When we understand an important equation for the first time, we glimpse deeper structures to the world than we suspected, in a way that reveals a deep connection between the way the world is and how we experience it. At such times, our reaction is not simply, ‘Yeah, that makes sense’, or even what is often called the ‘Aha!’ moment. This latter crude characterization goes hand in hand with the treasure-hunt picture of knowledge acquisition, for it simplifies and condenses the emotion of discovery into a single instant. The genuine emotion – wonder – is subtler, richer, and lengthier.
It is natural, though, even for scientists to stop wondering at equations, as they become more wrapped up in the world and their interests in it, and less attentive to the moments of disclosure in which its forms first appear. We lose wonder, indeed, at any instrument or object with which we grow too familiar. Equations can come to seem as just another set of tools that we find lying about in the world, or as onerous chores that we learn out of duty.
Pilots who learn too much about their craft, Mark Twain writes in Life on the Mississippi, often undergo a regrettable transformation. As they become increasingly skilled at reading the language of the river, they seem to grow correspondingly less able to appreciate its beauty and poetry. Features of the river – a floating log, a slanting mark on the water, a patch of choppy waves – that once aroused feelings of wonder and awe become increasingly appreciated only instrumentally, in terms of the use they have for piloting. Something similar is true of equations.
But great scientists are often still able to marvel at the breakthroughs of their predecessors. The physicist Frank Wilczek once wrote a series of articles on the simple equation expressing Newton’s second law of motion, F = ma, calling it ‘the soul of classical mechanics’, and exhibiting toward it the kind of appreciation that is appropriate to souls.8 The physicist and cosmologist Subrahmanyan Chandrasekhar wrote an entire book on Newton’s Principia, the book in which Newton proposed his second law of motion, comparing it to Michelangelo’s painting on the ceiling of the Sistine Chapel. And a listener to Richard Feynman’s famous Lectures on Physics can detect throughout his unabashed and spontaneous wonder at the equations he is trying to teach his students. These three Nobel laureates each knew enough to maintain their wonder at the world and at the equations through which we know it.
This book aims to show that there is much more to equations than the simple tools they seem to be. Like other human artifacts, equations have social significance and exert cultural force. This book takes some great equations and provides brief accounts of who discovered them, what dissatisfactions lay behind their discovery, and what the equations say about the nature of our world.
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‘The Basis of Civilization’:
THE PYTHAGOREAN THEOREM
c2 = a2 + b2
DESCRIPTION: The square of the length of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the lengths of the other two sides.
DISCOVERER: Unknown
DATE: Unknown
To this day, the theorem of Pythagoras remains the most important single theorem in the whole of mathematics.
– J. Bronowski, The Ascent of Man
The original journey to the Pythagorean theorem is forever shrouded in history. But we have countless stories of its rediscovery, both by people who taught it and by people who rediscovered it for themselves. These sometimes have been such powerful experiences as literally to have changed the lives and careers of those who have made them. The power and magic of the Pythagorean theorem arise from the fact that, while it is complex enough that its solution is not apparent at the outset, the proof process is condensed enough to constitute a single experience.
One person whose life it changed was the great political philosopher Thomas Hobbes (1588–1679). Until he was forty, Hobbes was a talented scholar who showed little originality. He was well versed in the humanities but dissatisfied with his erudition. His principal achievement was an elegantly written if sometimes inaccurate translation of the ancient Greek historian Thucydides. He had little exposure to science, despite the exciting recent breakthroughs of Kepler, Galileo, and others, which were then revolutionizing the scholarly world.
One day, while passing through the library of an acquaintance, Hobbes saw a copy of Euclid’s Elements displayed on a table. This was not unusual: a gentleman who owned a handsome and expensive volume of an important work, such as a Bible, would not store it out of sight but would prominently exhibit it for the benefit of visitors, usually opened to a famous passage or psalm.
Euclid’s Elements was indeed like a Bible. It set out much of the mathematical wisdom of its time in axioms and postulates; scholars had been analyzing it ever since its appearance in about 300 bc; and its knowledge remained current. No other book at the time, except the Bible, had been as frequently copied or studied. The particular chapter and verse that Hobbes saw was Book I, Proposition 47, the Pythagorean theorem.
Hobbes took a look at the claim: The square described upon the hypotenuse of a right-angled triangle is equal to the sum of the squares described upon the other two sides. He was so astounded that he used a profanity that his acquaintance and first biographer John Aubrey refused to spell out: ‘“By G—”, Hobbes swore, “this is impossible!”’1
Hobbes read on, intrigued. The demonstration referred him back to other propositions in the same book: Propositions 46, 14, 4, and 41. These referred to still others. Hobbes followed them and was soon convinced that the startling theorem was true.2
‘This made him in love with geometry’, writes Aubrey, adding that Hobbes was a changed man. He started obsessively drawing geometrical figures and writing out calculations on his bedsheets and even on his thigh. He began to devote himself to mathematics, showed some talent – though his abilities remained modest – and embroiled himself in controversies and hopeless mathematical crusades in a manner that still embarrasses his biographers and fans.3 These episodes are not terribly interesting. What matters is that the theorem transformed him and his scholarship. As one commentator wrote of Hobbes’s initial encounter with the Pythagorean theorem, ‘everything he thought and wrote after that is modified by this happening.’4
Hobbes began to chastise the moral and political philosophers of the day for their lack of rigor and for being unduly impressed by their predecessors. He compared them unfavorably with mathematicians, who proceeded slowly but surely from ‘low and humble principles’ that everyone understood and accepted. In books such as the Leviathan, Hobbes began to reconstruct political philosophy in a similar way, by first establishing clear definitions of terms, and then working out the implications in an orderly fashion. The Pythagorean theorem taught him a new way to reason, and to present the fruits of his reasoning persuasively, in ways that seemed necessary and universal.
Pythagorean Theorem: The Rule
The term, ‘Pythagorean theorem’, is popularly used to refer to two different things: a rule and a proof. The rule is simply a fact. It states an equality between the lengths of the sides of a right-angled triangle: the length of the hypotenuse squared (c2) is equal to the sum of the squares of the two other sides: (a2 + b2). That rule has a practical value: it allows us, for instance, to calculate the length of that hypotenuse if we know the lengths of the two sides. The proof is different. It’s the demonstration of how we know this fact to be true.
It is confusing that this phrase can refer to both. It’s a confusion embedded in the word ‘theorem.’ The word can mean a result that is (or is assumed to be) proven. It comes from the Greek for ‘to look at’ or ‘contemplate’, and has the same root as ‘theatre.’ When people like Hobbes see the Pythagorean theorem, they can pay attention to two very different things: to the product, rule, or thing proven – the hypotenuse rule – or to the process, the proving, or the way it is known.
The rule is extremely important, crucial to describing the space around us. It is invaluable to carpenters, architects, and surveyors
in small and large-scale construction projects. This is one reason Freemasons – the esoteric organization said to have been born in medieval stonemason guilds – adopted the Pythagorean theorem as a symbol. One piece of Masonic literature cites the Pythagorean theorem as ‘containing or representing the truth upon which Masonry is based, and the basis of civilization itself’,5 and a simplified version of the diagram accompanying Euclid’s proof, called the ‘Classic Form’, is often emblazoned on carpets in Masonic lodges. The rule characterizes celestial spaces as well, thus is essential to navigation and astronomy.
This rule was known long before Euclid or even Pythagoras. The fact that sides of specific lengths – 3, 4, and 5 units, say, or 6, 8, and 10 – create a ‘set square’ with a right-angled triangle between the two shorter sides was an empirical discovery known to ancient craftsmen. Such trios of numbers are called ‘Pythagorean triplets’, and their independent discovery in different lands is not surprising given their simplicity and practical importance. Another ancient discovery seems to have been the rule c2 = a2 + b2 for such triplets. A Babylonian cuneiform tablet of about 1800 BC, known as Plimpton 322 after the collection in which it resides at Columbia University, contains a table of fifteen rows of Pythagorean triplets. The tablet was evidently a trigonometric table or teaching aid for the rule to figure out hypotenuses of right-angled triangles. It contains no variables, but it seems to have been intended to communicate the rule via a list of examples.6
A Babylonian cuneiform tablet of about 1800 BC, known as Plimpton 322 after the collection in which it resides at Columbia University. The tablet, evidently a trigonometric table or teaching aid for the rule to figure out hypotenuses of right-angled triangles, contains a table of fifteen rows of Pythagorean triplets.
The rule was also known in ancient India. Applications of it are found in the Śulbasūtras, the texts that accompany the Sutras or ‘sacred teachings’ of the Buddha, which seem to have been written between 500 and 100 BC but clearly pass on knowledge of much earlier times. In their instructions for constructing ritual areas they display considerable geometrical knowledge, though it is expressed informally and approximately, and without much justification.7
The earliest existing Chinese writing on astronomy and mathematics, the Zhou Bi Suan Jing (‘Gnomon of the Zhou’, containing texts dating from the first century BC but whose contents are said to be centuries earlier), likewise exhibits knowledge of the rule. One application is in a calculation of how far the sun is from the earth. The reasoning process involves a bamboo tube and its shadow, and assumes that the earth is flat; the Zhou Bi is famous among historians of science for being ‘the only rationally based and fully mathematicised account of a flat earth cosmos.’8 The earliest extant version contains an often-reproduced diagram against a chessboardlike background from which one can readily see that the area of the square built on the hypotenuse is the same as the combination of the areas on the other two sides – but this almost certainly dates from a third century ad source, long after Euclid.
Diagram from a late edition of the Zhou Bi. The characters refer to the colors of the squares.
The Babylonian tablet, the Indian Śulbasūtras, and the Chinese Zhou Bi each exhibit knowledge of the rule as part of a body of mathematical knowledge applied to some other purpose: educational in the case of Plimpton 322, religious in the case of the Śulbasūtras, astronomical in the case of the Zhou Bi. In these and in other ancient texts the rule is presented without explicit justification, mainly as a way of finding distances and checking results, though occasionally with more formality.
Indeed, the Pythagorean theorem is surely unique among mathematical landmarks for the range of colorful practical illustrations, ranging from prosaic to poetic, over its thousands of years of history, involving the dimensions of fields, canals, clotheslines, footpaths, roads, and aqueducts. From an Egyptian manuscript: ‘A ladder of 10 cubits has its foot 6 cubits from a wall; how high does it reach?’ From a medieval Italian manuscript: ‘A spear 20 ft. long leans against a tower. If its end is moved out 12 ft., how far up the tower does the spear reach?’ An Indian text asks readers to compute the depth of a pond, swimming with red geese, if the tip of a lotus bud were about 9 inches above the water, but was blown over by the wind – its stem fixed to the bottom – and vanished beneath the water at a distance of about 40 inches. These kinds of exercises make mathematics fun!
The rule has become a model piece of knowledge, and knowing it is often symbolic of human intelligence itself. At the end of the movie the Wizard of Oz, the Scarecrow – to show he truly does have a brain – states a botched version: ‘The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.’ The levity is perfect, for it spares us in the audience from really having to follow it, and keeps what’s happening in the realm of fairy tale.
Pythagorean Theorem: The Proof
But proving a rule is much different from just knowing it. A proof demonstrates the general validity of a result based on first principles – for its own sake, not linked with a practical end, and with the focus less on the result than on how one arrives at it; on the process by which we come to trust it. A proof recounts the journey by which we know an equation. To provide the proof of a rule therefore involves a different perspective on mathematics than just stating the rule. For a proof is not an assertion of authority but an acknowledgement of intellectual democracy. It does not simply pass on a piece of wisdom from one’s precursors as a tour de force of intellect, a stroke of genius. It does not say, ‘This is a fact!’ or ‘This is how a genius told us to do it.’ Instead, the proof of a result says that the journey is something anyone can take, in principle at least, thanks to the matrix of mathematical definitions and concepts that we already possess. It therefore says in effect, ‘Follow this, and you’ll see that we know all the steps how to get there already!’ Giving the proof of a rule therefore establishes a landmark that anyone can get to by following the path indicated, and that one can trust to orient oneself while making further journeys in unexplored territory. Proofs of key equations transform mathematics from a complex terrain into a landscape by erecting landmarks. The rest of mathematics is still present, but in the background.
Although the first proof of the hypotenuse rule is traditionally ascribed to Pythagoras (ca. 569–475 BC), the claim that his proof was the earliest was first advanced half a millennium later, and is almost certainly untrue.9 The idea of proof seems to have originated in ancient Greece, and took hundreds of years to develop. It culminated in Euclid’s Elements, which presents mathematical knowledge entirely in the form of explicit, formal proofs. The proof of the Pythagorean theorem is the next-to-last one of Book I. In a right-angled triangle, the square on the side opposite the right angle is equal to the sum of the squares on the other two sides. Proposition 48, the last proof of Book I, is the converse: if the square on one side of a triangle is equal to the sum of the squares on the other two sides, it is a right-angled triangle. The proof is as follows: Build a square on each side of a right-angled triangle. Draw a line from the vertex of the right angle, perpendicular to the hypotenuse, to that square’s far side. This divides the big square into rectangles. Each rectangle turns out to be the same size as one of the squares: the sum of the smaller squares thus equaling the area of the square on the hypotenuse. Interestingly, Euclid’s proof is associated with the distinctive image created by its lines, and has been called the windmill, peacock, or bridal chair proof after fanciful images that it has been taken to suggest.
A classic diagram illustrating the proof in Euclid’s Elements.
Every great discovery seems to generate the irresistible urge to scour through records to see if anyone else discovered it earlier, discovered it but did not write it down, or brushed up against it without discovering it. The Pythagorean theorem, as we seem forever fated to call it, was no exception. For historians, showing how close a people came to proving the Pythagorean theorem appears
to be a way to try to show how advanced that civilization was – and claims have been made for the Babylonian, Indian, and Chinese discovery of the Pythagorean theorem based on Plimpton 322, the Śulbasūtras, the Zhou Bi, and other texts.10 But in the process, it is easy and tempting to confuse or ignore the difference between the Pythagorean theorem, the empirically determined rule, and the Pythagorean theorem, the proof of the equation.
New Proofs
Occasionally, humans have taken the journey on their own, discovering the Pythagorean theorem without the aid of teachers. One is the French mathematician and philosopher Blaise Pascal, whose father forbade any discussion of mathematics around the household, afraid that the subject might distract his child from the all-important studies of Greek and Latin. But the young Pascal began to explore geometry with the aid of a piece of charcoal, in the process discovering many of the proofs codified in Euclid’s Elements, including the Pythagorean theorem.11
It is also possible to discover new proofs of the theorem. For if the Pythagorean theorem is unique among mathematical landmarks for the range of its applications and examples, it is also unique for the range of ways that it has been proven. Most proofs are based on the same axioms, but follow different paths to the climax. Many – especially the earliest proofs such as Socrates’, Euclid’s, and in the later Chinese manuscript Zhou Bi – are geometrical, where a, b, and c refer to lengths of various sides of shapes, and the proof proceeds by manipulating the shapes and showing something about their areas. Other proofs are algebraic, or based on more complex mathematics where the numbers refer to abstract things, and can even refer to vectors. Some so-called proofs, though, assume results that are proven by the Pythagorean theorem, and so are really circular arguments. The algebraic approach – which the Babylonians understood – is what produced the c2 = a2 + b2 version of the rule.
A Brief Guide to the Great Equations Page 2
