starting ? Answer: It is not. This solution is considered to be the first theorem of graph theory and theory. Euler also introduced the notion now known as the of a and a formula relating the number of edges, vertices, and faces of a convex with this .

S. Lapin () Leonhard Euler 03/20/08 33 / 41 Contributions to mathematics Applied mathematics Outline

1 Biography Early years St. Petersburgh Berlin Return to St. Petersburgh

2 Contributions to mathematics Mathematical notation Analysis Number theory Graph theory Applied mathematics Physics and astronomy

3 Selected bibliography

S. Lapin () Leonhard Euler 03/20/08 34 / 41 Contributions to mathematics Applied mathematics Applied mathematics

Some of Euler’s greatest successes were in solving real-world problems analytically. He integrated Leibniz’s differential calculus with Newton’s Method of , developed tools that made it easier to apply calculus to physical problems, invented what are now known as the Euler approximations which leaded to: Euler’s method and the Euler-Maclaurin formula. Euler also facilitated the use of differential , in particular introducing the Euler-Mascheroni constant:

1 1 1 1 γ = lim 1 + + + + · · · + − ln(n) . n→∞  2 3 4 n 

S. Lapin () Leonhard Euler 03/20/08 35 / 41 Contributions to mathematics Applied mathematics Applied mathematics

One of Euler’s more unusual was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to incorporate musical theory as part of mathematics. However, this work did not receive wide attention and was described as too mathematical for musicians and too musical for mathematicians. Also, Euler became involved in when he was appointed director of the St Petersburg Academy’s geography section in 1735. As the result the Russian Atlas appeared in 1745, consisting of 20 maps. Euler, in Berlin by the time of its publication, proudly remarked that this work put the well ahead of the Germans in the art of cartography.

S. Lapin () Leonhard Euler 03/20/08 36 / 41 Contributions to mathematics Physics and astronomy Outline

1 Biography Early years St. Petersburgh Berlin Return to St. Petersburgh

2 Contributions to mathematics Mathematical notation Analysis Number theory Graph theory Applied mathematics Physics and astronomy

3 Selected bibliography

S. Lapin () Leonhard Euler 03/20/08 37 / 41 Contributions to mathematics Physics and astronomy Physics and astronomy

Euler helped develop the Euler-Bernoulli beam , which became a cornerstone of . He applied his analytic tools to problems in classical , and to celestial problems. His work in astronomy was recognized by a number of Paris Academy Prizes. His accomplishments in astronomy include determining the of comets and other celestial bodies, understanding the nature of comets, and calculating the of the sun. In , Euler made important contributions in optics.

S. Lapin () Leonhard Euler 03/20/08 38 / 41 Selected bibliography Euler’s bibliography

Euler has an extensive bibliography but his best known books include: Elements of Algebra. This text starts with a discussion of the nature of numbers and gives a comprehensive introduction to algebra, including formulae for solutions of equations. Introductio in analysin infinitorum (1748). English Introduction to Analysis of the Infinite by John Blanton (Book I, ISBN 0-387-96824-5, Springer-Verlag 1988; Book II, ISBN 0-387-97132-7, Springer-Verlag 1989). Two influential textbooks on calculus: Institutiones calculi differentialis (1755) and Institutiones calculi integralis (1768-1770).

S. Lapin () Leonhard Euler 03/20/08 39 / 41 Selected bibliography

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: 1, Volume 2 Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti (1744). (Method for finding curved lines enjoying properties of maximum or minimum, or solution of isoperimetric problems in the broadest accepted sense.) 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.

S. Lapin () Leonhard Euler 03/20/08 40 / 41 Selected bibliography The End.

THANK YOU.

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