“So a few years ago…” The survey, which I first saw in 2014, was conducted in 1988 by David Wells and published in 1990 in The Mathematical Intelligencer.
“It’s just that, inexplicably…” The calc books in question are Tom Apostol’s Calculus, volumes I and II, published in 1969. They’re still considered among the most rigorous, comprehensive calculus texts out there. Ironically, they’re highly regarded partly because they play up math history. Volume I even includes a mini-hagiography of Euler noting that he “discovered one beautiful formula after another.” Why don’t they highlight what is widely regarded as the most beautiful one of all? I’ll probably never know—Apostol, a former Caltech math professor, died in early 2016.
“Novelist Nicholson Baker memorably described…” Baker (2013).
“And I knocked it down.” Bouwsma (1965), v.
“Unlike the physics or chemistry or engineering of today…” Nahin (2006), xx.
“Leonhard Euler seemed as curious…” Calinger (2015).
“Euler was the Enlightenment’s greatest mathematician…” As Truesdell (2007) notes, Euler has long been known as one of history’s greatest mathematicians, but only in recent decades have historians recognized him as his century’s leading physicist too.
“Science historian Clifford Truesdell has estimated…” Quoted in the Euler Archive, “Europe in the 18th Century.” http://eulerarchive.maa.org.
“As mathematician William Dunham has noted…” Dunham (1999), 176.
“In 1752, for instance” Sandifer (2007).
“His ideas on how to construct achromatic lenses…” Calinger (2015), 383.
“He even proposed a design for a logic machine…” Sandifer (2004).
“… his writings clearly influenced Kant’s metaphysics.” Calinger (2015), 469.
“His work on the mathematics of population growth…” Klyve (2014).
“He had a hand in solving…” Calinger (2015), 389.
“Today, the mathematics he pioneered…” Stewart (2009), 104; and Nahin (2006).
“… perhaps only Voltaire…” Calinger (2015), 531.
“Mathematics textbooks call it Euler’s formula.” Euler didn’t publish the equation in the form shown here, which apparently first appeared in the nineteenth century. But in 1728, he wrote an equivalent formula when demonstrating a way to calculate the areas of circles. In 1748, he proved a general equation from which his famous formula follows as a specific case in the Introductio in analysin infinitorum (Introduction to the Analysis of the Infinite)—this proof is detailed in Appendix 1. So Euler is recognized as the formula’s primary author. However, some other mathematicians, including Johann Bernoulli, Euler’s mentor, came close to discovering it. And in 1714, England’s Roger Cotes developed an equation from which it could have been readily extracted. Cotes apparently didn’t see the full implications of what he’d done, however, and he died two years later, at age 33.
14 “Benjamin Peirce, considered America’s first world-class mathematician…” Quoted in Archibald (1925).
“Like a Shakespearean sonnet…” Devlin (2002).
“Richard Feynman was briefer…” Nahin (2006).
“And in 2014…” Zeki, Romaya, Benincasa, and Atiyah (2014).
“… the symbolic comprehension of the infinite…” Quoted in Fleron (2015), 21.
“Rigorously defining e requires…” Euler and his contemporaries knew that affixing a numerical value to e involves a calculation that, in principle, is infinitely long. Specifically, it entails adding (or, alternatively, multiplying together) an infinite number of similar-looking fractions. Euler was a dazzling master at manipulating infinite sums and products to make important math advances. But in his day, the conceptual machinery wasn’t available to do such manipulations without some fast and loose moves—ones that could lead to absurd results in certain situations. Even Euler, for all his genius, went astray at times because of this. As explained in Appendix 1, a carefully worded definition of limits, along with some other concepts that were developed after his era, helped eliminate the dubious maneuvering.
“… this is a high-altitude flyover…” Mathematician Arthur Benjamin offers a concise, readable introduction to calculus in The Magic of Math: Solving for x and Figuring Out Why.
“It required two great mathematicians…” While Newton and Leibniz are credited with inventing calculus, others came up with key pieces of the grand synthesis that became calculus, including Pierre de Fermat, Bonaventura Cavalieri, John Wallis, and James Gregory. Later mathematicians, prominently including Euler, helped flesh out and systematize the limited initial formulations of it by Newton and Leibniz. It seems that the history of ideas is almost always more complicated than it appears at first glance.
“The latter has been empirically confirmed…” See, for example, Allain (2009).
“… followers of Pythagoras…” Martinez (2012).
“… it’s an irrational number…” O’Connor and Robertson (2016), “Johann Heinrich Lambert.”
“… by German mathematician Carl Louis Ferdinand von Lindemann…” Ibid., “Carl Louis Ferdinand von Lindemann.”
“… it’s a transcendental number…” Interestingly, Euler’s formula played a key role in von Lindemann’s proof that π is transcendental—the most beautiful equation’s implications were deeper than anyone realized during Euler’s time.
“… by French mathematician Charles Hermite.” O’Connor and Robertson (2016), “Charles Hermite.”
“… when French mathematician Joseph Liouville…” Ibid., “Joseph Liouville.”
“Surely the population…” Wigner (1960).
“… this mysterious 3.14159…” Quoted in Gardner (2014).
“Historians believe the first discoverer…” Roy (1990).
“… after all, 2 − 3 + 4 equals 3…” Mathematically knowledgeable readers will realize that I’ve pulled a fast one here by supposing that infinite sums behave like finite ones when it comes to regrouping terms. They often don’t, of course. But that wasn’t clear during Euler’s era, and it makes sense to omit it in this historical example of confusion about the infinite.
“… never give out in our thought.” Clegg (2003), 30.
“May we not call them the ghosts…” Ibid., 124.
“… unsettling issues about the infinite once again.” Ibid.
“The most boggling implications of his theory…” In the last chapter of his book Journey through Genius: The Great Theorems of Mathematics, mathematician William Dunham offers a nice explanation of how Cantor proved that the irrationals have a larger degree of infinity than the rationals—although it’s a profound result, it’s no harder to understand than a Sudoku puzzle.
“… declared that Cantor’s theory…” Devlin (2013).
“… of Cantor’s theory of the infinite.” Ibid.
“This is why its symbol…” O’Connor and Robertson (2016), “John Wallis.”
“… some 4,000 years ago.” Blatner (1997).
“… a hypothetical polygon with 24,576 sides.” O’Connor and Robertson (2016), “Zu Chongzhi.”
“… the problem was first posed in 1644…” Sandifer (2007), “Euler’s Solution of the Basel Problem—The Longer Story.”
“… amateur British mathematician William Shanks…” O’Connor and Robertson (2016), “William Shanks.”
“One of the more notable records…” Preston (1992).
“… claimed Italian mathematician Gerolamo Cardano.” O’Connor and Robertson (2016), “Girolamo Cardano.”
“In the 1500s…” Nahin (1998).
“… actually a camouflaged real number.” Ibid.
“… between being and non-being.” O’Connor and Robertson (2016), “Quotations by Gottfried Leibniz.”
“Essentially throwing up his hands…” Quoted in Dunham (1999), 87.
“Let’s briefly veer off…” Unless otherwise noted, biographical details in this chapter are taken from Calinger (2015) and Fellmann (2007).
“My favorite description of Euler…” Thiebault’s appealing description of Euler may not be accurate, according to math historian Ronald Calinger, since the French linguist didn’t know Euler when his children were young. And it was likely that Euler’s wife mainly cared for the couple’s children while her husband worked. Still, Thiebault’s depiction accords with other, better established accounts of Euler’s personality and domestic life.
“… as mathematician William Dunham has noted…” Dunham (1999), xx.
“… marveled twentieth-century French mathematician André Weil.” Quoted in Dunham (1999), xv.
“… according to historian Eric Temple Bell.” Bell (1986).
“Bell added, because he possessed…” Ibid., 140.
“Still, when recently going over Euler’s derivations…” William Dunham presents lucid expositions of a selection of Euler’s important theorems in Euler: The Master of Us All. A person who’s completed a basic calculus course could follow most of it if willing to break into a minor sweat now and then.
“An ordinary genius is a fellow…” Quoted in Lemonick (1992).
“… as historian Clifford Truesdell sardonically put it.” Truesdell (2007), 22.
“If the stories told about Archimedes are true…” O’Connor and Robertson (2016), “Archimedes of Syracuse.”
“William Whiston, who assisted Newton…” Quoted in Keynes (1944).
“As historian Robert A. Hatch put it…” Hatch (1998).
“To top it off…” Ibid.
“After they quarreled…” Cajori (1899).
“One such case concerned…” O’Connor and Robertson (2016), “János Bolyai.”
“Gauss never met her…” Quoted in Rehmeyer (2008).
“… because he’d feared getting attacked…” Burris (2009), “Letter from Gauss to Bessel, January 27, 1829.”
“… remarked British mathematician Ian Stewart.” Stewart (1987).
67 “Gauss’s contemporaries less diplomatically…” Gray (2007).
“When author Alex Bellos surveyed…” Bellos (2014), 19.
“In other words, when the three enigmatic numbers…” After I wrote a draft of this book, I ran across a 2014 article in Wired by another fan of Euler’s identity, Lee Simmons, who also used a wormhole metaphor to describe Euler’s formula. So as not to seem derivative, I tried to think of an alternative metaphor. But then I decided to stick with wormhole and to note that this is an example of convergent evolution in meme-space, analogous to the evolving of sonar-like echolocation in bats, whales, certain birds, and shrews.
“Though wonderfully ingenious…” Arthur Benjamin offers a straightforward, calculus-based derivation of Euler’s identity in The Magic of Math. (By the way, don’t miss Benjamin’s “mathemagics” TED talk on YouTube, a crowd-pleaser that’s been viewed over seven million times.)
“… a no-choke mini-primer…” If you want a more comprehensive tutorial on trig, the online Khan Academy (www.khanacademy.org) is a good place to look.
“As he put it, functions are…” O’Connor and Robertson (2016), “History Topic: The Function Concept.”
“a forerunner to defining trig functions…” Ibid., “Hipparchus of Rhodes.”
“De Moivre is credited with…” Ibid., “Abraham de Moivre.”
“One of Euler’s intriguing findings…” Euler’s derivations of the infinite sums for cos θ and sin θ lie outside the conceptual boundaries of this book. If you’d like to see them, William Dunham’s book, Euler: The Master of Us All, covers them on pages 92–3.
“That is, he showed that…” Trig-related formulas involving infinite sums were known before Euler’s time. Impressively, for instance, Indian mathematicians of the fifteenth century worked out a number of them.
“Here’s the infinite sum…” Euler devised an elegant proof that the infinite sum 1 + x + x2/2! + … equals ex. But this math fact was prefigured in the work of Isaac Newton.
“An acquaintance described him…” Branner and Johansen (1999).
“… a fellow surveyor once wrote of him.” Ibid.
“Finally, in 1895…” A few mathematicians, in particular England’s John Wallis, adumbrated Wessel’s geometric interpretation by suggesting how to picture complex numbers geometrically on a two-dimensional plane. But these early beginnings didn’t lead anywhere.
“A few years after Wessel’s big idea…” O’Connor and Robertson (2016), “Jean Robert Argand.”
“… devised this rule in the 1500s…” Ibid., “Rafael Bombelli.”
“Math historians Edward Kasner and…” Quoted in Maor (1994), 153.
“In 1892 he joined…” Steinmetz (1893).
“Delighted by animals that…” Martin (2009).
“In later life he was dubbed…” Nahin (1998), 137.
“Today, Euler’s formula is a tool…” Nahin (2006) contains instructive examples of how Euler’s formula is used in electrical engineering.
142 “a much-cited observation by the great British philosopher and mathematician Bertrand Russell…” Russell (1959).
“English mathematician G. H. Hardy…” Hardy (1940).
“… there is no permanent place in the world for ugly mathematics…” Hardy (1940), 84.
“… he wrote that the sublime…” Tsang (1998), 3.
“French chemical engineer, writer, and amateur mathematician…” Quoted in Wells (1990).
“Keith Devlin, the mathematician who compared…” Devlin (1998), 134.
“This view has been held…” Hardy (1940), 124.
“And on other days…” Mazur (2003), 70.
“He argued that ‘if all…’” Gardner (1981).
“It takes enormous hubris…” Gardner (1997).
“Physicist Eugene Wigner famously dubbed…” Wigner (1960).
“‘… which can thus be said to inhere in the universe,’ he wrote.” Paulos (2002).
“Based on this logic…” Ofek (2001).
“… to represent basic arithmetic knowledge.” Dehaene (2001).
“He and colleague Marie Amalric…” Amalric and Dehaene (2016).
“Dehaene observes, for instance…” Dehaene (2001).
“Ramanujan was referring to the fact…” Bradley (2006).
“A memorable observation by German physicist…” O’Connor and Robertson (2016), “A Quotation by Heinrich Hertz.”
“Euler presented his initial proof…” Written in Latin, as was customary at the time, the Introductio’s two volumes were first published in English in 1988 and 1990, translated by John D. Blanton. Here, I’ve relied on William Dunham’s explication of Euler’s derivation in his book, Euler: The Master of Us All, and on an English translation of the Introductio by Ian Bruce, a retired Australian physics professor who has impressively translated several of Euler’s major works and posted them at www.17centurymaths.com.
“In a 1950 lecture, historian Carl Boyer…” Boyer (1950).
“One reason is that revisionist mathematicians…” Examples include the papers by Mark McKenzie and Curtis Tuckey; Jacques Bair, et al.; and Patrick Reeder that are cited in the bibliography.
“Further, some math educators have argued…” See, for example, Luis Moreno-Armella’s “An Essential Tension in Mathematics Education,” listed in the bibliography. David Tall makes similar points in his book, A Sensible Approach to Calculus, also cited in the bibliography.
“Largely because of that…” Pólya (1990).
“Mathematician William Dunham has similarly noted…” Dunham (1999).
“… according to eminent math historian Morris Kline” Kline (1972), 619.