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Monday, October 1, 2012

Leonard Euler


Leonhard Euler advanced every known field of mathematics in his day. A prolific author, among his greatest writings are treatises on analytic geometry, differential and integral calculus, and the calculusof variations. Euler developed spherical trigonometry, demonstrated the importance of convergence in algebraic series, proved important assertions in number theory, and made contributions to hydrodynamics, celestial mechanics, and optics. Euler also brought into common usage such mathematical notations "e" for the base of the natural logarithm, "i" for the square root of negative 1, and f(x) for a function of x. A variety of mathematical concepts bear his name, including Euler's characteristicin topology, Euler's trianglein geometry, Euler's polynomials, Euler's integrals, and Euler's constant. His accomplishments are especially remarkable in that many were made during the last quarter of his life, when he was totally blind.
Euler was born in Basel, Switzerland, on April 17, 1707 to Paul Euler and Marguerite Brucker. In 1708, his family moved to the nearby village of Reichen, where his father, a Calvinist pastor, had taken a parish. Before joining the clergy, Euler's father had studied mathematics under the tutelage of Jacob Bernoulli. Following in his father's footsteps, Euler also took his formal education in religion and mathematics, studying theology and Hebrew at the University of Basel, and taking weekly mathematics lessons from Johann Bernoulli, Jacob Bernoulli's younger brother. The Bernoullis recognized Euler's talent and when he received his master's degree from the University of Basel at age 17, they advised him to pursue a mathematical career. The advice was met with resistance from Euler's father, who wished for his son to inherit the pastorship in Reichen. Euler was to remain a devout Calvinist throughout his life, but the Bernoullis eventually convinced his father that Euler's true destiny was not with the church.
When Euler was 19 years old, he produced his first mathematical work, entering a contest sponsored by the French Académie Royale des Sciences. The object of the contest was to solve a problem related to the optimum placement of masts on sailing ships. Euler received an honorable mention for his effort, his solution suffering primarily in the area of practicality. Having not yet traveled outside of Switzerland, he had never seen a ship. Over the course of his career, Euler would eventually receive a total of twelve prizes from the French Académie for his mathematical solutions.
Inherits Top Mathematical Position at St. Petersburg Academy
Around the time that Euler was attempting to solve the ship mast problem, he was also trying, unsuccessfully, to obtain a post as a professor of mathematics at the University of Basel. Determined to hold an academic position, he corresponded with friends Daniel Bernoulli and his cousin, Nicolaus, who were members of the newly established St. Petersburg Academy of Sciences. They wrote to him about a post available in the medical section of the Academy, and, hoping to qualify, Euler immediately began studying physiology. Within three months he was considered sufficiently prepared for the medical post, and in 1727 he traveled to St. Petersburg to join the Academy. Euler's arrival in Russia coincided with the death of Catherine, the wife of Peter the Great. A period of political oppression ensued, lasting several decades, and in the initial turmoil Euler slipped quietly into the Academy's mathematical section. Academic and political freedoms eventually became so stifled that, in 1733, Daniel Bernoulli decided to leave Russia and return to Switzerland. Euler, then 26 years old, inherited Bernoulli's post, the top mathematical position in St. Petersburg. Two years later, he lost the vision in his right eye. According to some historians, Euler developed an eye infection while solving an astronomical problem that had been put forth by the French Académie. It is possible that he injured his eye by staring into the sun while working on the problem. Euler derived a solution in the course of only three days, and won the Académie's contest.
In 1736, Euler wrote a paper on the solution of the Königsburg Bridge Problem, a puzzle concerning attempts to cross seven different bridges in one journey. This work led to the development of the modern field of graph theory. Between 1736 and 1737, Euler wrote Mechanica , in which he demonstrated that mathematical analysis could be applied to Newtonian dynamics. This treatise and the wealth of articles he had already published secured his mathematical prominence. By the end of the 1730s, he had also established a reputation as a gifted educator, having written both elementary and advanced mathematical textbooks for the Russian schools. As a member of the Russian Academy, Euler was called upon to solve many practical problems for the benefit of the Russian government. He created a test for determining the accuracy of scales, developed a system of weights and measures, and supervised the government's department of geography. Although political oppression in Russia continued, Euler was never restricted in the pursuit of his own mathematical interests. However, he was growing increasingly weary of the injustices that surrounded him. In 1740, Euler accepted an invitation from Frederick the Great, the Prussian king, to join the Berlin Academy. He left Russia on sufficiently good terms, however, that throughout his tenure in Berlin the St. Petersburg Academy provided part of his salary. He was to remain at his Berlin post for the next 24 years.
Develops the Calculus of Variations
Frederick the Great, while lacking in his own mathematical ability, did appreciate the utility of mathematics. He directed Euler to work on calculations related to diverse practical matters, including pension plans, navigation, water supply systems, and the national coinage. In Berlin, Euler accomplished what many consider his most important work. He wrote Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, on the calculus of variations in 1744, and its publication led, in 1746, to his election as a Fellow of the Royal Society of London. This masterpiece was followed by several texts on calculus that were to become instant classics. These included Introductio in analysin infinitorum, in 1748, and Institutiones calculi differentialis, in 1755.
While in Berlin, Euler corresponded with many of his mathematical contemporaries, including Johann Lagrange and Jean d'Alembert. He was introduced to the field of number theory by Christian Goldbach, who presented him with the various challenges of Pierre de Fermat. Euler proved many assertions in the field of number theory and was the first mathematician to make serious progress in solving Fermat's Last Theorem. In a letter to Goldbach in 1753 he described a partial proof, ultimately shown to contain a fallacy, but which laid the foundation for its eventual solution.
Despite Blindness, Enters Most Prolific Period
In 1766, at the age of 59, Euler returned to Russia. He had fallen into gradual disfavor in King Frederick's court because of the positions he took in metaphysical arguments with contemporaries such as Voltaire. The king eventually concluded that Euler was unsophisticated, and took to calling him a "mathematical Cyclops," in reference to his partial loss of vision. When Euler went back to St. Petersburg, he was greeted with much greater esteem. Catherine the Great, now in power, provided him with a large estate and one of her personal cooks.
Not long after resettling in Russia, Euler developed a cataract in his left eye and totally lost his vision. He nonetheless entered one of the most prolific periods of his career. Nearly half of the 886 books and manuscripts Euler wrote were composed during this second tenure at the St. Petersburg Academy. From 1768 to 1770, he drafted a classical treatise on integral calculus, Institutiones calculi integralis. He went on to tackle the lunar theory problem, researching the phases of the moon and the tidal fluctuations on Earth. His calculations relating to the gravitational interactions among the moon, sun, and Earth won him a 300 pound prize from the British government.
Euler was known for his remarkable memory. As a boy, he had memorized the entire text of Virgil's Aeneid, and 50 years later could still recite it. His ability to perform complex calculations in his head was also renown. Once, when two of his students disagreed on the answer to a problem that required they sum a complicated convergent series to 17 terms, Euler settled the matter using only mental arithmetic. His memory and mental calculation skills undoubtedly allowed him to cope with the blindness during the latter part of his life.
Euler was married in 1733 to Catharina Gsell, the daughter of the Swiss painter Gsell, that Peter the Great had brought to Russia. They had 13 children, of whom only three sons and two daughters survived beyond their early years. Catharina died in 1776, and a year later Euler married her aunt and half-sister, Salome Abigail Gsell. He was known as a kind and generous man. Euler was especially fond of children, often writing mathematical treatises with a child on his lap. On September 18, 1783, while playing with his grandson, he suffered a stroke and died. Just before his death he had calculated the orbit of the newly discovered planet Uranus.