Leonhard Euler

Leonhard Euler was born on April 15, 1707, in Basel, a Swiss city on the Rhine with a proud university tradition ^[1]. His father, Paul Euler, was a Calvinist pastor who had studied some mathematics under Jakob Bernoulli, and his mother, Marguerite Brucker, came from a family of scholars and clergymen. The family soon moved to the nearby village of Riehen, where Paul took up a parish post and gave his son his first lessons at home ^[2].

The elder Euler intended his son for the ministry. Leonhard entered the University of Basel at thirteen, an age not unusual for the period, and completed a master's degree in philosophy in 1723 with a dissertation comparing the natural philosophy of Descartes and Newton ^[1]. What changed the course of his life was the attention of Johann Bernoulli, then among the most celebrated mathematicians in Europe. Bernoulli had no time to give private lessons, but he agreed to meet the young man on Saturday afternoons to answer questions about the difficult books Euler was reading on his own ^[2].

Bernoulli quickly recognized an extraordinary talent and persuaded Paul Euler that his son belonged in mathematics rather than theology. Euler never lost his religious conviction, and he remained a devout Calvinist throughout his life, but his working hours from that point forward belonged to calculation ^[3]. In 1726 he finished his studies at Basel, and in 1727, at nineteen, he entered the Paris Academy prize competition with a paper on the optimal placement of masts on a ship, taking second place despite having grown up in a landlocked country ^[1].

No academic post opened for Euler in Basel, so he looked abroad. Johann Bernoulli's sons Daniel and Nicolaus had taken positions at the newly founded Imperial Academy of Sciences in Saint Petersburg, established by Peter the Great, and they lobbied for their young friend to join them. Euler arrived in the Russian capital in May 1727, initially attached to the medical division of the academy before moving to the mathematical section ^[2].

The early Petersburg years were unsettled. Nicolaus Bernoulli had died shortly before Euler's arrival, and political turbulence after the death of Empress Catherine I made the academy's future uncertain. Euler kept his head down and worked. When Daniel Bernoulli returned to Switzerland in 1733, Euler succeeded him as head of the mathematics division at the age of twenty six ^[1]. His output during this first Russian period was astonishing: papers on number theory, analysis, and mechanics appeared in a steady stream, and in 1736 he published Mechanica, a two volume treatise that recast Newtonian dynamics in the language of calculus ^[4].

In 1735 Euler solved a problem that had defeated the leading mathematicians of the previous generation: the exact sum of the reciprocals of the square numbers, now called the Basel problem after his home city. His answer, pi squared divided by six, made his name across Europe ^[4]. That same decade he addressed the puzzle of the seven bridges of Königsberg, proving that no walk could cross each bridge exactly once. The 1736 paper that settled the question is generally counted as the founding document of graph theory ^[5].

In 1741 Euler accepted an invitation from Frederick the Great of Prussia and moved to Berlin, where he would spend the next twenty five years at the Berlin Academy. Relations with the king were never warm. Frederick preferred witty French philosophers and privately mocked the plain spoken Swiss mathematician, but he could not deny the value of the work Euler produced, which ranged from ballistics and canal engineering to the state lottery and pension schemes ^[3].

Any account of Leonhard Euler achievements has to begin with sheer scale. His collected works, the Opera Omnia, run to more than seventy large volumes, and the modern Euler Archive catalogs over eight hundred separate books and papers ^[6]. He wrote on analysis, number theory, geometry, mechanics, astronomy, optics, music theory, ship design, and cartography, and in most of these fields he did not merely contribute but reorganized the subject.

Euler gave mathematics much of its modern voice. He popularized the letter e for the base of natural logarithms, fixed the use of the Greek letter pi for the circle ratio, introduced i for the square root of negative one, and established the notation f(x) for a function ^[1]. The relation now called Euler's identity, linking e, i, pi, one, and zero in a single equation, follows from his formula connecting exponential and trigonometric functions, and it is regularly cited by mathematicians as the most beautiful equation in the subject ^[4].

His textbooks defined how the calculus would be taught for a century. The Introductio in analysin infinitorum of 1748 placed the concept of a function at the center of analysis, and it was followed by systematic volumes on differential calculus in 1755 and integral calculus between 1768 and 1770 ^[1]. In number theory he proved Fermat's little theorem, extended it with the function now named for him, and demonstrated that the sum of the reciprocals of the primes diverges, an early bridge between analysis and the study of whole numbers ^[4].

Beyond pure mathematics, Euler and Joseph Louis Lagrange built the calculus of variations, a method for finding curves and surfaces that make some quantity as small or large as possible. He derived the equations governing the motion of rigid bodies and, in 1757, the fundamental equations of fluid flow for an ideal fluid, still called the Euler equations and still central to aerodynamics ^[7]. His formula relating the vertices, edges, and faces of a polyhedron, V minus E plus F equals 2, opened the field later known as topology ^[5]. His lunar theory improved the tables navigators used to find longitude at sea, and his work on optics argued for a wave account of light against the prevailing Newtonian particle view ^[2].

In January 1734 Euler married Katharina Gsell, the daughter of Georg Gsell, a Swiss painter working at the Saint Petersburg academy. The couple had thirteen children, of whom only five survived early childhood, a loss rate painful even by eighteenth century standards ^[2]. Euler was by all accounts an affectionate father who claimed to do some of his best thinking with a baby on his lap and children playing around his desk ^[3].

His eldest son, Johann Albrecht Euler, became a mathematician and served as secretary of the Saint Petersburg academy; another son, Christoph, pursued a military career in Russia ^[2]. Katharina died in 1773, and three years later Euler married her half sister, Salome Abigail Gsell, who cared for him in his final years ^[1].

Euler's health gave way early. A near fatal fever in 1735 was followed a few years later by the loss of sight in his right eye, a decline he attributed to overwork on a cartographic project. Around 1766 a cataract destroyed most of the vision in his left eye as well, and after a failed operation in 1771 he was almost completely blind ^[1]. The blindness barely slowed him. Working with a prodigious memory, he could recite the entire Aeneid and carry out long calculations in his head, dictating papers to his sons and assistants. Roughly half of his lifetime output dates from the years after he lost his sight ^[6].

By the mid 1760s the situation in Berlin had become uncomfortable. Frederick the Great passed over Euler for the presidency of the academy after the death of Maupertuis, courted Jean le Rond d'Alembert for the post, and interfered in academy business in ways Euler found humiliating ^[3]. Catherine the Great of Russia, eager to restore the luster of the Petersburg academy, offered generous terms: a large salary, positions for his sons, and a fully furnished house. In 1766 Euler returned to Saint Petersburg, where he remained for the rest of his life ^[1].

The second Russian period brought hardship and productivity in equal measure. In 1771 a fire destroyed his house, and Euler, by then blind, was carried from the building; much of his library was lost, though many manuscripts were saved ^[2]. He kept dictating at a pace that strained the academy's ability to print his work. From these years came his complete lunar theory of 1772, the volumes on integral calculus, a widely read treatise on algebra composed by dictation, and the Letters to a German Princess, a survey of natural philosophy written for Frederick's niece that became one of the most popular science books of the century, translated across Europe ^[1].

On September 18, 1783, Euler spent the morning on calculations concerning the recently discovered planet Uranus and the mathematics of balloon flight, a subject in the news after the Montgolfier experiments. That afternoon, while talking with family, he suffered a brain hemorrhage and died within hours, at seventy six ^[2]. The Marquis de Condorcet, in his eulogy for the Paris Academy, recorded the moment with a phrase that became famous: Euler ceased to calculate and to live ^[3]. He was buried in Saint Petersburg, and in 1957 his remains were moved to the Alexander Nevsky Monastery cemetery in that city ^[2].

Mathematicians ran out of ways to name things after Euler long ago; the joke in the field is that discoveries are named for the first person to find them after him. Euler's number, Euler's identity, Euler's formula for polyhedra, the Euler equations of fluid dynamics, Euler angles for rotating bodies, and the Euler characteristic in topology only begin the list ^[4]. Pierre Simon Laplace is traditionally credited with the advice given to younger colleagues: read Euler, he is the master of us all ^[3].

The editorial project of publishing everything he wrote, begun by the Swiss Academy of Sciences in 1907, occupied scholars for more than a century, and the correspondence volumes are still appearing ^[6]. Switzerland placed his portrait on the ten franc banknote, and both Swiss and Russian institutions marked the tercentenary of his birth in 2007 with conferences and exhibitions. An asteroid and a lunar crater carry his name ^[5].

Among the more durable Leonhard Euler facts is this: no working scientist or engineer today gets through training without using his tools. The notation in every calculus textbook, the analysis behind signal processing and electrical engineering, the graph theory underneath network routing and social network analysis, and the fluid equations in weather and aircraft models all trace back to his papers ^[7]. Any Leonhard Euler biography ends up being, in part, a history of how modern mathematics learned to speak, because so much of its language was his invention ^[4].