For the truthful Newton, the best we have is the following response he made when once asked how he made discoveries such as the law of gravitation: ‘By always thinking [about] them’, Newton said. ‘I keep the subject constantly before me and wait till the first dawnings open little by little into the full light.’4 This rings far truer, and more closely resembles how other great discoveries are made, than the apple story. The dawnings involve more than seeing clearly, but conceptual shifts, transforming what he had inherited, and forging new things and new concepts. ‘The momentous discovery of universal gravitation, which became the paradigm of successful science, was not the result of an isolated flash of genius’, wrote I. Bernard Cohen, but a lengthy process involving the ‘transformation of existing ideas.’ Cohen added, ‘The discovery of universal gravity brings out what I believe is a fundamental characteristic of all great breakthroughs in science from the simplest innovations to the most dramatic revolutions: the creation of something new by the transformation of existing notions.’5
An apple may well have played a role in Newton’s thinking of gravitation. But if it did, it served a similar purpose to Socrates’ pointing to the diagonal did to Meno’s slave – it helped him recast what he already knew of the problem in a new light, helping to transform that thinking in the process.
4
‘The Gold Standard for Mathematical Beauty’:
EULER’S EQUATION
eiπ + 1 = 0
DESCRIPTION: The base of natural logarithms (an irrational number) raised to the power of pi (another irrational number) multiplied by the square root of negative one (an imaginary number) plus one is an integer: zero.
DISCOVERER: Leonhard Euler
DATE: 1740s
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.
– Keith Devlin
When the 14-year-old Richard Feynman first encountered eiπ + 1 = 0, the future Nobel laureate in physics wrote in big, bold letters in his diary that it was ‘the most remarkable formula in math.’ Stanford University mathematics professor Keith Devlin writes that ‘this equation is the mathematical analogue of Leonardo da Vinci’s Mona Lisa painting or Michelangelo’s statue of David.’ Paul J. Nahin, a professor of electrical engineering, writes in his book, Dr. Euler’s Fabulous Formula, that the expression sets ‘the gold standard for mathematical beauty.’ One of my correspondents said it was ‘mind-blowing’; another called it ‘God’s equation.’
This expression, discovered by the eighteenth-century Swiss mathematician Leonhard Euler, has become an icon – an object with special properties above and beyond the truths that it represents – for many people, even those with only a little mathematical training. Like other icons, it can become an object not only of fascination but also of obsession.
Consider first that it is surely one of the few mathematical expressions to serve as a piece of evidence in a criminal trial. In August 2003, an ecoterrorist assault on a series of car dealerships in the Los Angeles area resulted in $2.3 million worth of damage; a building was burned and over 100 SUVs were destroyed or defaced. The vandalism included graffiti consisting of slogans such as ‘GAS GUZZLER’ and ‘KILLER’; and, on one Mitsubishi Montero, the formula eiπ + 1 = 0. Using this as a clue and later as evidence, the FBI arrested William Cottrell, a graduate student in theoretical physics at the California Institute of Technology, on eight counts of arson and conspiracy to commit arson. At the trial that resulted in his conviction, in November 2004, Cottrell admitted having written that equation on the Montero. ‘I think I’ve known Euler’s theorem since I was five’, Cottrell said during the trial. ‘Everyone should know Euler’s theorem.’1
Another equation-turned-icon, and one certainly much better known than Euler’s, is E = mc2. This equation is a familiar part of popular culture, and has even been turned into a monument: during the 2006 World Cup, the six large outdoor sculptures that were erected in Berlin to illustrate Germany’s status as the ‘land of ideas’ included a car, a pair of football boots, and a gigantic representation of E = mc2.
But how is it possible for an equation to become an icon, anyway? After all, an equation is merely one step in the ongoing process of scientific inquiry. Euler’s expression was but one implication of his exploration of functions, while E = mc2 was an afterthought of Einstein’s development of special relativity. Aren’t equations just tools of science, of less intrinsic value and interest than the tasks they were developed to help us with? How do some of them acquire an inherent value or significance beyond the process of inquiry to which they belong? Tools surely can become icons, the way, for instance, a hammer and sickle became symbolic of the Soviet state – but a mathematical and technical object like an equation? What makes such an abstract thing able to stand literally alongside a pair of boots or a car?
The story of Euler’s formula helps to answer these questions.
The Neighbourhoods of Mathematics
Leonhard Euler (1707–1783) was the most prolific mathematician of all time; his collected works, when finished, will run to some seventy-five volumes. He calculated effortlessly, ‘just as men breathe, as eagles sustain themselves in the air.’2 It helped that he had a prodigious memory that spanned the swath of human knowledge, able to retain extensive mathematical tables and the entire text of Virgil’s Aeneid. It also helped that he had an eye for spotting deep connections between what seemed to be vastly different areas of mathematics, synthesizing them and making the result seem as obvious as 2 + 2 = 4. His equations about fundamental matters are of such simplicity and elegance that, one commentator remarked, their ‘form pleases the eyes as much as the spirit.’3 His famous formula eiπ + 1 = 0 was the most simple, elegant, and pleasing of all.
Euler was born in Basel, Switzerland. His father, a Protestant minister, awakened his earliest interest in mathematics by instructing him in the basics. Euler continued to receive private maths tutoring in high school, because the subject was not taught there. At age fourteen he entered the University of Basel and studied a wide range of topics, from theology to languages to medicine, but remained fascinated by maths. Saturday afternoons he was privately coached by the renowned mathematician Johann Bernoulli, and became friends with the latter’s sons Nicolaus and Daniel. After Euler received his degree, in 1723, he complied with his father’s wishes and tried to become a theologian, but soon turned back to mathematics.
Maths was not an easy career. Universities then were bastions of scholarship in the humanities, with few places for mathematicians or scientists. The rare available jobs for mathematicians were at a handful of royal academies.
Leonhard Euler (1707–1783)
Fortunately for Euler, Peter the Great of Russia and his second wife, Catherine I, one of history’s great ‘Renaissance couples’, were in the process of founding the Russian Academy of Sciences in St. Petersburg, and plucking for it leading scientists from all over Europe. Two early recruits were Nicolaus and Daniel Bernoulli, who in turn secured an invitation for their friend Euler. Both Peter the Great and Catherine died before Euler arrived, in 1727, and their successors were less enthusiastic about the academy; still, Euler was well cared for and supported. He was surrounded by first-rate scientists and was soon the academy’s chief mathematician. He was so productive that the editors of the academy’s journal stacked his manuscripts in piles, grabbing a few from the top when they had space. His 14 years at the academy were accompanied by some hardships, the worst of which was the loss of his right eye, probably through eyestrain due to overwork. But during these years he was free to calculate furiously, and reshaped the foundations of mathematics in the process.
Mathematics often grows in an indirect way, the way that many cities do. Certain scattered settlements spring up first, with little interaction among one another. These settlements eventually cluster around one anot
her, becoming neighbourhoods, but because they form almost at random they are poorly adapted and little commerce takes place. A visionary leader emerges who understands each neighbourhood, and by renaming some streets and building others between key centres allows them to grow into a greater structure that is simultaneously more simplified, organized, and unified.
This is the role Euler played in eighteenth-century mathematics.
At the time, mathematics had two thriving, well-developed neighbourhoods, geometry and algebra. Geometry is the study of points, lines, planes, and the properties of figures built from them. It had been systematized in antiquity by Euclid’s Elements (ca. 300 BC). One subdivision of geometry is trigonometry, concerned with the study of the relationships between the angles and lengths of sides in triangles, first developed as a tool of astronomy. Algebra is the study of equations with finite elements and discrete solutions, and largely concerned with rational numbers – numbers that can be expressed as integers or ratios of integers (in the form p/q) or, in what amounts to the same thing, numbers whose decimal representations repeat themselves. (Numbers like π, where the decimal values go on forever without repeating themselves, are said to be irrational.) Algebra had been largely organized, and given its name, in the Middle Ages by the Arab mathematician Mohammed ibn Musa al-Khowârizmî (ca. 780–850), thanks to his book Hisâb al-jabr wa’l muquâbalah (830). Al-jabr was al-Khowârizmî’s term for the process of adding equal quantities to both sides of an equation to simplify it; after the word was transliterated into Latin as ‘algebra’, it became the label for the entire field.
Unifying the Neighbourhoods
By the beginning of the eighteenth century, mathematics was evolving a new neighbourhood called analysis, or the study of – the collection of techniques for dealing with – infinities, for instance, series that include infinitely many numbers of terms. Analysis grew largely out of calculus, the study of continuous processes, which was developed by Gottfried Leibniz and by Newton (who called it the theory of fluxions). Analysis also involved the study of irrational numbers. And analysis treated imaginary numbers, or the square roots of negative numbers. These had been named by the philosopher and mathematician René Descartes, who seems to have thought them fictional – and his term stuck even as their uses and value to mathematics grew.
But it was Euler who organized analysis as a coherent body of knowledge, and transformed it into a thriving and organized area of mathematics: For instance, he carried out the first systematic study of functions. Functions are now-indispensable mathematical tools that pair or match one number with another (simple illustrations are formulas for calculating taxes, or for converting temperatures from Fahrenheit to centigrade). Euler also developed and expanded the tools that mathematicians had for summing infinite series of terms. Before him, mathematicians regarded summing infinite series of terms as an unpleasant duty that they sometimes had to do to solve problems when no other methods were available. But Euler showed that mathematicians need not be afraid of such series – they could be easy to work with, provided that the series converged. Euler was also the single most influential developer of mathematics notation in its history. Key symbols that he either introduced or standardized include:
π, for the ratio of a circumference to the diameter of a circle, perhaps named after the first letter of the Greek word for ‘perimeter.’
e, for the base of natural logarithms, probably named for the first letter of ‘exponential’; the logarithm is the power to which a base must be raised to get a certain number, and e is the base of natural logarithms (logey = x means ex = y).4
i, for the basic ‘imaginary number’, which is hardly fictitious as Descartes thought but extends the range of equations that can be solved.5
f(x), for a function of x, a function being a pairing or matching of one series of numbers with another.
sin, as an abbreviation for the sine function, pairing the measure of an angle in a right triangle with the ratio of the length of the opposite side to the hypotenuse.
cos, as an abbreviation for the cosine function, pairing the measure of an angle in a right-angled triangle with the ratio of the length of the adjacent side to the hypotenuse.
Σ, for the summation of a series of terms.
In 1741, after 14 years in St. Petersburg, Euler left for the Berlin Academy at the invitation of Frederick the Great, another Renaissance man, though Euler remained in close correspondence with his St. Petersburg colleagues. Euler found Berlin less congenial than St. Petersburg. Frederick the Great was accustomed to highbrows with more flair than the taciturn Euler, thought him an aberration among his collection of pundits, and called him a ‘mathematical Cyclops.’6 In 1766, after 15 years in Berlin, Euler returned to St. Petersburg at the invitation of Catherine II, Catherine the Great. Though he was well supported, his health woes increased. He learned that he had a growing cataract in his remaining eye that would ultimately lead to blindness. He gamely coped – ‘I’ll have fewer distractions’, he remarked – learned to write with chalk on a slateboard, and taught his children to copy his calculations. Fewer distractions indeed. Euler pressed on undaunted for 17 years more, calculating, revising, composing, talking while walking around a table, with sons and assistants copying down his words. In this way, completely blind, he produced almost half of his entire oeuvre.
In 1771, a fire destroyed much of St. Petersburg. With his house burning, Euler himself – weak and blind – was carried to safety on the shoulders of a friend. He calculated on. On September 18, 1783, he tutored one of his grandchildren in maths, worked out some problems regarding the paths of hot-air balloons, and considered possible orbits of the recently discovered planet Uranus, when his pipe suddenly dropped from his mouth. In the same breath, ‘he ceased to calculate and live.’7
Today, the modern metropolis of mathematics is much larger still than it was in Euler’s time. It is now laid out in huge boroughs, including analysis, algebra, and topology. Euler helped advance all three. His textbook on algebra, Vollständige Anleitung zur Algebra (Complete Instruction in Algebra, published in English as Elements of Algebra), presents the field essentially in the form it is today. He also made some of the first forays into topology, a field that did not exist yet, thanks to his famous solution to the Konigsberg bridge problem, involving the question of whether you could cross the seven bridges spanning the banks and islands of that city in a single walk without crossing any bridge twice – although topology would not be recognized as a borough for a hundred years or so.
But Euler was known as the master architect of analysis: scholars often called him ‘analysis incarnate.’ His most important single work in this field is a two-volume textbook, written during his Berlin years, entitled Introductio ad analysin infinitorum (Introduction to Infinite Analysis, 1748). In it, Euler unveiled numerous discoveries about functions involving infinite series, supplied proofs of theorems that others had left missing or incomplete, simplified many mathematical techniques, and proposed definitions and symbols that have since become standard, including π and e. ‘The Introductio did for analysis what Euclid’s Elements had done for geometry and al-Khowârizmî’s Hisâb al-jabr wa’l muquâbalah for algebra. It was a classic text from which whole generations were inspired to learn their analysis, especially their knowledge of infinite series.’8
But the Introductio did much more than reorganize analysis. By translating many mathematical terms and expressions into the language of infinite series, it transformed analysis from a newly developing area of mathematics, alongside the existing fields of geometry and algebra, into its principal region. It all but made analysis the centre city of mathematics.
The Deep Link
In the Introductio, Euler announced the dramatic discovery of a deep connection between exponential functions, trigonometric functions, and imaginary numbers. The proof grew out of his studies of exponential functions. In simplest terms, an exponential function involves a number called the base and another number se
t to the upper right of the base, called the exponent, with the exponent indicating how many times the base is multiplied by itself to produce the value of the function (this notation was invented by Descartes). A simple example of an exponential function is y = 2x, where 2 is the base and x the exponent. For any integer x, this gives rise to a finite series of terms and an integer product. For instance, 22 = 2 × 2 = 4, 23 = 2 × 2 × 2 = 8, 24 = 2 × 2 × 2 × 2 = 16, and so forth.
These integer pairs of numbers can be treated as belonging to a curve. On the infinite number of points on such a curve, only a few pairs are integers; the values on the dotted curve in between include decimals like 3.81 and even irrational numbers like and π. What does it mean to multiply a number such as 2 by itself 2.31 or or π times? For rational numbers expressible in the form p/q, this had to mean the pth root of 2 to the q. For example, 2 to the power 3.81 (3 381/100) would be the 100th root of 2 to the power 381. For irrational numbers, it would mean filling in the missing point on that curve, which can be calculated as the limit of an infinite sequence. Thus 2π is the limit of 23, 23.1, 23.14, ..., 23.1415926, ...as we take more and more decimal values of π.
In Chapter VII of the Introductio, Euler showed that, in choosing the base for an exponential function, there were numerous mathematical advantages to selecting the number created by adding up the following infinite series:
A Brief Guide to the Great Equations Page 9
