Leonhard Euler
Pierre-Simon Laplace told his colleagues to read Leonhard Euler, calling him the master of us all. The man he was describing was a Swiss polymath born in Basel on the 15th of April 1707, active not only as a mathematician but as a physicist, astronomer, logician, geographer, music theorist and engineer. Henri Poincare went further and named him the god of mathematics. Yet the boy who would earn these titles was raised to become a pastor, the son of Paul III Euler of the Reformed Church. How does a pastor's son end up founding entire fields of study, popularizing the symbol pi, and producing on average one mathematical paper per week into old age. How does a man who went almost totally blind become arguably the most prolific contributor in the history of mathematics and science. The answers run through Saint Petersburg, Berlin, a king's water fountain, and a memory that could recite every page of Virgil. To understand Euler is to follow a mind that calculated, in the words of Francois Arago, just as men breathe and as eagles sustain themselves in air.
In 1720, at the age of thirteen, Euler enrolled at the University of Basel, a young age that was not unusual for the time. His father Paul had taught him mathematics at home, having taken courses from Jacob Bernoulli years earlier. At the university the course on elementary mathematics fell to Johann Bernoulli, the younger brother of the deceased Jacob. Johann Bernoulli refused to give private lessons, pleading a busy schedule. Instead he told the young Euler to work through the hardest mathematical books on his own and to bring his difficulties to him every Saturday afternoon. Euler later recalled that when Bernoulli resolved one objection, ten others at once disappeared, which he called the best method of making happy progress. Backed by Bernoulli, Euler won his father's consent to become a mathematician rather than a pastor. In 1723 he received a Master of Philosophy with a dissertation comparing the philosophies of Rene Descartes and Isaac Newton, then enrolled in the theological faculty. In 1726 he completed a dissertation on the propagation of sound titled De Sono, attempting unsuccessfully to obtain a position at Basel. The following year he entered the Paris Academy prize competition for the first time, asked to find the best way to place the masts on a ship. Pierre Bouguer, later known as the father of naval architecture, won, and Euler took second. Over the years he would enter that competition fifteen times and win twelve.
On the 31st of July 1726, Nicolaus Bernoulli died of appendicitis after less than a year at the Imperial Russian Academy of Sciences in Saint Petersburg. That death set Euler's path in motion. When Daniel Bernoulli moved up to fill his brother's role, he recommended Euler for the vacated post in physiology. Euler accepted in November 1726 and arrived in Saint Petersburg in May 1727, where he was soon moved from the medical department to the mathematics department and even took a job as a medic in the Russian Navy. The academy had been established by Peter the Great to close the scientific gap with Western Europe, which made it especially attractive to foreign scholars. Its benefactress Catherine I died before Euler arrived, and Russian conservative nobility gained power under the twelve-year-old Peter II. Suspicious of foreign scientists, the nobility cut funding for Euler and his colleagues. Conditions improved after Peter II died in 1730 and Anna of Russia took power. Euler was made a professor of physics in 1731, left the Russian Navy that same year by refusing a promotion to lieutenant, and when Daniel Bernoulli left for Basel, Euler succeeded him as head of the mathematics department. In January 1734 he married Katharina Gsell, daughter of the painter Georg Gsell, and the couple bought a house by the Neva River. When Frederick II offered to pay 1600 ecus, the same as Euler earned in Russia, Euler agreed to move, requesting permission in 1741 on the grounds that he needed a milder climate for his eyesight.
Frederick the Great wanted a water jet in his garden at Sanssouci, and he blamed Euler when it failed. The king complained that his mill, built geometrically, could not raise a mouthful of water closer than fifty paces to the reservoir, and he cried out, Vanity of vanities, vanity of geometry. The disappointment was almost surely unwarranted, since Euler's calculations look likely to have been correct even if his interactions with Frederick were dysfunctional. Euler had left Saint Petersburg in June 1741 and would live twenty-five years in Berlin, writing several hundred articles. He published his Introductio in analysin infinitorum in 1748 and a text on differential calculus in 1755, the same year he was elected a foreign member of the Royal Swedish Academy of Sciences and the French Academy of Sciences. A simple and devoutly religious man, Euler never questioned the existing social order, which made him the polar opposite of Voltaire, who held a high place at Frederick's court. Euler was not a skilled debater and often argued subjects he knew little about, making him a frequent target of Voltaire's wit. Though Jean le Rond d'Alembert put him forward for the academy's presidency, Frederick named himself president instead. During his Berlin years Euler wrote 380 works, 275 of which were published, while supervising the library, the observatory, the botanical garden, and the publication of calendars and maps. In 1760, with the Seven Years' War raging, advancing Russian troops sacked his farm in Charlottenburg, and Empress Elizabeth of Russia added 4000 rubles in compensation, an exorbitant amount at the time.
Now I will have fewer distractions, Euler said of losing his sight. In 1738, three years after nearly dying of fever, he became almost blind in his right eye, a loss he blamed on the cartography he performed for the Saint Petersburg Academy, though the true cause remains the subject of speculation. As his vision worsened in Germany, Frederick took to calling him Cyclops. In 1766 a cataract was discovered in his left eye, and although couching the cataract temporarily improved his vision, complications left him almost totally blind. His productivity barely suffered. With the aid of his scribes, his output in many areas increased, and in 1775 he produced, on average, one mathematical paper per week. That endurance matched a memory that astonished those around him. Early in life he memorized Virgil's Aeneid, and in old age he could recite the poem and give the first and last sentence on each page of the edition he had learned it from. He knew the first hundred prime numbers and could give each of their powers up to the sixth degree. Nicolas Fuss noted that Euler had read all the best Roman writers and could recall the historical events of all times and peoples without hesitation. He was known as a generous and kind person who kept his good-natured disposition even after becoming entirely blind.
Most mathematicians today write f of x without knowing they are using Euler's hand. He introduced the concept of a function and was the first to write that notation for a function applied to its argument. He gave the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm now known as Euler's number, the capital Greek sigma for summations, and the letter i for the imaginary unit. The Greek letter pi for the ratio of a circle's circumference to its diameter he popularized, though it originated with the Welsh mathematician William Jones. Euler's analysis pushed past notation into deep results. His use of power series let him solve the Basel problem in 1735, finding the sum of the reciprocals of the squares of every natural number, a problem first posed by Pietro Mengoli in 1644 and popularized by Jacob Bernoulli. He defined logarithms for negative and complex numbers and discovered Euler's formula, which Richard Feynman called the most remarkable formula in mathematics, with the special case known as Euler's identity. He invented the calculus of variations and formulated the Euler-Lagrange equation. Drawing on his friendship with Christian Goldbach, godfather to his first son Johann Albrecht, Euler pioneered analytic number theory. He invented the totient function, generalized Fermat's little theorem into what is now Euler's theorem, and disproved Fermat's conjecture that all Fermat numbers are prime. By 1772 he had proved that 2,147,483,647 is a Mersenne prime, possibly the largest known prime until 1867.
In 1735, the residents of Konigsberg in Prussia wondered whether anyone could walk their city while crossing each of its seven bridges exactly once. The city sat on the Pregel River with two large islands joined to each other and the mainland by those bridges. Euler proved that no such path exists, and that solution is considered the first theorem of graph theory, partly a founding moment for topology as well. He also found that for a convex polyhedron with no holes, the number of vertices and faces minus the number of edges equals 2, a value now called the Euler characteristic. Euler's reach into the physical world ran just as wide. He reformulated Newton's laws of motion in his two-volume Mechanica to better explain the motion of rigid bodies, helped develop the Euler-Bernoulli beam equation, and gave structural engineers Euler's critical load, the buckling load of an ideal strut. In 1754 he was the first to predict cavitation, long before its first observation in the late 19th century, and in 1757 he published the Euler equations for inviscid flow. His optics papers of the 1740s helped the wave theory of light proposed by Christiaan Huygens become dominant over Newton's corpuscular theory. Even music drew his mathematics. In 1739 he wrote the Tentamen novae theoriae musicae, once described as too mathematical for musicians and too musical for mathematicians, in which he described eighteen genres dividing the octave using the prime numbers 3 and 5. He devised a graph he called the Speculum musicum, recalling his interest in the bridges of Konigsberg, a device that later drew renewed interest as the Tonnetz in Neo-Riemannian theory.
On the 18th of September 1783 in Saint Petersburg, after a lunch with his family, Euler was discussing the newly discovered planet Uranus and its orbit with his student Anders Johan Lexell when he collapsed and died of a brain hemorrhage. The Marquis de Condorcet wrote in his eulogy that Euler ceased to calculate and to live. He had returned to Russia in 1766 under Catherine the Great on exorbitant terms, a 3000 ruble annual salary, a pension for his wife, and high-ranking appointments for his sons, only to lose his home to a fire in 1771. His 866 publications, catalogued by Gustaf Enestrom from E1 to E866, were gathered into the Opera Omnia, a project still progressing with over 80 volumes published by 2022. Carl Friedrich Gauss judged that the study of Euler's works would remain the best school for the different fields of mathematics, with nothing else able to replace it. He was buried next to Katharina at the Smolensk Lutheran Cemetery on Vasilievsky Island, and in 1957, on the 250th anniversary of his birth, his tomb was moved to the Lazarevskoe Cemetery at the Alexander Nevsky Monastery. The asteroid 2002 Euler now carries the name of the pastor's son who chose mathematics instead.
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Common questions
Who was Leonhard Euler and what is he known for?
Leonhard Euler was a Swiss polymath born in Basel on the 15th of April 1707, active as a mathematician, physicist, astronomer, logician, geographer, music theorist and engineer. He founded graph theory and topology and is regarded as arguably the most prolific contributor in the history of mathematics and science, and the greatest mathematician of the 18th century.
What mathematical notation did Leonhard Euler introduce?
Leonhard Euler introduced or popularized much of modern mathematical notation, including the concept of a function and the notation for a function applied to its argument. He used the letter e for the base of the natural logarithm, the letter i for the imaginary unit, the Greek capital sigma for summations, and popularized the Greek letter pi, which originated with William Jones.
Where did Leonhard Euler spend his career?
Leonhard Euler spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. He arrived in Saint Petersburg in May 1727, lived twenty-five years in Berlin from 1741, and returned to Saint Petersburg in 1766, where he died in 1783.
How did Leonhard Euler solve the Seven Bridges of Konigsberg problem?
In 1735 Leonhard Euler proved that it was not possible to follow a path crossing each of Konigsberg's seven bridges exactly once. The city sat on the Pregel River with two large islands connected by the bridges, and his solution is considered the first theorem of graph theory.
Did Leonhard Euler go blind?
Yes, Leonhard Euler became almost blind in his right eye in 1738 and, after a cataract was discovered in his left eye in 1766, complications left him almost totally blind. His productivity barely suffered, and in 1775 he produced on average one mathematical paper per week with the aid of his scribes.
How did Leonhard Euler die?
Leonhard Euler died of a brain hemorrhage in Saint Petersburg on the 18th of September 1783, after a lunch with his family, while discussing the newly discovered planet Uranus and its orbit with his student Anders Johan Lexell. The Marquis de Condorcet wrote that he ceased to calculate and to live.