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 ma^ıtre a` tous. { Pierre-Simon Laplace S. Lapin () Leonhard Euler 03/20/08 2 / 41 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 3 / 41 Biography Early years 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 4 / 41 Biography Early years 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, who was then regarded as Europe's foremost mathematician. 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 University of Basel 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 Euler's own account given in his unpublished autobiographical writings: ...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 degree in philosophy having compared and contrasted the philosophical ideas of Descartes and Newton. Euler completed his PhD at the University of Basel in 1726. He had studied many mathematical works during his time 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 curves 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 Pierre Bouguer. Euler subsequently won this annual prize twelve times 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 Sciences in St Petersburg. In July 1726, Nicolas died of appendicitis and when Daniel assumed his brother's position 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 spring 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 Peter the Great, 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 Daniel Bernoulli 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 Mechanica (1736-37), which extensively presented Newtonian dynamics in the form of mathematical analysis for the first time, started Euler on the way to major mathematical work. 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 Frederick the Great of Prussia to take a post in Berlin Academy of Science. 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 Catherine the Great, came to the throne. The atmosphere 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 mathematicians 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 optics, algebra, and lunar motion. Amazingly after his return to St Petersburg he produced almost half his total works despite the total blindness! S.