A Brief Guide to the Great Equations (12 page)

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.

When the 14-year-old Richard Feynman first encountered
e
^i
π
+ 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
e
^i
π
+ 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
=
mc
²
. 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
=
mc
²
.

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
=
mc
²
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.

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
e
^i
π
+ 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 another, 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.

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 (log
_e
y =
x
means
e
^x
=
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.