source: P.J.Nahin. «Dr.Euler's fabulous formula» (Princeton, 2006).

P. J. Nahin

Euler: Tne Man and
the Mathematical Physicist

Leonhard Euler. Oil painting of E.Handmann, 1756

God, when he created the world, arranged the course of all events so that every man should be every instant placed in circumstances to him most salutary. Happy the man who has wisdom to turn them to good account!

— Leonhard Euler, whose own life is a testament to his words (from his Letters to a Princess of Germany, 3 vols., 1768–1772)

While there is a steady stream of biographies treating famous (or, even better from an entertainment point of view, infamous) persons in popular culture, there is still not even one book-length biography, in English, of Euler. There are a German-language biography (1929) and a French-language one (1927), as well as two more recent (1948, 1982) non-English works, but all are obsolete by virtue of the vast Eulerian scholarship that has occurred since they were written. Euler himself wrote so prodigiously that it would be a huge undertaking for a biographer to write, with true understanding, of what he actually did. Euler wrote, you see, more than any other mathematician in history. During his lifetime more than five hundred (!) books and articles by him were published (even more after his death), and his total work output accounts for a third of all that was published in Europe on mathematics, theoretical physics, and engineering mechanics from 1726 to 1800. In addition, there is another, equally enormous body of surviving personal letters (nearly 3,000 of them to and from hundreds of individuals), more thousands of pages of research notebooks, and voluminous diary entries that he continuously generated from his college days until his death. A dedicated biographer will have to read it all.

Still, scattered all about in the journal literature are numerous and (mostly) excellent essay-length biographical treatments on one of the greatest mathematicians who ever lived. Several have served as both my sources and my models. There is a lot of repetition among them, however, and to keep this essay from degenerating into an explosion of biographical citations I've simply listed my major references here in one note.1 Unless I am specifically quoting from a particular author, I've not bothered with biographical citations. Perhaps by 2107 (the quadri­centennial of Euler's birth) there will at last be an English-language, book-length biography of him that our grand­children can curl up with next to a fireplace on a cold night. [Книги-то есть, но, видимо, они не удовлетворяют автора листажом. Одна из последних — Emil A. Fellmann. "Leonhard Euler" (Birkhäuser, 2007), 185 p. Иллюстрации, которые вы видите в этом файле, взяты оттуда. E.G.A.]

Euler's life unfolded, in a natural way, in four distinct stages: his birth and youthful years in Switzerland, his first stay of fourteen years in Russia at the Imperial Russian Academy of Sciences in St. Petersburg, his departure from Russia for twenty-five years to join Frederick the Great's new Academy of Sciences in Berlin, and his return at Catherine the Great's invitation to the St. Petersburg Academy until his death. I'll treat each stage separately.

Euler's ancestors were present in Basel, a German-speaking area of Switzerland with a population of about 15,000, from 1594 onward. Most were artisans, but Euler's father, Paul (1670–1745), was a trained theologian (a 1693 graduate of the theological department of the University of Basel) and a Protestant minister. Paul's intellectual interests were broad, and while a student at the University he attended mathematics lectures given by Jacob (aka James) Bernoulli (1654–1705) — one of the founders of probability theory — at whose home he boarded. Another of the boarders Paul became friendly with was Jacob's brother Johann (aka John or Jean) Bernoulli (1667–1748), who would play an important role in the life of Paul's first son many years later.

Upon graduation at age twenty-three and after a short ministry at an orphanage, Paul was appointed pastor at a church next to the university. With that he felt secure enough to marry Margaret Brucker, herself the daughter of a minister, in April 1706. A year later to the month, on April 15, 1707, their first child and son Leonhard was born (Euler grew up with two younger sisters, but his only brother — Johann Heinrich who, following family tradition, became a painter — was born after Euler had left home by the age of twelve, if not sooner, for his first formal schooling). The year after his birth Euler's family moved a few miles from Basel, to the village of Riehen, where young Euler spent his youth.

It was not a life of luxury. His father's new parsonage had but two rooms; a study and the room in which the entire family lived. This simple life in the country with loving, educated parents certainly did nothing to make Euler either a snob or a fool. All through his life Euler impressed all by his calm disposition, his strong sense of practicality, and his deeply held religious views. He was strongly influenced by his parents, and his was at first a "home taught" education. His mother instructed him in classical subjects (blessed with what must have been a photographic memory, Euler could recite all 9,500 verses of Virgil's Aeneid by heart!), while his father introduced him to mathematics. In a short, unpublished autobiographical sketch he dictated to his eldest son in 1767, Euler recalled that his very first mathematical book — almost certainly given to him by his father — was an old (1553) edition of a text on algebra and arithmetic, and that he had faithfully worked through all 434 problems in it. There was, however, only so much that his parents could do, so, seeing that he was clearly an unusually talented boy, it was decided that young Euler needed a more formal educational setting. He was therefore sent back to Basel, to live with his now widowed grandmother (on his mother's side) while a student at Basel Latin school. This education was supposed to lead, at least in Paul's mind, to the son following in the father's theological footsteps.

Entering Basel Latin must have been quite a shock to the newly arrived country boy. Corporal punishment was employed on a regular basis and in abundance; no one would have said the rod was spared, as it wasn't unheard of for an unruly student to be beaten until the blood flowed. If the teachers weren't hitting the schoolboys, then the students did it themselves, with classroom fist-fights mixed in between parental assaults on the teachers! Curiously, the one subject that Euler would perhaps have thought would make such brutality tolerable — mathematics — was not part of the school's curriculum. Instead, Euler's parents found it necessary to employ a private mathematics tutor for their son, who did manage somehow to survive Basel Latin in one piece.

In the fall of 1720, at the age of thirteen, Euler entered the University of Basel. To be so young at the University was not at all unusual in those days, and it wasn't as if he had been admitted to the local equivalent of Princeton, either. There were only a hundred students or so, along with nineteen underpaid and mostly (but not all, as you'll see) second-rate faculty. It should be no surprise that Euler stood out in such a crowd, and in June 1722 he graduated with honors with an undergraduate thesis in praise of temperance. He followed that with a master's degree in philosophy granted in June 1724, when a mere seventeen years old, with a thesis written in Latin comparing the philosophical positions of Descartes and Newton. While all of this would hardly seem the stuff of a proper education for one who would not too much later become the greatest mathematician of his day, the pivotal intellectual event in Euler's life was soon to take place.

Two years before Euler's birth, his father's mathematics teacher at Basel, Jacob Bernoulli, had died. He was succeeded as professor of mathematics at Basel by his younger brother Johann, Paul's old friend and fellow boarder in Jacob's home. (Johann's youngest son, Johann II [1710–1790], was similarly a friend of young Euler and a fellow master's degree recipient in 1724.) While many of the colleagues of the Bernoulli brothers at Basel may have been second-raters, Jacob and Johann certainly were not. Both were mathematicians with international reputations and, while young Euler of course never met Jacob, Johann was to have a profound influence on him.

Johann Bernoulli could have been at several far more prestigious places than the University of Basel, which was pretty much a backwater institution. He had, in fact, declined multiple attractive offers of professorial chairs in Holland because of the wishes of his wife's family. But just because he was at Basel didn't mean he had to like it, and he didn't — he was notorious for giving short-shrift to his elementary mathematics classes. He did, however, offer private instruction to the few he felt promising, and Euler joined that select group — the fact that Bernoulli knew Euler's father certainly didn't hurt — sometime around 1725. Probably because of his friendship with Johann II — and, through him, with the older Bernoulli sons, Nicholas (1695–1726) and Daniel (1700–1782) — Euler came to the attention of the Basel professor of mathematics. As Euler recalled in his 1767 autobiography,

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 Saturday afternoon and he kindly explained to me everything I could not understand ... and this, undoubtedly, is the best method to succeed in mathematical subjects.

Before coming under the influence of Bernoulli Euler had, in obedience to his father's wishes, devoted himself to the study of theology, Hebrew, and Greek, all in preparation for a life of ministry. Once Bernoulli came to realize the promise of his young student, however, Paul was eventually convinced that his son should follow in the footsteps of Johann, and not his. It says much to the love and compassion Paul must have had for his son that, even when faced with what had to be a great personal disappointment, he stood aside in favor of what his son so dearly wanted. Bernoulli's ever increasing admiration for Euler's blossoming genius can be measured by how he addressed his former student in their correspondence: from 1728 until Bernoulli's death, the salutations evolved through "The Very Learned and Ingenious Young Man" (1728), "The Highly Renowned and Learned Man" (1729), "The Highly Renowned and by Far Most Sagacious Mathematician" (1730), and "The Incomparable L. Euler, the Prince among Mathematicians" (1745). Johann Bernoulli was a man given to much professional competition and even jealousy, even with his own brother Jacob and with his own sons,2 but not once does it seem that he challenged Euler's superiority as a mathematician. It was just so obvious that Euler was of a different class altogether (and Bernoulli was, himself, world class).

Once he had reached the ripe old age of nineteen (!) Euler was finished with his formal schooling, and he began to think about starting a career as an academic. He hoped to remain in Basel, near his family, and so when the professor of physics there died in September 1726 Euler applied for the job. Johann Bernoulli was delighted at the prospect of Euler joining him on the faculty, and encouraged him in his application. As part of that application, Euler prepared what is now considered to be equivalent to a doctoral dissertation, titled "Dissertatio physica de sono" ("A Physical Dissertation on Sound"). Short it was, a mere sixteen pages, but it was to become famous, and was cited by scholars for a century. It was essentially a research program in acoustics, which ended with discussions on six then hotly debated special problems of physics. As an example, one of them is now a classic included in just about all calculus-based college freshman physics texts: what would happen, if we ignore air friction and the earth's motion, to a stone dropped into a straight tunnel drilled to the center of the earth and onward to the other side of the planet?3

In this attempt at a professorship at Basel, Euler failed — he was simply too young — but yet another far more wonderful possibility existed for him, again because of his friendship withjohann Bernoulli (as well as with his sons Nicholas and Daniel). At almost the same time that he applied for the Basel position, Euler received an offer to join the new Imperial Russian Academy of Sciences in St. Petersburg. Founded by Peter the Great in 1724, the year before his death, the Academy was part of Peter's efforts to improve education and to encourage the development of western scientific thought in peasant Russia. The Russian Academy was created in the image of the existing scientific academies in Berlin and Paris, not in the image of the Royal Society in London, which, in Peter's view, was far too independent. The Russian Academy had direct governmental oversight. That was a feature that wouldn't bother Euler until after his arrival. Upon Peter's death his widow Empress Catherine I — not to be confused with Catherine II, better known as Catherine the Great — who shared his desire to improve education in Russia, became the Academy's benefactor.

Euler's invitation to go to Russia came about due to the cleverness of his Basel mentor Johann Bernoulli, who was himself the original choice of the Academy. Declining the invitation, he instead suggested that either of his two oldest sons would be good alternatives and — of course! — neither could go without the other. So, in 1725 Nicholas (appointed in mathematics) and Daniel (appointed in physiology) were off to Russia, thereby giving Euler two friends in St. Petersburg "on the inside." They promised their young friend that, at the first opportunity, they would champion him for an appointment. That opportunity soon came in a totally unexpected manner, when, in the summer of 1726, Nicholas suddenly died from appendicitis. Daniel assumed his late brother's mathematics appointment, and recommended that Euler be invited to fill his own now vacated spot in physiology. And so it happened that in the fall of 1726 Euler received his invitation to join the St. Petersburg Academy as a scholar not in mathematics but in physiology!

In November Euler accepted; it was, for a young man, not an unattractive proposition, carrying a small salary combined with free lodging, heat, and light, as well as travel expenses. In addition to Daniel Bernoulli, the Swiss mathematician Jakob Hermann (1678–1733), who was a second cousin of Euler's mother, would be a colleague at St. Petersburg. Euler's only condition of acceptance was to ask for a delay in his departure until the spring of 1727. His letter of acceptance cites weather concerns as the reason for the delay, but his real reasons were twofold. First, of course, was his desire to remain in Basel where the vacant physics position decision had not yet been made. And second, assuming that job failed to materialize (as it did not), Euler needed time to learn some anatomy and physiology so as not to arrive in Russia an ignoramus. Euler spent so much time in his early years studying the "wrong stuff"!

The academic year 1726–1727 was a "holding action" year for Euler, but, being Euler, he didn't just sit around with his fingers crossed hoping for a possible physics position in Basel and studying anatomy for the fall-back Russian job. He also wrote and submitted a paper to the Paris Academy of Sciences prize competition for 1727, in which the problem was to determine the best arrangement and size for the masts on ships, where "best" meant achieving the maximum propelling force from the wind without tipping the ship over. It is astonishing that Euler, still a teenager, nevertheless took second place, losing only to Pierre Bouguer, a professor well on his way to becoming a leading French nautical expert. Indeed, the Paris competition problem had been selected purposely to give Bouguer a huge head start on any competitors — he had been working on the problem for years. It must have been a disappointment to Euler to lose — later in his career Euler would win the Paris Academy competition a total of twelve times — but in fact Bouguer's win was a blessing in disguise. If Euler had won, maybe he would have gotten the Basel job, and would have taken a pass on Russia, where he would find compelling reasons to dedicate himself, totally and without distraction, to his academic work. In April 1727, as he turned twenty, Euler left for Russia and never set eyes on Basel again.

After an arduous seven-week journey by boat, wagon, and on foot, Euler arrived in St. Petersburg (then the capital of Russia) in late May, only to learn that Catherine had just died. The future of the Russian Academy was suddenly very much in jeopardy. The new tsar, Peter II, was a twelve-year-old boy, and the real power in Russia lay in the shadows. The nobility, who liked peasant Russia just as it was, ignorant and pliant, resented all of the foreign German, Swiss, and French intellectuals who had been recruited for the Academy and so withdrew financial support. The Academy appeared to be on the verge of physical collapse when the nobles moved the imperial court back to Moscow, and took the Academy's President with them to serve as the boy tsar's tutor. A number of the Academy's members despaired and, as soon as they could, returned home. But not Euler. His studies of anatomy and his second place finish in the Paris masting competition finally paid off for him — they brought him to the attention, of all things, the Russian Navy, which offered him a position as a medical officer. Even if the Academy sank, at least Euler's ship would still be afloat, perhaps even literally! With the medical appointment, along with his "doctoral" dissertation on sound, we might legitimately think of Euler as "Doctor" Euler!

Turmoil at the Academy continued until 1730, when, with the death of Peter II, Empress Anna Ivanovna's rise to power brought some stabilizing influence to the political situation. Euler's distant relative Jakob Hermann had resigned to return home to Switzerland, but Daniel Bernoulli had replaced Hermann as professor of mathematics at the Academy. Two years later Anna returned the capital of Russia to St. Petersburg. Euler's life flourished thereafter and, at age twenty-three, he was made professor of physics. When Bernoulli resigned in 1733 to accept a professorship back in Basel, Euler was selected to replace him as the premier mathematician of the St. Petersburg Academy. His personal life took a happy turn, as well, with his marriage to fellow Swiss Katharina Gsell (1707–1773), the daughter of a painter who taught at the school attached to the Academy. This event was celebrated by an Academy poet, who at one point gives us a hint as to how Euler's dedication to mathematics was viewed by others:

Who would have thought it,

That our Euler should be in love?

Day and night he thought constantly.

How he wanted more to calculate numbers,

. . .

Euler didn't think always of numbers: the first of Euler's thirteen children was born late the next year. [Только не надо думать, что Эйлер встретил свою старость, окружённый многочисленными потомками: детская смертность в те времена была фантастически высока (см. далее табличку из книги Феллмана), а медицина, очевидно, наоборот (см. примечание 13 про то, как Эйлера лечили от катаракты).

Leonhard Euler's children
(The three underlined sons outlived their father.)
1 Johann Albrecht 27.11.1734   Petersburg 18.9.1800   Petersburg
2 Anna Margaretha 8.6.1736   Petersburg 2.7.1736   Petersburg
3 Maria Gertrud 9.5.1737   Petersburg 1.5.1739   Petersburg
4 Anna Elisabeth 5.11.1739   Petersburg 19.11.1739   Petersburg
5 Karl Johann 15.7.1740   Petersburg 16.3.1790   Petersburg
6 Katharina Helene 15.11.1741   Berlin 4.5.1781   Wiborg
7 Christoph 1.5.1743   Berlin 3.3.1808   Wiborg
8 Charlotte 12.7.1744   Berlin 13.2.1780   Hückelhoven
9 Hermann Friedrich 8.5.1747   Berlin 12.12.1750   Berlin
10 Ertmuth Louise 13.4.1749   Berlin 9.8.1749   Berlin
11 Helene Eleonora 13.4.1749   Berlin 11.8.1749   Berlin
12 August Friedrich 20.3.1750   Berlin 10.8.1750   Berlin
13 NN probably died before baptism or stillbirth

Кстати, в «Кванте» № 5 (1974), с. 26 портрет старшего сына Эйлера был ошибочно объявлен портретом самого Эйлера. E.G.A.]

Johann Albrecht Euler. Oil painting by E.Handmann, 1756

Happily married, and the first among mathematicians in St. Petersburg, Euler was a contented man. Both the Academy President and its Secretary, the Prussian Christian Goldbach, were his close friends, and his job security seemed assured. And, as long as he kept his nose firmly planted in mathematics, his life would be equally sheltered from the outside world of Russian political intrigue. Euler had, at last, "arrived," and the first period of his enormously productive career began to really take-off. Indeed, it already had. In a letter dated December 1, 1729, for example, Goldbach brought one of Fermat's conjectures (that 22ⁿ+ 1 is prime for all nonnegative integers n) to Euler's attention. By 1732 (and probably earlier) he had shown the conjecture is false by factoring the n=5 case. During that same period, 1729–30, Euler discovered how to generalize the factorial function for the nonnegative integers to the gamma function integral, which holds for all real numbers. Before Euler, writing (–½)! would have been without meaning, but after Euler the world knew4 that (–½)! = √π.

By 1735 Euler had solved a problem that had stumped all mathematicians — including both of the Basel professors of mathematics, Jacob and Johann Bernoulli — for almost a century. He calculated the exact value of
 ζ(s) =     1 

 ns

 n=1

— what we today call the zeta function — not only for s=2 (the original problem) but for all even integer s. This wonderful calculation made Euler's reputation across all of Europe as news of it spread through the mathematical world. Euler's old mentor back in Basel, Johann Bernoulli, was moved to say of his brother Jacob, who had tried so hard and failed to do what Euler had done, and had died not knowing the elegant solution Johann would live to see, "if only my brother were still alive." In 1735 Euler defined what has been called the most important number in mathematical analysis after π and e

 lim  ( 1 +   1 

2

 + ... +   1 

 n

 – ln n ) ,
n → ∞

called Euler's constant or gamma — and calculated its value to fifteen (!) decimal places. In an age of hand computation, that was itself an impressive feat. The year 1735 wasn't all glorious, however; he nearly died from a fever. Once recovered, though, he soon hit his stride again; for example, in 1737 he found a beautiful connection between the primes and the zeta function, which gave him the first new proof since Euclid of the infinity of the primes.5

It was in the 1730s, too, that Euler began his fundamental studies in extrema theory. In the "ordinary" calculus of Newton and Leibniz one learns how to find the values of the variable x such that the given function  f (x) at those values has a local minimum or maximum. In the calculus of variations, one moves up to the next level of sophistication: what function  f (x) gives a local extrema of  Jf (x)}, where J (called a functional) is a function of the function  f (x)? One of the pioneers in this sort of so-called variational problem was Johann Bernoulli, who in 1696 posed the famous brachistochrone problem: what is the shape of the wire (connecting two given points in a vertical plane) on which a point mass slides under the force of gravity, without friction, so that the vertical descent time from the high point to the low point is minimum? An even older question is the classic isoperimetric problem: what closed, non–self-intersecting curve of given length encloses the maximum area? Everyone "knew" the answer is a circle, but nobody could prove it! Those two problems,6 and others like them, were all attacked by specialized techniques, different for each problem. There was no general theory.

Until Euler. In 1740 he finished the first draft of his book Method of Finding Curves that Show Some Property of Maximum and Minimum (it was published in 1744, after he had left St. Petersburg for Berlin). In it appears, for the first time, the principle of least action, about which I'll say more in just a bit. A line in an appendix to this work displays both the religious side of Euler and the deep attraction such problems had for him: "[S]ince the fabric of the universe is most perfect, and is the work of a most wise Creator, nothing whatsoever takes place in the universe in which some relation of maximum and minimum does not appear."

It wasn't all pure mathematics at St. Petersburg, however. In 1736 his two-volume book on mechanics appeared (Mechanics, or the Science of Motion Set Forth Analytically), in which he made extensive use of differential equations. This work was almost immediately recognized as a worthy successor to Newton's 1687 masterpiece, Principia; for example, Johann Bernoulli stated that the book showed Euler's "genius and acumen." Not everybody felt that way, however, notably the English gunnery expert Benjamin Robins (1707–1751), who thought the use of differential equations to be an admission of failure (to do experiment), and to represent an uncritical obedience to calculation. This is, of course, a distinctly odd position to the modern mind! Robins was no fool — he was the inventor of the ballistic pendulum, studied by every first-year college physics student to this day — but even in the 1730s Robins's negative view was that of a tiny minority. Euler and Robins would cross swords, of a sort, a few years later over a text authored by Robins, and again you'll see that Robins was singularly unappreciative of Euler.

In this short essay I can't even begin to do justice to what Euler did during his first stay in St. Petersburg, and my comments so far are a mere sampling of his technical accomplishments, out of dozens that could have been cited. But let me also mention here that he provided great immediate practical service to Russia with his astronomical observations at the St. Petersburg Observatory, work that played an important role in bringing the science of cartography (mapmaking) in Russia up from a primitive state to then modern standards. It was during that period that Euler's earlier, near-fatal brush with fever came back to haunt him — he began to lose vision in his right eye. Euler wrote to Goldbach in 1740 to say "Geography is fatal to me," believing that eyestrain from detailed attention to correcting landmaps was the cause of his difficulty. (Today it is believed that an eye abscess resulting from the earlier fever was the more likely cause.) By the time he wrote to Goldbach, Euler was nearly blind in the right eye. Later, a cataract in his left eye would leave him totally blind for the last twelve years of his life.

While in St. Petersburg Euler also worked on practical engineering problems involving naval ship design and propulsion, again using differential equations to study the motion of objects in a fluid. He brought all of that work together in his book Naval Science, mostly completed by 1738 while he was still at St. Petersburg, but not published until 1749, after he had left for Berlin.

That same year saw Euler's path cross, indirectly, that of a man he would be involved with in a most unpleasant encounter years later. That episode would be a war of words, and the "other side," Voltaire, was one of the great literary figures of those times and as much a master of the poison-pen as Euler was of mathematics. Voltaire was the pen name adopted in 1719 by the French writer/poet François-Marie Arouet (1694–1778), who, during a forced exile in London from 1726 to 1729 (as an alternative to a prison sentence for the crime of exchanging insults with a man of higher social station than Voltaire's), became swept up by Newton's theories. Voltaire had attended Newton's funeral in 1727, talked of that impressive event for the rest of his life, and threw himself into writing what became a famous popularization of Newton's philosophy (Éléments de la philosophie de Newton), which appeared in 1738. It was a time during which the conflict between Newton's physics and Leibniz's metaphysics was a hot issue, and in Éléments Voltaire praised Newton while later (in 1759), in his famous satire Candide, he spoofed Leibniz (in the form of that work's character Dr. Pangloss). Candide is an attack on the view championed by Leibniz that we live in the "best of all possible worlds," and that all that happens is "for the best": the quote from Euler that opens this essay would have earned Voltaire's deepest scorn.

In the 1738 near-miss encounter with Euler the battleground was scientific, however, not literary, and while the prose of Éléments was elegant it is clear Voltaire did not have a deep understanding of Newton's mathematical and scientific concepts. As one writer put it, Éléments "made Newton's mathematics known to others if not to its author."7 In the arena of analytical reasoning Voltaire, great writer of literary prose that he might be, was no match for Euler. The near-encounter with Euler was the result of the Paris Academy of Sciences prize competition, announced in 1736, to be awarded in 1738. The Academy's problem was for competitors to discuss the nature of fire. This was a time before there was any concept of a "chemical reaction," and philosophers still talked of the Aristotelian elements of air, earth, water, and fire as if they were fundamental entities. It was, not to be too tongue-in-cheek, also a hot topic. Being fascinated by science apparently convinced Voltaire that he could do science, even though he was completely without formal training. Voltaire was not a modest man. Euler's entry shared first place, but Voltaire did manage to snare an honorable mention, as did his lover Émilie du Châtelet (1706–1749), who, by all accounts, understood science and mathematics far better than did Voltaire. She prepared, for example, the first French translation of Newton's Principia, published after her death.

Émilie, with whom Voltaire had begun an affair in 1733 that would last until her early death shortly after childbirth (by a man neither Voltaire nor her husband), was an intelligent woman who employed experts in mathematics and physics as tutors. This is important in our story of Euler because one of her instructors (and yet another lover) was the French mathematician and astronomer Pierre Louis de Maupertuis (1698–1759) who in 1736 led an expedition to make measurements of the earth's shape and in 1738 published the book La figure de la terre, which supported the conclusion the planet is oblate and made Maupertuis famous as "the earth flattener." Another tutor was Samuel König (1712–1757), who had also studied for three years in Basel with Euler's old mentor Johann Bernoulli. While taking lessons from König, du Châtelet wrote a book titled Institutions de physique (published in 1740), treating the philosophical ideas of Descartes, Newton, and Leibniz, as well as the concepts of the natures of space, matter, force, and free will. König and du Châtelet fell out over her book, which König felt was simply a rehash of what he had taught his pupil. He essentially charged her, in private conversations with others, with stealing his work. Ten years later König would make a similar charge directed at Maupertuis, a charge that resulted in Voltaire, Maupertuis, and König clashing in a conflict that has been called one of the ugliest in the history of science. Euler would be swept up into it as well, and none of the four men would emerge unscathed.

The events that eventually led to that conflict started in mid-1740, when the new Prussian monarch Frederick II ("the Great") attempted to entice Euler away from St. Petersburg to join his newly energized Berlin Academy of Sciences.8 Frederick neither knew nor appreciated mathematics but wanted Euler in his circle anyway, just because he knew others thought Euler was a genius. Euler was simply a prize to be bought as an ornament for his court (a type of faculty recruitment not unheard of in modern academia). Indeed, the entire Academy may have been just for show, at least at first: in a letter dated July 1737, to Voltaire, Frederick wrote that a "king needed to maintain an Academy of Sciences as a country squire needed a pack of dogs."

Euler initially declined the Berlin offer; several months later, when Empress Anna died leaving only an infant heir, which threw Russia once more into political turmoil and resulted in all the "foreigners" at the St. Petersburg Academy again being viewed with hostile suspicion, Euler (at his wife's insistence) reconsidered the king's invitation. He told Frederick what it would take to get him to come to Berlin, and in February 1741 the deal was struck. Euler's official reason to the St. Petersburg Academy for his wish to resign was that of health — he claimed he needed a less harsh climate, and that he was concerned for his eyesight. The Academy seemed to accept that, and Euler managed to leave Russia on good terms (which would work to his advantage in the future). His real reason for leaving was revealed shortly after he arrived in Berlin in late July 1741, when Frederick's mother, puzzled at why Euler seemed unwilling to answer any questions at length, bluntly asked him why he was so reserved, almost timid, in his speech. Euler's answer was equally blunt: "Madam, it is because I have just come from a country where every person who speaks is hanged."

When Euler came to Berlin Frederick's new Academy was still very much in a formative stage — there was not yet even a president. Frederick had offered the job the year before to another person, who had declined. Euler was therefore, at least in his mind, a candidate for the job, but so was Maupertuis, who had also been invited (at Voltaire's suggestion) to Berlin by the Francophile Frederick. It would only be after many years and disappointments that Euler would come to understand that a social snob like Frederick would never consent to a mere Swiss burgher being the head of his Academy, no matter how brilliant and accomplished he might be. Whenever possible — that is, when competence was not required — Frederick filled openings in government and military positions with nobility, and excluded commoners no matter how talented they might be. It did Euler's cause no good either that he also failed in the king's eye at being a witty conversationalist or the writer of French poetry. (So enamored with French culture was Frederick that in 1744 he ordered all the memoirs of the Berlin Academy to be published in French, not the usual Latin or even German.)

Simply being a mathematician hurt Euler, too. Frederick had written (January 1738), while still crown prince, to Voltaire — with whom he had corresponded already for two years — to tell the French writer what his plan of study would be: "to take up again philosophy, history, poetry, music. As for mathematics, I confess to you that I dislike it; it dries up the mind." Time did nothing to change the king's mind. Years later (January 1770) he wrote to Jean D'Alembert — the French mathematician Frederick wanted to be president of the Berlin Academy — to say "An algebraist, who lives locked up in his cabinet, sees nothing but numbers, and propositions, which produce no effect in the moral world. The progress of manners is of more worth to society than all the calculations of Newton."

This was the man to whom the naive Euler bowed his head. Of his Berlin appointment he wrote to a friend to say "I can do just what I wish [for his technical studies].... The King calls me his professor, and I think I am the happiest man in the world." Later, Euler would change his opinion. Things got off to an unsettled start when, nearly simultaneous with Euler's arrival in Berlin, Frederick was off to war with an invasion of neighboring Austria. His mind was not on either the Academy or Euler's possible role in it. The issue of the presidency would, in fact, remain unresolved for five years! The king's correspondence with Maupertuis shows that Euler was never, ever, in the running: more than a year before Euler's arrival, in a letter dated June 1740, Frederick wrote to the Frenchman to express his "desire of having you here, that you might put our Academy into the shape you alone are capable of giving it. Come then, come and insert into this wild crabtree the graft of the sciences, that it may bear fruit. You have shown the figure of the Earth to mankind; show also to a King how sweet it is to possess such a man as you" (my emphasis). When Maupertuis finally accepted in 1746, he was, in the words of Frederick himself, to be "the pope of the Academy." Euler received the consolation prize of being Maupertuis's chief deputy and director of the mathematics class of the Academy.

Euler's Berlin years were a time of stunning brilliance. The list of his accomplishments is simply enormous (he prepared 380 works, of which 275 were published!), but to select just a few, let me mention analyses of proposed government-supported lotteries, annuities, and pensions, studies that led to Euler's writings in probability theory; translation from English to German of Benjamin Robins's (mentioned earlier) 1742 book New Principles of Gunnery, a work of tremendous interest to the warrior-king Frederick (Robins was greatly irritated with Euler because he added supplementary material five times longer than the original work!); authorship of the book Introductio in analysin infinitorum (in which he clearly states what I have called "Euler's formula" all through this book [Т.е. eπi = –1. E.G.A.]), a text one prominent historian9 of mathematics has ranked as important as Euclid's Elements; studies in the technology of constructing optical lenses, toothed gears, and hydraulic turbines; and finally (for this list), assorted studies in differential geometry, hydrodynamics, and lunar/planetary motion. Euler's extraordinary intellectual and physical powers were at their peak in his Berlin period.

While denied the presidency he so desired, Euler's administrative responsibilities at the Academy were nevertheless extensive. He served as the de facto president during Maupertuis's absences, selected Academy personnel, oversaw the Academy's observatory and botanical gardens, and provided oversight of numerous financial matters (most important of which was the publication of calendars, maps, and almanacs, the sale of which generated the entire income of the Academy). On this last matter, in particular, Euler learned early on that while Frederick might be a mathematical novice, when it came to money the king could count. In January 1743 Euler wrote to the king to suggest more money could be raised by selling almanacs in the newly conquered territory in Austria. In reply, Frederick wrote "I believe that, being accustomed to the abstractions of magnitude in algebra, you have sinned against the ordinary rules of calculation. Otherwise you would not have imagined such a large revenue from the sale of almanacs." Two decades later the matter of almanac revenue would drive the final wedge between Euler and the king.

Frederick's private view of Euler, briefly hinted at in the above response, was more openly expressed in his correspondence with others. In an October 1746 letter to his brother, for example, the king called Euler a necessary participant in the Academy because of his prodigious abilities, but, said that persons such as Euler were really nothing more than "Doric columns in architecture. They belong to the under-structure, they support the entire structure." That is, good enough to hold the roof up, but that was it. What Voltaire and Maupertuis had, that Euler didn't and Frederick valued most, was the ability to generate light-hearted, clever conversation and correspondence (often at the expense of others). The fact that Euler couldn't compose a minuet or a flowery poem was a fatal lacking, in the king's view. Despite all this, Euler's life under Frederick seems to have been a full one, as well as one of increasing financial well-being. Since 1750, for example, his now widowed mother had lived with Euler, and in 1753 he had the resources to purchase an estate on the outskirts of Berlin that she managed for him.

Then, in 1751, we can see the beginning of the end of Euler's hopes for a lifelong career in Berlin. A few years earlier, just after assuming the presidency of the Academy, Maupertuis put forth what he claimed to be a new scientific principle, called least action.10 The fact that Euler had enunciated essentially the same ideas in 1744 seems to have escaped him, and Maupertuis claimed the principle of least action in his 1750 book Essai de cosmologie. As he wrote there, "Here then is this principle, so wise, so worthy of the Supreme Being: Whenever any change takes place in Nature, the amount of action [a term most ambiguously defined by Maupertuis] expended in this change is always the smallest possible." Despite Maupertuis first laying claim to the presidency that Euler so wanted, and then to a technical concept that Euler had mathematically refined far beyond Maupertuis's mostly theological statement and which Euler surely felt was really his, Euler remained supportive of Maupertuis. Then Samuel König entered.11

König, who since 1749 had been the librarian to the royal court at The Hague, had been proposed by Maupertuis for election to the Berlin Academy, which was done in 1749. Nonetheless, König then accused Maupertuis of having stolen the least action concept from an October 1707 letter by Leibniz to the Swiss mathematician Jakob Hermann (Euler's distant relative who had been at St. Petersburg with him from 1727 to 1730), a copy of which König claimed to have seen. Accusing the president of the Berlin Academy of plagiarism was a serious charge, and he was of course asked to substantiate the charge. König wasn't able to produce the copy, and a search of the surviving letters of Leibniz to Hermann failed to produce the original. An Academy committee, headed by Euler, was formed to investigate this awkward mess, which concluded that it was König who was the fraud. (Modern historians generally believe König was in the right on this matter, and that there was indeed such a letter from Leibniz, but it still has not been found to this day.) But that wasn't the end of the matter. Voltaire, who had earlier fallen out with Maupertuis over both a squabble on filling a vacancy at the Academy and Maupertuis's refusal to provide a false alibi to help Voltaire escape blame in a stock swindle (!), felt he had reason to take revenge on his previous friend. He claimed that Maupertuis had earlier been in a lunatic asylum and, in his opinion, was still crazy! Maybe König, suggested Voltaire, wasn't a fraud after all.

When Frederick publicly sided with Maupertuis, Voltaire was stung by the royal rebuff and decided to really retaliate. "I have no scepter," he wrote, "but I have a pen." The result was the 1752 satire Diatribe du Dokteur Akakia, in which a thinly disguised Maupertuis was plainly portrayed as an idiot: he is finally "reduced" to the principle of least action, that is, death, by a bullet going at the square of its speed! Diatribe made Maupertuis the laughing-stock of Europe, and the ridicule was devastating to him. In 1753 Maupertuis returned to France and then, only at Frederick's demand that he return because the Academy was in chaos with his absence, he came back the next year — only to leave again in 1756 for good. As a supporter of Maupertuis, the episode did Euler no good, either. In a sequel to Diatribe, Voltaire inserted a snide reference to Euler by name. At one place Maupertuis and König are imagined to sign a peace treaty, which includes the following passage:

our lieutenant general L. Euler hereby through us openly declares I. that he has never learnt philosophy and honestly repents that by us he has been misled into the opinion that one could understand it without learning it, and that in future he will rest content with the fame of being the mathematician who in a given time has filled more sheets of paper with calculations than any other.

While it is said the king laughed until he cried at reading Voltaire's cruel spoof (thus showing he loved what passed for satiric wit more than friendship), Voltaire's book nonetheless was a public insult to the head of Frederick's Academy. The king had a bonfire made of copies of Diatribe and Voltaire, too, found it expedient to return to France. He, like Maupertuis, never returned to Berlin. The assassin had brought himself down along with his victim.

To Euler's despair, even with the downfall of Maupertuis the king continued to overlook Euler as the logical person to be the next Academy president. Frederick clearly preferred the disgraced Frenchman to the one-eyed Swiss mathematician, even though it was Euler who was now keeping the king's Academy from total disintegration. There was simply no spiritual connection between the king and the half-blind man he mocked (behind Euler's back) as a "limited cyclops." So, again, the Academy went for years without a president, with Euler again serving de facto in that role. Then, in 1763, Frederick offered the presidency to the French mathematician Jean D'Alembert, who declined. If offering the job to another wasn't enough of an insult to Euler, the year after D'Alembert's refusal to come to Berlin the king named himself president! The hurt to Euler must have been enormous. And yet he remained in Berlin for two more years. What finally made Euler's decision to leave was yet one more insult from Frederick.

At the end of 1763 the King believed the Academy's income from the sale of its almanacs could be increased by changing the administrative structure of the Academy. That is, Euler would no longer be the man in charge, but would be just one voice on a committee. Euler wrote in protest, with the king replying in a sharp, unpleasant manner. Frederick's decision stood — Euler was, in no uncertain terms, in a certain sense "demoted." Euler had at last had enough with this sort of treatment, and looked for a way out. He didn't have to look far. His way out of Berlin had been laid, in fact, years earlier. In early July 1763 Euler had received a letter from Grigorij Teplov, Assessor of the St. Petersburg Academic Chancellery, sent by the authority of Russian Empress Catherine the Great, offering Euler the position of Director of the Mathematical Division of the St. Petersburg Academy. In addition, he offered Euler the post of Conference Secretary of the Academy, and positions for all of his sons. Euler quickly wrote back to Teplov to say

I am infinitely sensitive to the advantageous offers you have made by order of Her Imperial Majesty and I would be particularly happy if I were in a position to profit from it immediately... if... Mr.D'Alembert or another Frenchman had accepted the President's position of the [Berlin] Academy, nothing could have stopped me from my immediate resignation and I could not have been refused under any pretext. It is understood that everyone would have blamed me for submitting to such a President.... However, not only did Mr. D'Alembert refuse this offer, but he subsequently did the wrong thing by highly recommending me12 to the King, and if I wished to give my resignation I would be met with the most obstinate refusal. This would place me in a decidedly difficult, if not impossible, situation for any subsequent steps. (my emphasis)

Euler's comments seem to indicate that at the time he wrote he still thought he had a chance to be named president of the Academy. By 1766 those hopes were finally dead, and Euler revisited the St. Petersburg offer. Proving himself a tough negotiator, he got everything he asked for including, at last, the directorship of the St. Petersburg Academy. It took four letters of request to Frederick to obtain permission to leave Berlin, but in May 1766 the king finally relented and let Euler go, and in June Euler and his family left for Russia. Of their unhappy parting, the king wrote at the end of July to D'Alembert to say "Mr. Euler, who is in love even to madness with the great and little bear, has travelled northward to observe them more at his ease." To replace Euler as director of the mathematical class in Berlin, D'Alembert recommended to the king that the position be offered to the Italian-born Frenchman Joseph Lagrange (1736–1813), who accepted. In his July letter to D'Alembert, the king thanked D'Alembert for his aid in replacing Euler, and also got in one last insulting shot (which he somehow imagined to be funny) at Euler, writing "To your care and recommendation am I indebted for having replaced a half-blind mathematician by a mathematician with both eyes, which will especially please the anatomical members of my academy." Such a wit was Frederick.


Leonhard Euler's house in St. Petersburg from 1766 to 1783. Originally, the house had two floors, the top floor was added in the 19th century.

The final stage of Euler's life, his last seventeen years in St. Petersburg, was the mirror image of the Berlin years. In St. Petersburg he was a celebrity, and there was no greater admirer of him than the Empress herself. His personal life, however, was not so uniformly rosy. Soon after his arrival he lost nearly all the vision in his remaining eye, and a failed cataract operation13 in 1771 left him almost totally blind. That same year saw a fire destroy his home; he escaped serious injury, perhaps death, only with the aid of a heroic rescue. And in late 1773 his wife died; three years later he married again, to his first wife's half-sister. The powerful Euler intellect was not to be stopped by these events, however, and his scientific output continued to be enormous. About half of his total lifetime output was generated after his return to St. Petersburg. He started off with a bang by publishing what today we would call a bestseller, his famous Letters to a Princess of Germany.14 This work found its origins in lessons, in the form of letters, given by Euler to a fifteen-year-old second cousin of Frederick's. Those letters covered a wide range of topics, including general science, philosophy, and physics. Letters was a huge success, with many editions in French, English, German, Russian, Dutch, Swedish, Italian, Spanish, and Danish. His more advanced work in Russia included other books and papers on algebra, geometrical optics, calculus, and the probability mathematics of insurance.

Perhaps a true sign of fame, of 'having arrived,' is when people start making-up stories about you. There is a famous example of this, famous, at least, in the mathematical world, in the case of Euler. To quote a well-known historian of mathematics,

The story goes that when the French philosopher Denis Diderot paid a visit to the Russian Court, he conversed very freely and gave the younger members of the Court circle a good deal of lively atheism. There upon Diderot was informed that a learned mathematician was in possession of an algebraical demonstration of the existence of God, and would give it to him before all the Court, if he desired to hear it. Diderot consented. Then Euler advanced toward Diderot, and said gravely, and in a tone of perfect conviction: "Monsieur, a + bn/n = x, donc Dieu existe: répondez!" Diderot, to whom algebra was Hebrew, was embarrassed and disconcerted, while peals of laughter rose on all sides. He asked permission to return to France, which was granted.15

This story is absurd on the face of it — Denis Diderot (1713–1784) was not a mathematical illiterate, and it is unimaginable that a man like Euler would have participated in such a stupid stunt. Modern historians have demonstrated quite convincingly that this tale is a fairy tale, probably started by Frederick (who greatly disliked Diderot) or one of his sycophants.16

Euler had a long, almost unbelievably productive life, but it all came to an end on September 18,1783. As his biographical entry in the Dictionary of Scientific Biography describes his final hours,

Euler spent the first half of the day as usual. He gave a mathematics lesson to one of his grandchildren, did some calculations with chalk on two boards on the motions of balloons; then discussed with [two colleagues] the recently discovered planet Uranus. About five o'clock in the afternoon he suffered a brain hemorrhage and uttered only "I am dying," before he lost consciousness. He died about eleven o'clock in the evening.

Analysis incarnate, as Euler was known, would calculate no more, and he was buried with great fanfare. He lies today in the Alexander Nevsky Lavra cemetery in St. Petersburg,17 beneath an enormous headstone erected in 1837. His tomb is near some of the greatest Russian musical talents, including Mussorgsky, Rimsky-Korsakov, and Tchaikovsky.

The end came suddenly for Euler, but, as a deeply religious man, I suspect that even if he had had some advance warning he would have been at peace. Just after writing the words in the quote that opens this essay, Euler went on to write "[The] idea of the Supreme Being, as exalted as it is consolatory, ought to replenish our hearts with virtue the most sublime, and effectually prepare us for the enjoyment of life eternal." Euler clearly believed there is something beyond the grave, and it is comforting to imagine him now, vision restored with pen in hand, finishing new calculations that have at last revealed to him the value of ζ(3). But, no matter. Euler will never die. The brilliance of his mind, the clarity of his thought, lives everywhere in mathematics.18


Notes
1. 

David Brewster, "A Life of Euler," in Letters of Euler on Different Subjects in Natural Philosophy Addressed to a German Princess, J.&J. Harper, 1833; J. J. Burckhardt, "Leonhard Euler, 1707–1783," Mathematics Magazine, November 1983, pp. 262–73; A. P. Yushkevich's entry on Euler in the Dictionary of Scientific Biography, vol. 4, pp. 467–84; C. Truesdell, "Leonhard Euler, Supreme Geometer (1707–1783)," in Irrationalism in the Eighteenth Century (edited by Harold E. Pagliaro), Press of Case Western Reserve University, 1972, pp. 51–95; Ronald Calinger, "Leonhard Euler: The Swiss Years," Methodology and Science, vol. 16-2, 1983, pp. 69–89; Ronald Calinger, "Leonhard Euler: The First St. Petersburg Years (1727–1741)," Historia Mathematica, May 1996, pp. 121–66.

2. 

For some discussion on this aspect of Bernoulli's personality, see my book When Least is Best (Princeton University Press, 2004), pp. 211, 244–45.

3. 

One historian (Calinger, 1983 — see Note 1) says Euler "proposed the first correct solution" to this question, and that "Euler rejected both existing opinions that the stone would either make a dead stop at the center [of the earth] or travel beyond it. Instead, he argued that it would rebound at the center and return by the same path to the surface." This is puzzling since that is not the correct answer! A reference is given to a discussion of how this problem perplexed even the great Newton, but that discussion reveals that the "puzzle" was essentially over the nature of the gravitational force inside the earth. If R is the radius of the earth, and r is the distance of the stone from the center of the earth, then of course gravity famously varies as the inverse square of r if r>R. By 1685 Newton knew that the variation of gravity with r (for a uniformly dense spherical earth) is directly as r if r<R, and from that it is easy to show that the stone will execute what is called simple harmonic motion, as it oscillates sinusoidally from one side of the earth, through the center, all the way to the other side, and then all the way back to its starting point. Then it does the trip all over again, endlessly. It is easy to calculate that for a uniformly dense sphere of radius R=4,000 miles and a surface acceleration of gravity of 32 ft/second2 (that is, the earth) one complete round trip takes 85 minutes, and that as the stone transits the earth's center it is moving at 26,000 ft/second towards the other side of the planet (there is no "center rebound"). For a complete, modern discussion on this issue, see Andrew J. Simoson, "Falling Down a Hole Through the Earth," Mathematics Magazine, June 2004, pp. 171–89.

4. 

See my book An Imaginary Tale (Princeton University Press, 1998), pp. 176–178, and Philip S. Davis, "Leonhard Euler's Integral: A Historical Profile of the Gamma Function," American Mathematical Monthly, December 1959, pp. 849–69.

5. 

See An Imaginary Tale, pp. 150–152.

6. 

Both the isoperimetric and brachistochrone problems are discussed at length, and solved in my When Least is Best (Note 2); in particular, on pp. 200–78 there is an introduction to the calculus of variations.

7. 

David Eugene Smith, "Voltaire and Mathematics," American Mathematical Monthly, August-September 1921, pp. 303–5. An example of the emptiness of Voltaire's mathematics in Éléments is found in his "explanation" of Snel's law of the refraction of light — he tells his readers that there is a relationship between a light ray's angle of incidence and angle of refraction at the boundary of two distinct regions (e.g., air and glass, or air and water) and that this relationship involves some thing called a sine, but he fails to tell his readers what a sine is, because that's too technical! Voltaire didn't actually think much of his readers, in fact — in a letter to a friend before he began to write Éléments, he said his goal was "to reduce this giant [Newton's Principia] to the measure of the dwarfs who are my contemporaries."

8. 

My primary sources of historical information on the Berlin phase of Euler's life include Ronald S. Calinger, "Frederick the Great and the Berlin Academy of Sciences," Annals of Science 24, 1968, pp. 239–49; Mary Terrall, "The Culture of Science in Frederick the Great's Berlin," History of Science, December 1990, pp. 333–64; Florian Cajori, "Frederick the Great on Mathematics and Mathematicians," American Mathematical Monthly, March 1927, pp. 122–30; and Ronald S. Calinger, "The Newtonian–Wolffian Controversy (1740–1759)," Journal of the History of Ideas, July-September 1969, pp. 319–30.

9. 

C. B. Boyer, "The Foremost Textbook of Modern Times," American Mathematical Monthly, April 1951, pp. 223–226.

10. 

See When Least Is Best, pp. 133–34.

11. 

See Calinger, 1969 (Note 8).

12. 

The relationship between Euler and D'Alembert was a complicated, professional one, far different from the personal friendship Euler enjoyed with, for example, Daniel Bernoulli. You can find more on this matter in the paper by Varadaraja V. Raman, "The D'Alembert–Euler Rivalry," Mathematical Intelligencer 7, no. 1, 1985, pp. 35–41. This paper, while quite interesting, should be read with some care because the author occasionally overstates his criticism of Euler. For example, he takes Euler to task for not citing (in one of his works) Johann Bernoulli's publication of a derivation of ζ(2). Well, why should Euler have have cited Bernoulli? — it was Euler, not Bernoulli, who first calculated ζ(2).

13. 

Just the thought of an eye operation in the 1770s is probably enough to make most readers squirm. Euler's operation was a procedure called couching, which can be traced back to at least 2000 B.C. With the patient's head clamped in the hands of a strong assistant, the doctor would use a sharp needle to perforate (!) the eyeball and push the cataractous lens aside to let light once again reach the retina (today, the lens is totally removed and replaced with an artificial one). The risk was that lens proteins would be dispersed in the eye, which would cause a severe intraocular inflammation that could result in blindness. That's just what happened to Euler.

14. 

Ronald Calinger, "Euler's 'Letters to a Princess of Germany' as an Expression of his Mature Scientific Outlook," Archive for History of Exact Sciences 15, no. 3, 1976, pp. 211–33.

15. 

F. Cajori, A History of Mathematics (2nd edition), Macmillan, 1919, p. 233.

16. 

See Dirk J. Struik, "A Story Concerning Euler and Diderot," Isis, April 1940, pp.431–32; Lester Gilbert Krakeur and Raymond Leslie Krueger, "The Mathematical Writings of Diderot," Isis, June 1941, pp. 219–32, and B. H. Brown, "The Euler–Diderot Anecdote," American Mathematical Monthly, May 1942, pp. 302–3.

17. 

You can find a photograph of Euler's tomb in Frank den Hollander, "Euler's Tomb," Mathematical Intelligencer, Winter 1990, p. 49. Hollander says the tomb is in Leningrad, Russia, which was true in 1990 — St. Petersburg had its name changed to Leningrad in 1924 (since 1914 it had been known as Petrograd). But in 1991 the original name was restored. Euler was buried in St. Petersburg, and that is where he is today. Another photograph of his tomb, as well as one of his house at #15 Leytenant Schmidt Embankment, along the Neva River, is on p. 18 of the March 2005 issue of Focus, the newsletter of the Mathematical Association of America.

18. 

For much more on the mathematics of Euler, see the really outstanding book by William Dunham, Euler: The Master of Us All (Mathematical Association of America, 1999).



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