CHAPTER 5
Portrait of the Master
The fact that Euler could effectively grant citizenship to an entire class of troubling numbers thought to resemble ghostly salamanders gives an indication of his great sway in math. He even turned the imaginaries into fetching little playthings. What made him so influential? (True, he was a genius, but there have been many math geniuses in history who weren’t as important as he was—and still is—in mathematics.) Let’s briefly veer off our excursion’s main path to get a better look at the one who blazed it.
My favorite description of Euler was offered by Dieudonné Thiébault, a French linguist who met him in mid-life: “A child on his knees, a cat on his back, that’s how he wrote his immortal works.” Euler loved accompanying his children to marionette shows, where, it is said, he laughed robustly along with the kids at the puppets’ antics. He liked joking around with his children and grandchildren and teaching them about math and science. He also liked to take them to the zoo, where he gravitated to the bears—he loved watching the cubs play. He enjoyed having visitors drop by to talk about anything under the sun and was adroit at shifting from deep technical discussions to casual conversation as called for by the occasion. I suspect that cats generally purred in his presence.
Born in 1707 to a Swiss pastor and his wife, Euler seemed likely to follow his father into theology until he got hooked on math as a teenager at the University of Basel. It was a mediocre school overall, but luckily one of the world’s greatest mathematician at the time, Johann Bernoulli, taught there. After Euler’s talent came to Bernoulli’s attention, he took the youth under his mighty wing, giving him special tutorials every Saturday afternoon. Bernoulli assigned Euler increasingly difficult problems to work out by himself, reserving the Saturday sessions to help his student with the ones he had trouble with. But after a while, as mathematician William Dunham has noted, “it was Bernoulli who more and more seemed to become the pupil.” A few years after the senior mathematician began mentoring Euler, he was addressing his young protégé in letters with a Latin phrase translated as “The Most Famous and Learned Man of Mathematics.” It should be noted that Bernoulli wasn’t a humble man, nor was he given to jests.
Euler gained public recognition at age 19 when he first entered the international contest held annually by the Paris Academy of Sciences. That year the challenge was to determine the best placement of masts on ships to capture the wind’s pushing power. Euler’s submission tied for second place—not bad for a teenager competing against the top mathematicians and scientists of Europe. (And the Swiss youngster had never even seen a large sailing ship.)
Leonhard Euler
As broad as he was deep, Euler made seminal advances in just about every area of math—number theory, calculus, geometry, probability, you name it. He “seems to have carried in his head the whole of the mathematics of his day,” marveled twentieth-century French mathematician André Weil. He also put new areas of math on the map.
The breadth partly reflected his prodigious memory. Even in old age Euler could readily recite the 9,500-plus lines of Virgil’s Aeneid from memory. He knew five languages: Latin, Russian, German, French, and English. It’s said that he could reel off the first six positive integer powers of any number between 1 and 100. (In case you’d like to master that trick yourself, here’s a start on the 600 numbers you’d need to know: 991=99; 992=9,801; 993=970,299; 994=96,059,601; 995=9,509,900,499; and 996=941,480,149,401.)
Euler apparently could write a groundbreaking mathematics paper in the half hour between the first and second calls to dinner, according to historian Eric Temple Bell. He could do that, Bell added, because he possessed “all but supernatural insight into apparently unrelated formulas that reveal hidden trails leading from one territory to another” in math.
To be fair, it was somewhat easier for Euler to be prolific than it was for the later practitioners of mathematics—they were compelled to be more painstaking than he was as the bar was raised on rigor, and math’s branches bore less low-hanging fruit after the eighteenth century’s rich harvest. Still, when recently going over Euler’s derivations of several landmark equations, including his famous formula, I found myself thinking that, like the imaginary numbers, he must have arrived here from a different dimension—as Bell said, he had a truly uncanny knack for sensing the presence of hidden trails. (Or, as twentieth-century mathematician Mark Kac once put it, “An ordinary genius is a fellow that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is different with the magicians… the working of their minds is for all intents and purposes incomprehensible.”)
This book can present only a superficial glance at Euler’s overall achievements. But let me offer a sports analogy that occurred to me as I perused a tiny fraction of his vast output.
The math game in Euler’s day resembled track and field in the early twentieth century, a freewheeling era for the sport that was memorably portrayed in the Oscar-winning movie Chariots of Fire. A champion runner at the time might have strolled up to the starting line of a race puffing a cigar, nonchalantly set it down by the track, taken off like a shot at the gun to handily win, then picked up his still-smoldering stogie and sauntered off to the locker room. It was obviously easier to set records in those days, just as it was to make math advances in the eighteenth century.
But if Euler had been an early track star, he wouldn’t have just won a race now and then. He’d have regularly trotted into track meets with a child in his arms and a cat reposing on his back, and, without putting them down, proceeded to win the discus, the hammer throw, the shot put, the javelin, the long jump, the high jump, the triple jump, the steeplechase, the 100-meter dash, the 200, the 400, the 800, the 1,500, and the mile. The kid and cat would have been napping by then, but because he always did his thing with incredible gusto he’d have gone on to set a world record in the 5,000-meter race running backwards in bedroom slippers with a blindfold, and then finished up by topping his own previous world record in the pole vault—somehow without waking the child or the cat.
Euler is said to be the greatest math explainer ever. He authored a classic guide to algebra that has been called the second most popular math book in history after Euclid’s Elements. (The latter, by the way, is thought to be second only to the Bible as the most frequently printed book.) Euler also wrote widely-used textbooks on calculus and the laws of motion, and a primer on science, philosophy, music theory, and other topics that became an eighteenth-century bestseller: Letters of Euler on Different Subjects in Natural Philosophy Addressed to a German Princess. Dedicated to the Princess of Anhalt Dessau, the niece of Prussia’s King Frederick II, the book made Euler something of a pioneer in supporting the education of women on technical topics. (It consisted of letters because Euler had acted as the princess’s remote tutor when the Prussian royal court had fled Berlin during the Seven Years’ War.) The Letters explained such things as why it is cold on top of mountains in the tropics, why the moon looks larger when it’s near the horizon, and why the sky is blue. Euler wrote it in French but it was soon translated into all the other major languages of Europe and widely used to teach basic science. The book got rave reviews from, among others, Kant, Goethe, and Schopenhauer.
Euler suffered an infection that blinded his right eye when he was in his 20s, and later a failed cataract operation in his left eye rendered him unable to make out people’s faces and even nearby objects. This great loss didn’t slow him down a bit. In fact, he cheerfully talked of losing his eyesight as “one fewer distraction.” With the help of assistants, he produced more than half of his entire life’s work during the last 17 years of his life, after he had lost most of his vision. Ever resourceful, he found a way to get exercise by repeatedly walking around a large round table in his study while running his hand along the edge.
He overcame a number of setbacks and personal tragedies besides going blind. Only five of the 13 children he had with his wife Katharina survived to adulthood, and only three, all sons, outlived him. His house burned down when he was 64, destroying his library and some of his unpublished work. Euler, who was by then nearly blind, was stranded on the second floor during the fire until his Swiss handyman, one Peter Grimm, heroically climbed a ladder and brought him down over his shoulder. Katharina, his wife of 40 years, died when he was 66; three years later he married Katharina’s widowed half-sister so that he wouldn’t be totally dependent on his children.
After getting established at the St. Petersburg Academy of Sciences in Russia early in his career, Euler was basically forced out as a wave of anti-foreigner sentiment swept through Russia. It hadn’t helped that the main interest of the academy’s effective director, Johann Schumacher, “lay in the suppression of talent wherever it might rear its inconvenient head,” as historian Clifford Truesdell sardonically put it.
Capitalizing on the situation, Prussia’s King Frederick hired Euler to help bolster the Berlin Academy of Sciences. But Frederick never regarded Euler, a quiet, pious family man, as the kind of drawing-room wit he wanted for the academy. The king, despite his pretensions to intellectual breadth, apparently suffered from incurable math phobia. Mathematics “dries up the mind,” he once wrote in a letter. Frederick was deeply annoyed whenever Euler attended the theater in his presence—the mathematician was known for getting noticeably distracted from the play as he jotted notes about the hall’s optics, sound effects, and other features that could be mathematically modeled.
As the years passed, Euler became the butt of bumpkin jokes made by the academy’s favored ornaments, such as Voltaire, whom Frederick had lured from France by offering to pay him some 20 times as much as he’d proposed paying Euler as a starting wage. Since the mathematician’s right eye was blind, Frederick smirkingly referred to him in a letter to Voltaire as “our great Cyclops.” In the same letter the king joked that he would be willing to trade Euler to Voltaire’s consort, Émilie du Châtelet, in return for Voltaire. In a letter to his brother, Frederick observed that people like Euler are “useful… but otherwise are anything but brilliant. They are used as are the Dorian [sic] columns in architecture. They belong to the subfloor, as support… .”
Euler actually did serve as a crucial part of the Berlin academy’s support system for a quarter century. He supervised its observatory and botanical gardens, oversaw its finances, managed the publication of its calendars and maps (the academy’s main source of income), advised the government on state lotteries, insurance, pensions, and artillery, and even oversaw work on hydraulic pipes at Frederick’s summer residence. But finally he got fed up with being treated with contempt and petitioned Frederick for permission to resign from the Berlin academy. At first the king refused to even acknowledge the request, but Euler stubbornly persisted and was set free at last when the king sent him a curt, cold note: “With reference to your letter of 30 April, I permit you to quit in order to go to Russia.” Frederick apparently never realized that he had driven away one of history’s greatest minds. After later filling Euler’s position at the Berlin academy, he commented in a letter that “the one-eyed monster has been replaced by another who has both eyes.” (The two-eyed monster was the great French-Italian mathematician Joseph-Louis Lagrange.)
Euler returned in 1766 at age 59 to Russia’s St. Petersburg Academy, where the anti-foreigner movement had faded during the reign of Catherine II, and spent the rest of his incredibly productive life there.
EULER IS REGARDED as having had only three equals in the history of mathematics: Archimedes, Isaac Newton, and Carl Friedrich Gauss. I find it interesting to compare their personal characteristics with his. Such comparisons usually say little to nothing about great creators’ achievements. But this is an exception. In my view, Euler’s tranquil temperament, fairness, and generosity were integral to his greatness as a mathematician and scientist—he was never inclined to waste time and energy engaging in petty one-upmanship (like his mentor, Johann Bernoulli, who was known for getting into the eighteenth-century version of flame wars with his older brother, mathematician Jakob Bernoulli, and even with his own son, Daniel, over technical disputes), brooding about challenges to his authority (like Newton), or refusing to publish important findings because of the fear that they might be disputed (like Gauss).
If the stories told about Archimedes are true, he was a colorful character. It is said that he once jumped from his bath and ran naked through the streets shouting “Eureka!” when he realized how to measure volumes of irregular solids by submerging them in water. According to Plutarch, when Roman soldiers overran Syracuse, where Archimedes lived, the mathematician told one of them he had to finish some calculations before he could meet the conquering Roman general. The enraged soldier drew his sword and killed the ancient world’s most brilliant man on the spot.
Newton was a shy, prickly loner who held grudges. He was given to outbreaks of near-psychotic rage when challenged or contradicted. When compiling a list of his sins at age 19, he noted that one of them was “threatening my [step]father and mother Smith to burn them and the house over them.” William Whiston, who assisted Newton and then succeeded him as Lucasian Professor of Mathematics at the University of Cambridge, reported that “Newton was of the most fearful, cautious and suspicious temper that I ever knew.”
He was notably tyrannical as president of Britain’s Royal Society—contradicting his opinions or directives simply wasn’t allowed (which is an odd position to take for a scientist).
Newton conceived an especially passionate hatred for the scientist Robert Hooke, who challenged Newton’s ideas on the nature of light. Some years later an exchange of letters between the two helped inspire Newton’s breakthrough ideas about gravity and planetary motion. After Hooke later suggested that he’d had a role in bringing forth the famous concepts, Newton furiously deleted every reference to Hooke in his forthcoming magnum opus, Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy). As historian Robert A. Hatch put it, Newton’s “hatred for Hooke was consumptive.”
Newton was even more consumed by his famous dispute with Leibniz about which one of them deserved credit for inventing calculus. Newton first developed the basic ideas of calculus, but Leibniz came up with them independently and was the first to publish them. While Newton pretended to be above the fray, he secretly oversaw attacks by his English allies on Leibniz and, behind the scenes, he dominated a supposedly impartial investigation by a committee of the Royal Society that was convened to decide on the priority dispute. The committee didn’t bother to give Leibniz a hearing before pronouncing in favor of Newton, who wrote its report himself. To top it off, he then tried to ensure that the report would be widely noticed by writing an anonymous review of it for the Philosophical Transactions of the Royal Society. Because of the dispute, English mathematicians spent the next century chauvinistically ignoring math advances on the Continent, where Leibniz and his main allies were, thereby losing their innovative edge in mathematics.
Gauss was similarly forbidding. He disliked teaching, had few friends, and alienated one of his sons, Eugene, by trying to control his life. Gifted with languages, Eugene had wanted to study philology as a youth, a choice opposed by his father. After they quarreled about a dinner party that Eugene had held with his friends and had asked his father to pay for, the 19-year-old son abruptly left for America and never returned. In the United States, he learned the Sioux language and wound up working for a fur company in the Midwest.
Gauss withheld many of his results because he felt they weren’t perfect enough to release. But after other mathematicians independently discovered and published the same findings, he wasn’t averse to letting people know that he’d privately made them first. One such case concerned the Hungarian mathematician Wolfgang Farkas Bolyai and his son, János. The elder Bolyai was a friend of Gauss, and in 1816 he asked the famous German mathematician if he would let János, then 14, live with him and become his student. Bolyai couldn’t afford to send his gifted son to a prestigious university, and Gauss could have been a great help to him. But Gauss refused.
Nevertheless, János went on to help pioneer “non-Euclidean” geometry while he was in his early 20s. A century later this revolutionary development in mathematics would inform Einstein’s General Theory of Relativity, which suggests that space-time is curved. While working out the new ideas, the hopeful young man excitedly wrote his father, “I myself have made such wonderful discoveries that I am myself lost in astonishment.” A paper he authored on the findings was published as an appendix to a math book by his father that appeared in the early 1830s. Clearly proud of his son’s work, Wolfgang sent a copy to Gauss.
Gauss replied that to praise János’s work “would be to praise myself”—meaning that he had already discovered everything that János had worked out. That was a severe blow to János. Soon after, his mental and physical health deteriorated. While continuing to work fitfully on mathematics, he never lived up to his early promise and eventually gave up trying to win recognition in math. He died in obscurity, and his innovative work was ignored until after Gauss’s death in 1855; the posthumous publication of Gauss’s private notes and letters on non-Euclidean geometry made the topic seem worth studying to mathematicians, who then recognized Bolyai’s contributions.
Gauss wasn’t always so dismissive of young talent. For several years during the early 1800s, he sent encouraging letters to France’s Marie-Sophie Germain, one of the first prominent female mathematicians. As a young teen she’d developed a passion for math but was discouraged from studying it by her parents—that was considered unladylike. They’d even tried to discourage her from surreptitiously working math problems during cold nights by denying her warm clothes and a bedroom fire. But they’d finally relented after finding her asleep at her desk one morning next to a frozen inkwell and a slate full of calculations. Although she was barred from attending France’s leading school for aspiring mathematicians, she went on to do pioneering work in number theory and other areas.*
Gauss never met her, but in one of his letters to her, he sympathetically (and accurately) wrote that when “a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarizing herself with [number theory’s] knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius.”
It seems that Gauss himself didn’t show much in the way of noble courage. In 1829, for example, he confessed in a letter to a friend that he’d long withheld his results on non-Euclidean geometry because he’d feared getting attacked for espousing radical new ideas. And whenever he did disclose his indisputably brilliant discoveries, he often put them in a form that made it difficult for others to follow his thought processes. “He reworked his mathematical proofs to such an extent that the path whereby he had obtained his results was all but obliterated,” remarked British mathematician Ian Stewart. Gauss’s contemporaries less diplomatically described his writing style as “thin gruel.”
In contrast, Euler was about the nicest, most forthcoming person you could hope to meet in the annals of genius. Cats and children weren’t the only ones who gravitated to him—the record shows that almost everybody who met him found him charming. The main exceptions were King Frederick and the people who thought his cyclops joke was funny. Some years after Euler left Prussia, however, he and Frederick had a friendly exchange of letters when the king got interested in a tract by the mathematician on how to set up and calculate pensions—Euler seemed incapable of holding grudges. This isn’t to say that he was always mild-mannered. He held strong opinions and wasn’t afraid to contradict peers when he thought they were wrong. But his disputes with them were generally cordial arguments rather than bitter fights.
Not surprisingly, the great explainer had a passion for teaching and, according to one story, taught elementary algebra to his tailor by employing him as a scribe when dictating his famous algebra text. Euler’s son, Johann, later remarked that the man was capable of solving complicated algebra problems after the experience.
One day in 1763, a young Swiss man, Christoph Jezler, arrived at Euler’s house in Berlin. Jezler explained that he wished to copy Euler’s yet-to-be-published textbook on integral calculus, page by page. It turned out that as a youngster he’d aspired to be a mathematician, but family pressures had forced him to become a furrier, a trade he’d recently put aside after his father had died. Euler not only took Jezler in but also offered to help him understand parts of the text that he had trouble with. After some months as Euler’s guest, during which Jezler furiously copied away while his mother sent cherries, apple wedges, and plums to the Euler household (Euler especially loved the plums), the young man returned home and later became a professor of physics and mathematics.
Euler carefully cited other mathematicians’ contributions in papers he wrote related to their work. Sometimes he gave others more credit than they deserved. Once he generously put aside his rapidly advancing work on hydrodynamics, a branch of physics dealing with fluids, so that he wouldn’t risk upstaging a friend—mathematician Daniel Bernoulli, the son of Euler’s former mentor—who he knew was laboring on a major book about the topic. (Euler also probably wanted to steer clear of a messy dispute between the younger and older Bernoulli concerning which one of them first came up with key ideas on the topic. Daniel’s imbroglio with his irascible father, who took credit for much of Daniel’s work, made the son thoroughly miserable, and at one point he commented that he wished he’d become a cobbler instead of a mathematician.)
Another time, Euler translated a book on ballistics by England’s Benjamin Robins, who held eccentric views on math and physics that had earlier prompted him to mount a ludicrous public attack on a related work by Euler. Euler made the translation much better than the original, adding commentaries and corrections that turned the book into a major work as well as making it much longer than it originally was. As a Euler admirer later noted, the only revenge the great mathematician took on Robins for attacking him was to make the Englishman’s book famous.
Euler’s even-tempered generosity even extended to those with whom he differed on religion. In 1773, the French philosopher Denis Diderot visited St. Petersburg for several months at the invitation of Catherine II. Diderot was best known as an editor of the Encyclopédie, a pioneering, 28-volume encyclopedia that covered everything from mathematics to music to medicine. But it was most famous for advocating Enlightenment ideals, including reason-based challenges to religious dogma.
Euler was a devout Protestant who held daily prayers in his home, which seemingly would have made for conflict between him and Diderot. Indeed, according to a story widely circulated in Europe after Euler’s death, the mathematician embarrassed the visiting Frenchman in a public debate about God’s existence by exclaiming to Diderot, “Sir, a+bn/n = x, therefore God exists—Respond!” Diderot, who supposedly knew little about math, stood astonished by this absurdity as onlookers burst into laughter. Deeply mortified, he soon after returned to France.
There’s no evidence at all that this story is true, according to historian Ronald Calinger. It appears to have been invented by Prussia’s King Frederick, or a member of his court, to belittle Diderot, who had incensed the king by publicly criticizing some of his military policies. In fact, soon after Diderot had arrived in St. Petersburg, Euler had arranged for him to be inducted into the Russian Academy of Sciences as a “corresponding member” and had subsequently presided over the induction ceremony. Diderot had responded by sending a letter to the academy declaring that “everything that I created I would gladly trade for a page of the writings of Monsieur Euler.” Contrary to the scurrilous story, Diderot was well-versed in mathematics and apparently regarded Euler as one of the leading lights of the age. And for his part, Euler was much too gracious to publicly attack the visiting scholar.
The most-repeated quote on Euler is attributed to French mathematician Pierre-Simon Laplace, who advised fellow mathematicians to “read Euler, read Euler. He is the master of us all.” Gauss commented that “studying the works of Euler remains the best school in the various fields of mathematics and cannot be substituted for anything else.” Twentieth-century Swiss mathematician and philosopher Andreas Speiser had this to say: “If one considers the intellectual panorama open to Euler, and the continual success of his work, he must have been the happiest of all mortals, because nobody has ever experienced anything like that.” (Emphasis added to highlight that this extraordinary statement is probably not an exaggeration.) To put that another way, Euler wasn’t at the cutting edge—he was the cutting edge, one unlike any we’ve seen before or since thanks to the fortuitous combination of his unsurpassed genius and the immense opportunities for progress opened up by the mathematicians and scientists who preceded him. Given that, it is only to be expected that Euler’s formula is deep and strange. In fact, it’s a lot like Euler’s mind.
* Given all the doors closed to women through the ages, it is striking (and hugely instructive) that accomplished female mathematicians have not been terribly rare. A short list would include Hypatia of Alexandria, Maria Gaetana Agnesi, Sofia Kovalevskaya, Alice Boole Stott, Julia Robinson, Emmy Noether, and Mary Lucy Cartwright.