An Euler Bookshelf: No Ordinary Genius
Siobhan Roberts on books about the brilliant 18th-century mathematician Leonhard Euler.
A few years ago, a mathematician and a neuroscientist led a study investigating “the experience of mathematical beauty and its neural correlates.” The methodology rolled 14 mathematicians into a functional magnetic resonance imaging machine and asked them to view and rate a collection of 60 mathematical formulas that they had previously assessed as beautiful, neutral or ugly. (Detractors of this sort of study call it “neurotrash,” but no matter.) While viewing the more aesthetically pleasing specimens, the mathematicians’ fMRI results showed activity in the “emotional brain,” specifically field A1 of the medial orbito-frontal cortex—the same area stimulated by moral, musical and visual beauty. Sometimes the mathematicians exited the machine weeping.
The equation that consistently rated the most beautiful was a famously compact specimen devised in the 18th century by the Swiss mathematician Leonhard Euler : e iπ + 1 = 0.
Euler’s equation links—via three basic arithmetic operations, each deployed only once—five fundamental mathematical constants: 0, 1, i (the square root of -1, aka the “unit imaginary number”), π and e (“Euler’s number”—2.71828 . . . —which is linked to exponential growth). It is sometimes called Euler’s identity, or Euler’s formula, but by whatever name it is currently having something of a moment.
Two new books pull apart the equation—deconstructing it technically and historically—and celebrate its niftiness: “A Most Elegant Equation” (Basic, 221 pages, $27) by David Stipp and “Euler’s Pioneering Equation” (Oxford, 162 pages, $19.95) by Robin Wilson. Mr. Stipp’s roving account is propelled by his folksy sense of humor, and, as the author himself admits at one point, by “giddy metaphorical overreach.” Mr. Wilson’s account is more no-nonsense, proceeds on a shorter mathematical tether and has a quieter epigrammatic levity.
PHOTO: PETER AND MARIA HOEY
Both books, by way of introduction, mention the neuroscience study, and both lean on the Stanford mathematician Keith Devlin for this pronouncement: “Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.” Both also quote the physicist Richard Feynman, who at age 14 wrote in a notebook that Euler’s equation was “the most remarkable formula in math.”
Convinced yet? If you aren’t by all this anecdotal testimony about the formula’s pure beauty, then consider its applied incarnations. Euler’s equation is a special case of a more general equation, and this “conceptual parent” (as Mr. Stipp calls it) is useful for modeling phenomena such as growth, shrinkage, rotations and oscillations—for instance, the oscillations of alternating current, the electrical current that periodically reverses direction (versus direct current, flowing in one direction). As Mr. Stipp writes, with one of his signature metaphors: “Today, Euler’s formula is a tool as basic to electrical engineers and physicists as the spatula is to short-order cooks. It’s arguable that the formula’s ability to simplify the design and analysis of circuits contributed to the accelerating pace of electrical innovation during the twentieth century.” This pure-applied transformation—an evolution of sorts whereby a result derived through abstraction comes to bear on the real world—is what the physicist Eugene Wigner termed “the unreasonable effectiveness of mathematics in the natural sciences.”
Leonhard Euler (pronounced “oiler”) was born in Basel, Switzerland, in 1707. By age 17, he had completed a master’s degree in philosophy at the University of Basel, comparing the ideas of Descartes and Newton. Then, benefiting from the private tutelage of the world-class mathematician Johann Bernoulli, he switched to mathematics, where he left such a mark that only Archimedes, Newton and Gauss are considered his equal. According to the slightly hyperbolic account of the mathematical historian Eric Temple Bell, Euler could write a momentous math paper in the 30 minutes between the first and second calls to dinner, possessing as he did “all but supernatural insight into apparently unrelated formulas that reveal hidden trails leading from one territory to another.”
Euler’s big best seller was “Letters of Euler on Different Subjects in Natural Philosophy Addressed to a Princess of Germany” (1768-74), in which he explained, among other things, why it is cold on mountaintops in the tropics and why the moon looms larger at the horizon. This book made Euler, as Mr. Stipp puts it, “something of a pioneer in supporting the education of women on technical topics.” It was published in French, English, German, Russian, Dutch, Swedish, Italian, Spanish and Danish and got rave reviews from Kant, Goethe and Schopenhauer.
Failing to get the professorship he wanted at his alma mater in Basel, Euler began his career in 1727 at the St. Petersburg Academy of Sciences. In 1741 he published his solution to a famous open problem about whether it was possible to cross seven bridges in the Prussian city of Königsberg with a connected route, traversing each bridge only once, and returning to the same spot. Euler proved that there was no such route (and also offered a general solution that held for any number of bridges and landmasses) and in doing so inaugurated the mathematical field of graph theory, along with the subfield of network theory, which is relevant to social, transportation and internet networks, among others.
Euler left St. Petersburg the same year amid political turmoil. He took up a position at King Frederick II’s Berlin Academy of Sciences, where he was underappreciated—the king’s view was that mathematics “dries up the mind.” Nevertheless, while in Berlin, Euler derived his famous equation and all in all prepared nearly 400 works. But by 1766, he’d had enough of Frederick’s insults and mockery. (Euler was by then half-blind, and in a letter to Voltaire the king called him “our great Cyclops.”) He returned to the St. Petersburg Academy at the invitation of the Russian Empress Catherine the Great, among his greatest admirers. He became a celebrity and continued his prolific ways, despite personal misfortune. Only five of his 13 children lived to become adults, and only three survived him; his house burned down when he was 64, and his wife died two years later. When he became completely blind, he gratefully accepted his fate as offering him “one fewer distraction” (though into old age he still recited from memory 9,500 or so lines from the “Aeneid”). With the help of assistants, he produced more than half his life’s work in his last 17 years—in total some 900 books and papers, 500 of them published posthumously.
Euler produced more than any other mathematician in history. Mr. Stipp points out that, in terms of mathematical results, the 18th century yielded a “rich harvest” of “low hanging fruit” and thereafter “the bar was raised on rigor.” But all things considered, Euler was no ordinary genius. He possessed a vast intellectual panorama and is considered an unrivaled mathematical expositor. As the French mathematician Pierre-Simon Laplace famously said: “Read Euler, read Euler, he is the master of us all.” And he’d keep us all reading for life.
To that end, for even more Eulerian reading, there is also a new paperback edition of “Dr. Euler’s Fabulous Formula” (Princeton, 380 pages, $22.95) by Paul J. Nahin, first published in 2006, as well as the first full biography, Ronald S. Calinger’s “Leonhard Euler: Mathematical Genius in the Enlightenment” (Princeton, 669 pages, $55), from 2015. While Mr. Stipp’s book forewarns that, “if you’re a long-time math lover, you’ll probably find most of [the book] too elementary,” Mr. Nahin acknowledges by contrast that his telling involves “more advanced mathematical arguments . . . on issues that I think could fairly be called the ‘sexy part’ of complex numbers . . . [that will] leave more than a few otherwise educated readers out in the cold.” Mr. Wilson’s 162-page version falls somewhere in between. Mr. Calinger’s biographical treatment is exhaustive, with more than 100 pages given over to end matter, including a dense register of principal names that runs more than 30 pages.
In offering consolation for his heady account, Mr. Nahin borrows from Winston Churchill, who wrote in his autobiography: “I had a feeling once about Mathematics, that I saw it all—Depth beyond depth was revealed to me—the Byss and the Abyss. I saw, as one might see the transit of Venus, . . . a quantity passing through infinity and changing its sign from plus to minus. I saw exactly how it happened and why the tergiversation was inevitable: and how the one step involved all the others. It was like politics. But it was after dinner and I let it go!”
—Ms. Roberts is the author of “Genius at Play: The Curious Mind of John Horton Conway. ”
