CHAPTER 1
LEONHARD EULER AND HIS THREE “GREAT” FRIENDS
We have become accustomed to hyperbole. Television commercials, billboards, sportscasters, and popular musicians regularly throw around sensational words such as greatest, best, brightest, fastest, and shiniest. Such words have lost their literal meaning—they are employed in the normal process of selling a product or entertaining a viewer. So, when we say that Leonhard Euler was one of the most influential and prolific mathematicians the world has ever seen, the reader’s eyes may glaze over. We are not overselling the truth. Euler is widely considered, along with Archimedes (287–211 BCE), Isaac Newton (1643–1727), and Carl Friedrich Gauss (1777–1855), to be one of the top ten—or top five—most important and significant mathematicians in history.
In his life of seventy-six years, Euler created enough mathematics to fill seventy-four substantial volumes, the most total pages of any mathematician. By the time all of his work had been published (and new material continued to appear for seventy-nine years after his death) it amounted to a staggering 866 items, including articles and books on the most cutting-edge topics, elementary textbooks, books for the nonscientist, and technical manuals. These figures do not account for the projected fifteen volumes of correspondence and notebooks that are still being compiled.
Euler’s importance is due not to his voluminous output but to the deep and groundbreaking contributions he made to mathematics. Euler did not specialize in one particular area. He was one of the great generalists: he had expertise that spanned the disciplines. He published influential articles and books in analysis, number theory, complex analysis, calculus, calculus of variations, differential equations, probability, and topology. This list does not include his contributions to such applied subjects as optics, electricity and magnetism, mechanics, hydrodynamics, and astronomy. Furthermore, Euler possessed a trait that was and is rare among the top scholars: he was a first-rate expositor. Unlike the mathematicians who preceded him, Euler wrote using clear, simple language that made his work accessible to experts and students alike.
Figure 1.1. Leonhard Euler.
Euler was a gentle, unpretentious man whose life was centered around his large family and his work. He lived in Switzerland, Russia, Prussia, and then again Russia, and corresponded extensively with many important thinkers of the eighteenth century. His professional life was linked to three “Great” rulers of Europe—Peter the Great, Frederick the Great, and Catherine the Great. The legacies of these leaders include the creation or revitalization of their countries’ national academies of science. These academies supported Euler so that he could spend time on pure research. The only repayment they expected was the occasional use of his scientific expertise for matters of the state and the recognition that his celebrity brought to the nation.
Leonhard Euler was born in Basel, Switzerland, on April 15, 1707, to Paul Euler and Marguerite Brucker Euler. Shortly afterward, the family moved to the nearby town of Riehen, where Paul took a job as the minister of the local Calvinist church.
Leonhard’s earliest mathematical training was provided by his father. Although Paul was not a mathematician, he had studied mathematics under the famed Jacob Bernoulli (1654–1705). This instruction took place while Paul and Jacob’s younger brother Johann (1667–1748), both students at University of Basel, boarded at Jacob’s home. Jacob and Johann Bernoulli were members of what was to become the most esteemed family in mathematics. For over a century the Bernoulli clan played an important role in the advancement of mathematics, with at least eight Bernoullis making lasting contributions.
Leonhard began his formal studies at the University of Basel at the age of fourteen. This was not an unusually young age for a university student at that time. The university was quite small—it had only a few hundred students and nineteen professors. Paul hoped that his son would follow a career path into the ministry, so Leonhard studied theology and Hebrew. But his mathematical abilities were undeniable, and he quickly attracted the attention of his father’s friend Johann Bernoulli. By this time Johann was one of the leading mathematicians in Europe.
Johann was an arrogant, brusque man with a competitive streak that produced storied rivalries (with, among others, his brother and one son). Yet he recognized the boy’s remarkable talents and encouraged him to pursue mathematics. Euler wrote in his autobiography, “If I came across some obstacle or difficulty, I was given permission to visit him freely every Saturday afternoon and he kindly explained to me everything I could not understand.”2 These lessons played a valuable role in the maturation of Euler’s mathematical skills.
Even as Leonhard excelled in his private mathematical studies, Paul held out hope that his son would enter the ministry. At the age of seventeen, Euler earned his master’s degree in philosophy. Johann feared that mathematics might lose his protégé to the Church, so he intervened and told Paul in no uncertain terms that Leonhard had the potential to become a great mathematician. Because of his fondness for mathematics, Paul relented. Even though Euler abandoned the ministry, he remained a devout Calvinist for his entire life.
Euler’s first independent mathematical accomplishment came at the age of nineteen. His theoretical work on the ideal placement of masts on a ship secured him an accessit, or “honorable mention,” in a prestigious competition sponsored by the French Académie des Sciences. This feat would be incredible for any teenager, but was especially so for a Swiss youth who had never seen a ship on the ocean. Euler did not win the top prize for this competition, which would have been roughly equivalent to winning a Nobel Prize today, but in the years to come he won the highest honor on twelve occasions.
Figure 1.2. Peter the Great of Russia.
At the time of Euler’s birth, a thousand miles to the northeast of Basel, the Russian tsar Peter the Great (1672–1725) was building the city of St. Petersburg. It was founded in 1703 on the marshy swampland where the Neva River flows into the Baltic Sea. Peter used forced labor to construct both the city and the strategically located Peter and Paul Fortress on an island in the Neva. He loved this new city, calling it his “paradise” and naming it after his patron saint. Despite the fact that most Russians, especially government officials, did not share Peter’s feelings about this cold, wet place, he moved the Russian capital from Moscow to St. Petersburg. The young Euler had no way of knowing that this city would be his home for much of his life.
Peter the Great, a physically imposing figure who stood nearly seven feet tall, was the energetic, self-taught, determined leader of Russia from 1682 to 1725. Known as a ruthless reformer, he began the transformation of his country from an agrarian and feudal nation dominated by the Church into a powerful empire. His goal of modernizing—that is to say, Westernizing—Russian government, culture, education, military, and society was largely realized. As one Russian historian wrote, “All of a sudden, skipping entire epochs of scholasticism, Renaissance, and Reformation, Russia moved from a parochial, ecclesiastical, quasi-medieval civilization to the Age of Reason.”3
As part of the process of Westernization, Peter wanted to reform Russia’s educational system, which was nonexistent before his reign except for minimal teaching by the powerful Orthodox Church. As a result, Russia had no scientists. Because of the strong presence of the Church, Russians were fearful of scientific explanations of the world, preferring instead the traditional religious explanations. Peter recognized the need to improve Russia’s international image and to dispel the notion that Russians hated science. He also knew that having a science program was crucial for creating and maintaining a powerful state.
Peter visited the Royal Society of London and the Académie des Sciences in Paris, both of which were founded in 1660. He was impressed with what he saw. He also admired the new Berlin Academy of Sciences, which was founded in 1700 upon the advice of Gottfried Leibniz (1646–1716). Leibniz is the famed mathematician who, along with Isaac Newton, is credited with the invention of calculus. These academies were not universities; they were “dedicated to the search for new knowledge and not the dissemination of existing wisdom.”4 The members of the academies were scholars, not teachers; their prime objective was the advancement of knowledge.
Peter wanted to create an academy such as the ones in Paris, London, and Berlin, and he wanted to establish it in his new city of St. Petersburg. For advice he turned to Leibniz. For nearly two decades, Peter and Leibniz had extensive conversations, both through letters and in face-to-face meetings, about education reform and the creation of an academy of science.
In 1724 Peter finalized his plans for the creation of the Academy of Sciences in St. Petersburg; it was the final and most ambitious project in his quest to improve Russia’s educational system. However, he could not model his academy exactly on the European academies. Because Russia had no native scientists, he would have to persuade talented foreign scientists to relocate to St. Petersburg. Also, because Russia had no university system, the Academy of Sciences must function as a university. Part of the mandate of the Academy was to train Russians in science so that the Academy would not always have to rely on the foreigners.
Peter never saw the fruits of his labor; he died in early 1725. Thanks to the new empress, Peter’s second wife, Catherine I (1684–1727), plans for the Academy continued. Foreign scholars began arriving within months of Peter’s death, and the Academy of Sciences held its first meeting before the end of the year. Peter was fortunate that Catherine embraced the idea of the Academy. In the years that followed, the Academy was not always blessed with such sympathetic leaders. During the thirty-seven years between Peter’s death and the coronation of Catherine the Great (1729–1796), Russia was led by six rulers, and the Academy was always at the mercy of these opinionated and powerful people.
Initially the Academy was staffed by sixteen scientists: thirteen German, two Swiss, one French—and no Russians. The large German presence and the absence of Russians would later be a source of tension.
Because of the cold climate, the remote location, and the academic isolation, it was necessary to offer high salaries and provide comfortable accommodations to lure these scientists to St. Petersburg. The new academy was small, but it quickly fulfilled its promise of being an important, internationally renowned scientific institution. Eventually it became the center of all scientific scholarship in Russia. The Academy of Science has had several name changes, but it is still exists today and is known as the Russian Academy of Sciences.
Two of the foreign scholars who were stars of this new institution were Euler’s friends, and Johann Bernoulli’s sons, Nicolaus (1695–1726) and Daniel (1700–1782) Bernoulli. The two brothers had spoken to Euler about the Academy before they left Switzerland, and they promised to secure him a position as soon as possible. Immediately upon their arrival in Russia, they began lobbying the administrators of the Academy to hire their bright young friend. Their campaigning paid off quickly. In 1726 Euler was offered a position in the medical and physiology division. Unfortunately, Euler could not fully enjoy nor celebrate this exciting job offer. He was hired to fill an opening created by the tragic and untimely death of Nicolaus.
Euler was grateful for the job, but he did not move to Russia immediately. He had two reasons for remaining in Basel and putting his new job on hold. First, he had accepted a job in medicine, but he possessed only minimal knowledge in that area. So he decided to remain at the University of Basel and study anatomy and physiology. Second, he was stalling for time as he waited to hear if he would be offered a faculty position in physics at the University of Basel. In the spring of 1727, when he heard that he was not chosen for the job, he left for Russia. So began his life in St. Petersburg, where he lived for the next fourteen years and then again for the last seventeen years of his life.
Euler’s journey to St. Petersburg by boat, foot, and wagon took seven weeks. On the day that he set foot in Russia, Empress Catherine I died after ruling for only two years. The fate of the new Academy was uncertain. Those who ran the country on behalf of the eleven-year-old tsar Peter II (1715–1730), the grandson of Peter the Great, saw the Academy as an expendable luxury and contemplated closing its doors. Fortunately, the school remained open, and in the confusion that ensued, Euler ended up where he rightly belonged—in the mathematical-physical division, not in the medical division. This first year of Euler’s mathematical career, 1727, was also the year the mathematical giant Isaac Newton died.
Life at the Academy was difficult under Peter II, so the members of the Academy hoped that their fortunes would improve following the death of the fifteen-year-old tsar in 1730. The Academy did fare slightly better during the ten-year reign of Anna Ivanovna (1693–1740), but conditions in Russia turned bleak. Anna brought into her government a strong German influence, most notably her lover Earnst-Johann Biron (1690–1772). Biron was a ruthless tyrant who executed several thousand Russians and exiled tens of thousands more to Siberia. Those targeted by Biron included common criminals, Old Believers (of the Russian Orthodox Church), and Anna’s political opponents. Later, when in Berlin, Euler was asked by the Queen Mother of Prussia why he was so taciturn. He replied, “Madam, it is because I have just come from a country where every person who speaks is hanged.”5
In 1733, having had enough of the difficult Russian lifestyle and the internal politics at the Academy, Daniel Bernoulli moved back to Switzerland, and, at the age of twenty-six, Euler stepped into Daniel’s role as head mathematician.
At this point, Euler realized that he might be in Russia for a long time, perhaps for the rest of his life. With the exception of hardships imposed by the political climate in Russia, Euler found a comfortable life. He acquired a good command of the Russian language, and he felt financially secure thanks to the higher salary that came with his promotion. So, in 1733 he decided to marry Katharina Gsell, the daughter of the Swiss-born painter Georg Gsell who had been brought to Russia by Peter the Great. Leonhard and Katharina started a family, eventually producing thirteen children. As was not uncommon in that day, only five of them lived past childhood, and just three outlived their parents.
Being a husband and a father did not slow down Euler’s stream of publications. Now, and in every period of his professional life, he was an extremely active researcher. It is difficult to overstate Euler’s massive output. Mathematical folklore says that he could write mathematics papers while bouncing a baby on his knee, and that he could compose a treatise between the first and second calls for dinner. He wrote about anything and everything. He produced masterpieces, short notes, corrections, explanations, partial results, ideas of proofs, introductory texts, and technical books.
No setback was able to slow Euler down. Even blindness could not impede the flood of his mathematical output. In 1738 he became ill after spending three long days working on an astronomical challenge. Although modern medicine casts this claim into doubt, it was long believed that this illness caused the deterioration and eventual loss of vision in his right eye. Euler took the loss of vision in stride. In his typically modest way he remarked, “Now I will have fewer distractions.”6 He later lost the vision in his other eye, and he lived in near-total darkness for the last seventeen years of his life. Despite his loss of sight he continued to make important mathematical contributions until the day of his death.
Euler’s brain seemed hard-wired for mathematics in a way that other brains are not. He was able to juggle many abstract notions in his head simultaneously, and he was able to perform staggering mental calculations. In one famous story, two of Euler’s students were adding seventeen fractional terms only to discover a discrepancy in their sums. Euler performed the sum in his head and settled the dispute by providing the correct answer. As the mathematician François Arago (1786–1853) famously wrote, “Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind.”7 Modestly, Euler stated that his ability to manipulate symbols substituted for cleverness, saying that his pencil surpassed himself in intelligence.
Euler was also blessed with an amazing memory. He memorized countless poems; from the time he was a child until his old age he was able to recite the entire text of Virgil’s Aeneid, being able to recall the first and last sentences on any page. A more mathematical example of his prodigious memory was his ability to give the first six powers of the first hundred numbers. To put this in perspective, the sixth power of 99 is 941,480,149,401.
During his stay in St. Petersburg Euler devoted some of his time to projects for the state. In 1735 he was appointed the director of the Academy’s geography section and subsequently made important contributions to the creation of a much-needed map of Russia. He also wrote a two-volume book on shipbuilding that was so valuable that the Academy doubled his salary for that year.
Even while Euler enjoyed remarkable productivity, a happy family life, and a sizable income, conditions in Russia continued to worsen. The atmosphere at the Academy had become very tense, even hostile. Most of the senior faculty were German, and there was still very little Russian involvement. In the first sixteen years of the Academy’s existence, only one Russian was given membership, and he was an adjunct who was never promoted to professor. The Russians resented the power possessed by the Germans and openly voiced anti-German opinions. Fortunately, the calm and reserved Euler was able to remain neutral in the internal politics at the Academy, but it made his working life stressful.
With the presence of Biron and the “German party” in Anna’s government, fear and hatred of Germans continued to rise among the Russian people. In late 1740, shortly before her death, Anna appointed Biron regent for her successor, the two-month-old Ivan VI (1740–1764). After Anna’s death the Russians’ animosity toward the Germans came to a head— within a month Biron was overthrown, and a year later Ivan and the entire “German party” were removed from power. Peter the Great’s daughter Elizabeth I (1709–1762) became the next empress.
During this period, life in Russia was dangerous, especially for non-Russians. Foreign academics were viewed suspiciously as possible spies for the West. Euler reacted to these conditions by staying quiet and by devoting all of his time to his work and family. In 1741 Euler was unable to tolerate life in Russia any longer, so he decided to leave St. Petersburg for Berlin.
The Berlin Academy of Sciences was founded in 1700 with the name Societas Regia Scientiarum. Leibniz had grand designs for the Academy. Like those in Paris and London, the Berlin Academy focused on science and mathematics, but unlike the others, it broadened its scope to include history, philosophy, language, and literature.
Despite Leibniz’s high expectations, the Berlin Academy was slow to take off. Its difficulties were due in part to continual underfunding and to the internal French-German tensions. Conditions worsened following the accession of Frederick William I (1688–1740) in 1713. Under this anti-intellectual ruler, the Academy was completely neglected. The Berlin Academy showed none of the success of those in Paris and London. It was not a significant factor in the advancement of scientific knowledge; in fact, it was given the title “the anonymous society.”
When Frederick William I died in 1740, his son Frederick II (1712–1786), later known as Frederick the Great, came to power. Although Frederick William I deliberately groomed his son for leadership, Frederick was in many ways his father’s opposite. Tensions between the two ran deep. When he was eighteen Frederick was caught attempting to flee the country. His father forced Frederick to witness the execution of his friend and coconspirator (and, some say, homosexual lover).
Figure 1.3. Frederick the Great of Prussia.
Frederick was determined to extend the German territories, but he was also artistically and philosophically inclined. He aspired to be an enlightened ruler-philosopher. The revival of the Academy played an important role in his plan for revitalizing his country.
Unlike his father, Frederick disdained German culture and loved all things French. He changed the official name of the Berlin Academy to Académie Royale des Sciences et Belles Lettres. He insisted that French be the official language of the Academy, and he demanded that every article published in its journal be written in, or translated into, French. He preferred to be in the company of witty Frenchmen rather than calm and unemotional Germans. Voltaire (1694–1778) was one of his favorite correspondents and one of his closest advisors for matters relating to the Academy. It was Voltaire who first suggested to Frederick that he entice Euler to leave Russia and join the Berlin Academy.
Frederick had an intense aversion to the mathematical arts. In 1738 he wrote to Voltaire, “As for mathematics, I confess to you that I dislike it; it dries up the mind. We Germans have it only too dry; it is a sterile field which must be cultivated and watered constantly, that it may produce.”8 He viewed mathematics—and the sciences in general—as servants of the state. He judged the success of his scientists by their usefulness in practical matters. The scientists at the Academy were free to pursue their own projects so long as they attended to the requests made by the king.
At this time Euler was the most distinguished scholar in St. Petersburg, with a reputation that spread across Europe. Frederick set out to woo Euler. Despite the fact that Euler was troubled by the dangerous conditions in Russia, it took repeated contacts by Frederick to convince the Swiss mathematician to abandon St. Petersburg. In 1741 Euler assented, and by citing his declining health and the need for a warmer climate, he was able to take his leave of St. Petersburg.
At first Euler was satisfied in Berlin; he wrote to a friend in 1746, “The king calls me his professor, I think I am the happiest man in the world.”9 Unfortunately, this contentment did not last. In many ways life in Berlin was better than life in Russia, but Euler’s experience was soured by Frederick’s peculiar and surprising disdain for him. He referred to Euler as his “mathematical cyclops,” ungraciously alluding to Euler’s one good eye. Frederick’s coolness was due in part to his dislike of mathematics, but that was not all. Euler’s low-key and quiet demeanor did not sit well with Frederick, who viewed him as a simpleton. Frederick preferred the company of the witty, sophisticated, boisterous Voltaire. Also, Euler was a devout Calvinist. He read scripture to his family every evening, often accompanied by a sermon. Publicly Frederick espoused tolerance of religion, but privately he was a deist and had little respect for the pious Euler or his deeply held spiritual beliefs.
Euler harbored hard feelings toward Frederick as well. His greatest frustration while in Berlin was Frederick’s refusal to make him president of the Academy. For several busy years, while he was fighting the Seven Years’ War, Frederick was unable to find a suitable person to fill this position. During the interim Euler held the unofficial role of “acting president,” but time and again Frederick passed him over as the permanent replacement. Euler performed well as interim president, but because he was not a philosopher capable of sharp, lively conversation, he would never gain Frederick’s favor. The ultimate insult occurred in 1763 when Frederick admitted that he could not find a suitable replacement and declared himself president of the Academy.
There developed a further animosity between Euler and Frederick when in 1763, the king refused to allow one of Euler’s daughters to marry a soldier because of the soldier’s low rank. Perhaps the last straw was a series of heated exchanges between Frederick and Euler from 1763 to 1765. At issue was the sale of the state calenders (almanacs). These were made at great expense by the members of the Academy and were sold to the public to fund its operations. It was discovered that the chief commissioner had been pocketing funds from the sale of the calendars. Frederick and Euler disagreed over how to handle the corruption and mismanagement of this fundraising endeavor. It ended with Frederick sending a sharp rebuke to Euler.
While living in Berlin, Euler maintained good relations with his former colleagues in St. Petersburg. He remained editor for their journal, and he sent a total of 109 of his articles to be published in it. He tutored Russian students who were sent to Berlin. In exchange for the editing and tutoring, he was paid a regular stipend by the Russians. A more remarkable example of the Russians’ respect for Euler occurred during the Seven Years’ War. In the march on Brandenburg in 1760 the Russian army entered Carlottenburg. They came upon and pillaged a farm that belonged to Euler. When this action was discovered, the Russians—first the general, then Empress Elizabeth—paid Euler reparations that far exceeded the cost of the damages.
During the twenty-four years that Euler was in Berlin, the Russians were eager to get him to return to St. Petersburg. They approached him in 1746, in 1750, and in 1763, making generous offers to lure him back. Each time he refused, but he never closed the door completely. Finally in 1765, fed up with Frederick’s hostility and seeing improved political conditions in Russia, he decided to return to St. Petersburg.
Despite his personal feelings, Frederick recognized Euler’s prominent place in the international scientific community. Euler had published over two hundread works during his stay in Berlin. In 1749 he had been elected a Fellow of the Royal Society of London. In 1755 he had been appointed the ninth foreign member of Académie des Sciences in Paris, even though the number of foreign members was limited to eight. He had also served the state well; in addition to the creation of the calendars, Euler worked on coinage for the national mint, the placement of canals, the design of aqueducts, the creation of pensions, and the improvement of artillery.
Frederick attempted to prevent Euler’s departure. Euler was forced to make repeated requests for permission to leave. In 1766 Frederick finally relented and allowed Euler to depart; at the age of 59, Euler and his eighteen dependents returned to St. Petersburg.
Later that year, on the recommendation of the French mathematician Jean D’Alembert (1717–1783), Frederick replaced Euler with Joseph-Louis Lagrange (1736–1813), a rising star who turned out to be a stellar mathematician. In his typically caustic way, the king wrote to D’Alembert thanking him “for having replaced a half-blind mathematician by a mathematician with both eyes, which will especially please the anatomical members of the academy.”10 Ironically, despite Frederick’s distaste for mathematics and his love of philosophy, his Academy will forever be remembered for its impressive roster of mathematicians, and not for its philosophers.
Figure 1.4. Catherine the Great of Russia.
At the end of his stay in Berlin, while Euler was clashing with Frederick, Russia was under the rule of Peter III (1728–1762), a miserable, psychologically unstable, pro-German leader who was known to “fear and despise Russia and the Russians.”11 In 1762 his rule came to an abrupt end when he was overthrown by his wife, who then took the throne as Catherine II. Shortly thereafter, perhaps under Catherine’s orders, Peter was murdered by guards holding him in custody.
Catherine, who later became known as Catherine the Great, was empress until 1796. Just as the eighteenth century began under the rule of the powerful and influential Peter the Great, so was the century’s end marked by the distinguished leadership of Catherine the Great. She was an intelligent, strong-willed, ambitious, and energetic leader. As the French philosopher Denis Diderot (1713–1783) said after visiting Catherine’s court, she “is the soul of Caesar with all the seductions of Cleopatra.”12 Under her rule the quality of life in Russia showed a marked improvement. Education, which had largely been ignored since the time of Peter the Great, was again a priority for the Russian government.
During the early days of the Academy, the institution shone due to Euler’s brilliance. When he moved to Berlin, so did the mathematical spotlight. This loss, together with the years of rapid political turnover, made it difficult for the institution to attract talented foreign scholars. The Academy was on shaky ground. One of Catherine’s projects for educational reform was to revive the St. Petersburg Academy and elevate it to its earlier heights. As the mathematician André Weil (1906–1998) wrote, “This was almost synonymous with bringing Euler back.”13
Catherine saw to it that Euler’s stiff demands were met and exceeded. He received double the salary that was offered to him in 1763, his wife received a stipend, his eldest son was hired by the Academy, and his younger sons were guaranteed future employment. In addition, Catherine provided Euler a fully furnished house, and he was given one of her own cooks. Upon his arrival in St. Petersburg, Euler was greeted warmly by the Empress. His return to the Academy refocused the attention of the mathematical community on St. Petersburg and ensured the continued success of the Academy.
There are similarities between Catherine the Great and Frederick the Great; they were, after all, prime examples of “enlightened despots.” However, Euler’s relationships with the two leaders were very different. His experience in Catherine’s St. Petersburg was much more positive than was his experience in Frederick’s Berlin. Catherine was a lover of science, and she welcomed Euler as a celebrity. He was the ranking academician with the most administrative power of any scholar.
In his lifetime, Euler saw many changes in the capital city of St. Petersburg. The city was only twenty-four years old when he first arrived, sixty-three years old when he returned, and eighty years old when he died. By the end of the eighteenth century the population had grown to more than 166,000. St. Petersburg was home to some of the wealthiest noblemen in the empire and some of the poorest peasants. Nearly one quarter of the population was military.14 St. Petersburg continued to be loved by some Russians and hated by others (this is still true today). In keeping with Peter the Great’s plan, the city was full of beautiful European-style architecture. It was the most European of all Russian cities. It came to be known as the “Venice of the north” because of its many islands and waterways.
Euler’s second stay in St. Petersburg was a time of professional success, but it was peppered with episodes of personal loss. In 1771 his house burned to the ground. The quick action of a selfless servant who carried him out of the burning building saved Euler’s life. His entire library was destroyed by the fire, but thankfully for science, his manuscripts were rescued. In response to the tragedy Catherine gave him new housing and replaced his losses. In 1776 Euler’s beloved wife Katharina died. A year later he married Katharina’s half-sister Salome Abigail Gsell.
Almost immediately after he left Berlin, cataracts stole the vision from his left eye. An operation in 1771 briefly returned vision to this eye, but an infection caused a relapse and he became blind again. During this time Euler continued to publish mathematics, primarily by dictating his work to his son. Amazingly, Euler’s mathematical output continued unabated. During this time of total blindness he proved some of his most important theorems and wrote some of his most influential books.
There is a widely held belief that a mathematician’s most productive years are found in his youth; by the time he reaches forty—or thirty—all creativity and genius disappears. In his well-known memoir, A Mathematician’s Apology, the British mathematician G. H. Hardy (1877–1947) wrote, “No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man’s game.”15 While this does describe the declining quality of professional accomplishments for many mathematicians (and for practitioners of other creative fields), it does not represent the trajectory of Euler’s career. His return to St. Petersburg was celebrated with fanfare, and he did not disappoint. As one historian wrote, Euler “demonstrated at once that he had not returned to Russia to retire but was, on the contrary, at the peak of his productivity.”16
Just as Beethoven overcame the seemingly insurmountable obstacle of deafness to compose symphonies, so was Euler able to create deep, beautiful, and often “visual” mathematics from his darkened world. It is one of the great triumphs of the human spirit.
In addition to his pure mathematical research, Euler continued to make key contributions to applied mathematics. One of the most important problems of the day was to devise an accurate and reliable method of navigating at sea. Celestial navigation was only as good as the nautical tables that gave the locations of the heavenly bodies at any given time. The moon is the most conspicuous object in the nighttime sky, but since the motion of the moon is determined by the gravitational interaction of three bodies—itself, the earth, and the sun—it is an extremely difficult mathematical task to determine, in advance, its location at a specific time. Even today we do not fully understand the infamous three-body problem. Newton’s theory of gravitation described planetary motion, but this work did not provide a computational algorithm for predicting this motion. In 1772 Euler developed a mathematical model of the motion of the moon that was computable and that yielded a very accurate approximation to the moon’s motion. Extremely dependable sets of lunar tables were assembled from Euler’s model. Expressing their thanks for his contribution, the Board of Longitude in France and the British parliament both rewarded Euler handsomely.
Euler’s mathematical output continued until the day of his death, at the age of seventy-six. His last day was described by the Marquis de Condorcet (1743–1794) in his eulogy to Euler:
He had retained all his facility of thought, and, apparently, all his mental vigour: no decay seemed to threaten the sciences with the sudden loss of their great ornament. On the 7th of September, 1783, after amusing himself with calculating on a slate the laws of the ascending motion of air-balloons, the recent discovery of which was then making a noise all over Europe, he dined with Mr. Lexell and his family, talked of Herschel’s planet [the recently discovered planet Uranus], and of the calculations which determine its orbit. A little after he called his grand-child, and fell a playing with him as he drank tea, when suddenly, the pipe, which he held in his hand dropped from it, and he ceased to calculate and to breathe.17
Leonhard Euler is buried in St. Petersburg, Russia.
It is difficult to list Euler’s greatest accomplishments in the subject of mathematics. We could quote one of his many theorems. We may also point to the successful textbooks that he penned, such as Introductio in analysin infinitorum, which the historian Carl Boyer called the most influential textbook in the modern era of mathematics. It might be his work in applied mathematics, such as his book Mechanica in which, for the first time, techniques of calculus are systematically applied to physics. It may be his writings for nonspecialists, such as the extremely popular Letters to a German Princess, which consists of a collection of lessons for Frederick the Great’s niece, the Princess of Anhalt-Dessau. Perhaps it was his ability to organize and frame isolated results and seemingly unrelated ideas into a cohesive and ordered body of mathematics. Maybe it was the elegant and useful notation he created: Euler introduced e as the base of the natural logarithm; he made popular the use of the symbol π; at the end of his life he used i to denote 
(this notation was made popular by Gauss); he used a, b, and c to denote sides of a typical triangle with A, B, and C the angles opposite; he used Σ for sums; he denoted finite differences by ρx; and he began the use of f (x) for a function.
It is difficult to single out one of Euler’s many, many theorems as the most important. Some contend that it is the relation that brings 0, 1, 7, π and i into one concise into one concise formula,
eπi + 1 = 0.
Perhaps it is one of his wondrous infinite series formulas, which showed the power of calculus. It might be one of his theorems in number theory, such as those that brought closure to famous conjectures of Pierre de Fermat (1601–1665).
We, of course, will focus on the simple formula for polyhedra that relates the number of vertices, edges, and faces by
V − E + F = 2.
A recent survey of mathematicians showed that in their eyes, Euler’s polyhedron formula is the second-most beautiful theorem in all of mathematics. The theorem voted most beautiful was Euler’s formula eπi + 1 = 0.18
In order to understand Euler’s polyhedron formula we must look a little closer at polyhedra. What is a polyhedron?