by Ellen Kaplan
HIDDEN
HARMONIES
THE LIVES AND TIMES
OF THE
PYTHAGOREAN THEOREM
ROBERT KAPLAN AND
ELLEN KAPLAN
Illustrations by Ellen Kaplan
L’arc et la fleche
Mesure du carré de l’hypotenuse
Mont-St-Michel 12th century
FOR
BARRY MAZUR
A hidden connection is stronger than an apparent one.
—HERACLITUS
Contents
An Outlook on Insights
CHAPTER ONE
The Mathematician as Demigod
CHAPTER TWO
Desert Virtuosi
CHAPTER THREE
Through the Veil
CHAPTER FOUR
Rebuilding the Cosmos
CHAPTER FIVE
Touching the Bronze Sky
CHAPTER SIX
Exuberant Life
CHAPTER SEVEN
Number Emerges from Shape
CHAPTER EIGHT
Living at the Limit
CHAPTER NINE
The Deep Point of the Dream
CHAPTER TEN
Magic Casements
AFTERWORD
Reaching Through—or Past—History?
Acknowledgments
Notes
Footnotes
Selected Bibliography
A Note on the Authors
By the Same Authors
An Outlook on Insights
Man is the measure of all things. While Protagoras had the moral dimension in mind, it is also true that we are uncannily good as physical measurers. We judge distances—from throwing a punch to fitting a beam—with easy accuracy, seeing at a glance, for instance, that these two lines
add up to this one:
What’s odd is that we are much less accurate at estimating areas. How do the combined areas of these two squares:
compare to the area of this?
It’s hard to believe that they are the same—which is why the Pythagorean Theorem is so startling:
Where did this insight come from—and how do mathematical insights in general surface? How did the tradition evolve of measuring the worth of our insights by proofs, and what does this tell us about people and times far distant from ours—yet with whom we share the measures that matter?
We share with them too the freedom of this shining city of mathematics, looking out where we will from its highest buildings or in astonishment at the imaginative proofs that support them, made by minds for minds. You can safely no more than glance at these proofs in passing, knowing they wait, accessibly, for when you choose to explore their ingenious engineering.
Come and see.
CHAPTER ONE
The Mathematician as Demigod
The Englishman looked down from the balcony of his villa outside Florence. Guido, the peasant’s six-year-old son, was scratching something on the paving-stones with a burnt stick. He was inventing a proof of the Pythagorean Theorem; what we remember as the algebraic abstraction a2 + b2 = c2 he saw as real squares on the sides of a real triangle—
“‘Do just look at this. Do.’ He coaxed and cajoled. ‘It’s so beautiful. It’s so easy.’” And Guido showed the Englishman’s son how the same square could be filled with four copies of a right triangle and the squares on its sides,
or the same four triangles and the square on the hypotenuse—so that the two squares of his first diagram must equal in area the one square in the second.
And the Englishman thought “of the vast differences between human beings. We classify men by the colour of their eyes and hair, and the shape of their skulls. Would it not be more sensible to divide them up into intellectual species? This child, I thought, when he grows up, will be to me, intellectually, what a man is to a dog. And there are other men and women who are, perhaps, almost as dogs to me.”
But the child never grew up: he threw himself to his death in despair at being snatched from his family by a well-meaning signora, who forced him to practice his scales and took away the Euclid that the kind Englishman had given him.
A true story? Emphatically not. It is Aldous Huxley’s “Young Archimedes”,1 published three years after Sir Thomas Heath’s History of Greek Mathematics came out in 1921, with its quotation from ancient Callimachus: “By a happy chance Bathycles’s son found old Thales scraping the ground and drawing the figure discovered by Pythagoras.”2
The falsity of this story isn’t just in Huxley’s having patched it together from what he had read in Heath, and from the legend of the brutal centurion who, sent to fetch Archimedes, killed him instead, because he wouldn’t stop drawing his diagrams in the dirt. Huxley’s story is much more deeply false: false to the way mathematics is actually invented, and false to the universality of mind. The picture of super-humans in our midst—living put-downs to our little pretensions, yet testimony to more things in heaven and earth—is certainly dramatic, but the actual truth has greater drama still, woven as it is of human curiosity, persistence, and ingenuity, with relapses into appeals to the extraterrestrial. How this truth plays out is the story we are going to tell.
A touch of taxes makes the whole world kin. Around 1300 B.C., Ramses II (Herodotus tells us)3 portioned out the land in equal rectangular plots among his Egyptian subjects, and then levied an annual tax on them. When each year the flooding of the Nile washed away part of a plot, its owner would apply for a corresponding tax relief, so that surveyors had to be sent down to assess just how much area had been lost.
This gave rise to a rough craft of measuring and duplicating areas. Perhaps rules of thumb became laws of thought among the Egyptians themselves; in any event, such problems and solutions seem to have been carried back by early travelers, for “this, in my opinion,” Herodotus says, “was the origin of geometry [literally ‘land-measure’], which then passed into Greece.”
Did their rules, long before Pythagoras, include the theorem we now attribute to him? For Egyptian priests claimed that he, along with Solon, Democritus, Thales, and Plato, had been among their students.4 Whoever believes that all things great and good should belong to a Golden Age, and that Egypt’s was as golden as its sands, would like to think so. But a theorem is an insight shackled by a proof (such as Guido’s) so that it won’t run away, and not the least line of a proof has been found on Egypt’s walls or in its papyri. Well, but might they not have had the unproven Insight, which would have been glory enough?
Here’s a shard of what seems like evidence. The Berlin Papyrus 6619,5 dating from the Middle Kingdom (roughly 2500 to 1800 B.C.), has a problem that amounts to finding the side of a square whose area, along with that of a square with ¾ its side-length, sums to a square of area 100. Since the solution to what we would write as is x = 8, giving squares of sides 6, 8, and 10; and since these three numbers are doubles of 3, 4, 5—the most basic, and famous, triple of whole numbers that fulfill the Pythagorean relation—this could be muffled testimony to knowledge of how the squares on the sides of at least a particular right triangle were related.ap Of course no triangle appears in this problem, much less a right triangle; but if you hear a faint “Yes” when you ask “Is anyone there?”, it would be natural to assume that someone indeed was.
Or it might be that 3, 4, and 5 came up just through learning your times tables. After you found out that 3 × 3 = 9, and then that 4 × 4 = 16, you might well have been struck, on discovering that 5 × 5 = 25, that this was the sum of the two so recently learned (our pattern-making instinct seizes on such slantwise connections). A discovery like this might fall into the background of thought when the pattern failed to continue (42 + 52 62) and surface again when constructing framing for doors—or accessible pro
blems for students. To disguise what might have been the too obvious (3, 4, 5) as (6, 8, 10) would have been a teacher’s familiar dodge. But if the only Pythagorean triples we find among the Egyptians are multiples of (3, 4, 5), the likelihood grows that this generator was for them no instance of a broader truth, but just an attractive curiosity.
Here, however, is a second shard—needing more imagination to decipher than the first, and therefore more intriguing. Early in the fifth century B.C., the philosopher Democritus bragged that none had surpassed him in geometric constructions and proofs, “not even the rope stretchers of Egypt”—apparently referring to the surveyors who, from as early as 2300 B.C., used ropes and pegs to lay out the corners of sacred precincts. Long after this, ancient Indian and perhaps Chinese geometers made right angles by stretching ropes knotted in lengths of 3, 4, and 5 (so that a triangle made with them would have to have been a right triangle). By thus stretching imagination itself back to the time of Amenemhat I, we might feel convinced that the full Pythagorean Theorem, and not just this mingy triple, had been a permanent feature of the Egyptian landscape. You could go on to let living tradition testify to its antiquity: a stonemason of our acquaintance blithely uses what he calls the 3-4-5 rule for checking that his foundations are square. And is not his craft directly descended from the Egyptians, Indians, and Chinese?aq
Fantasy more than imagination is needed to interpret a third offering of evidence, stretching this rope still farther. The engineer and archeological theorist Alexander Thom surveyed, in the middle of the last century, hundreds of megalithic standing stones in northern Europe, and concluded that a standardized ‘megalithic yard’, and the properties of Pythagorean triangles, had been used in their placing. Followers spread his conclusions southward, so that, as one of them writes, “It should therefore perhaps not come as a surprise that in 1997, a stone circle was found in Egypt, in Nabta, which was dated to roughly the same period as the stones at Carnac. It introduced a megalithic dimension in Egypt. At the same time, the Nabta circle had clear astronomical components.” Unfortunately, a statistical analysis of Thom’s data for 465 possible triangles led to the conclusion that “the ancient megalithic builders did not understand the properties of Pythagorean triangles . . . and the idea that they might have is really an artifact of modern interpretation, reasoned a posteriori according to our modern-day knowledge of Pythagoras’ theorem.”6
To whatever slight extent the Egyptians had the Pythagorean Insight, where did it come from? Or if their surveyors had neither knotted ropes nor megalithic yards in their backpacks, how did the Greeks then manage to come by this knowledge? A desert of ignorance drifts for us over these vast centuries; yet turn northeast from Egypt and backward in time, and look across the sands to Mesopotamia . . .
CHAPTER TWO
Desert Virtuosi
The deeper in time the Golden Age is set, the more romantically it gleams. Some four thousand years ago, between the Tigris and Euphrates rivers, the Old Akkadians and then the Old Babylonians developed a way of life that bustled with pride and commerce. They did things with numbers and shapes of a finesse and intricacy that will take your mind’s breath away. These people were the contemporaries of your great-great-[some 150 of these] . . . great-grandparents; they stood two-thirds as tall and lived half as long as we; had a hundredth of our comforts and none of our safeguards. They had no Twitter, no dentists, no Big Macs—but their sense of humor puts them just down the street from us.
FATHER: Where did you go?
SON: Nowhere.
FATHER: Then why are you late?
This snatch of dialogue has been deciphered from an ancient cuneiform tablet, startlingly small, and indented with the neat bird-tracks of wedges that read as easily to them as our letters do to us.1
The ancestors of these people had kept their accounts with clay tokens through the four thousand preceding years, but as a temple-based bureaucracy developed, the growing complexity of life, and of the bookkeeping that recorded it, led to symbols for these tokens, and signs for 1, 10, and 60, which they iterated to make the other numbers. Once again we recognize ourselves in them: abstraction from things to names, and names to numbers, is the way we mark our turf.
An organic interplay between mathematics and administration continued through most of the third millennium B.C., developing into a 60-based system of whole numbers and fractions (60 has enough whole number divisors to make calculations easier than in our base 10 system). Then something important seems to have happened around 2600, when a class of scribes emerged. For them—perhaps in the slack periods when goods were not being brought in—writing broadened from making inventories to recording epics, hymns, and proverbs, and mathematics from the practical to the precious. We set our young mathematicians difficult theorems to prove; they gave their rising scribes horrendously long calculations to carry out, with as little connection to reality as had the highly artificed poems that Mandarin officials were required to write in ancient China.
Everything changed again near 2300, with the invasion of an Akkadian-speaking dynasty. Sumerian stiffened into an administrative language only (playing the ennobling role that Latin once did for us), and new sorts of mathematical problems arose, centering on area. In the blink of an eye from our perspective—two hundred years for them—this dynasty fell, and a neo-Sumerian state took its place in 2112. That 60-base number system now began to work itself out on the backs of the people and the brains of the scribes. Our lawyers punch in a client’s time every fifteen minutes, but these scribes, acting now as overseers, had to keep track of their laborers through a day of ten-minute quotas.
This state in turn collapsed—probably under its own administrative weight—within a century, and the four-hundred-year glory of the Old Babylonian period began, epitomized by the famous lawgiver Hammurabi. This was a time of high scribal culture, featuring ideals we recognize as humanistic, and calculations whose balance of cleverness and painstaking tedium we gasp at.
A Hittite raid around 1600, then an overwhelming invasion by warrior Kassites, suddenly brought down a curtain a thousand years thick, hiding away almost all traces of this Hobbit-like people, whose glass-bead game culture and animated bureaucracy we see far away down the wrong end of a telescope.2 Our story turns into history.
Both are deceptive. This narrative, made to appear seamless, is actually stitched together from so much of so little—as far as the mathematics goes, mountains of clay tablets recording hardly more than school exercises or teachers’ trots. The context of the society at large, and of the scribal community within it, is just guesswork, attempts at rational reconstruction baffled by the distortions of historical foreshortening: events spread over vast stretches of time and space are collapsed to aperçus, and anything is taken to stand for everything. The conclusion, for example, that a thuggish regime shut down intellectual pursuits for a millennium might be skewed by the economics of excavation: with funding scarce, who would dig up schoolrooms, despite their possibly valuable evidence of evolving thought, when there are palaces waiting to emerge?
We’re not only sitting at the far end in a game some call Chinese Whispers and others Telephone, but those in front of us each have their own agendas and personalities to promote, while they indulge in the peculiar practice of letting their civility be seen as no more than veneer. Perhaps they would be less vituperative, and hence more enlightening, were they a more expansive community (squabbling seems to breed in close quarters), or had they evidence rather than speculation to go on, and a logic founded on ‘only if’ rather than ‘if only’. As it is, the hum of knives being whetted may serve the ends of mean fun, but can be distracting. From a recent scholarly work:
The pretentious and polemical attempt by Robson in HM 28 (2001) to find an alternative explanation of the table on Plimpton 322 is so confused and misleading that it should be completely disregarded, with the exception of the improved reading of the word I-il-lu-ú in the second line of the heading over the first preserved
column, and the dating of the text. . . . Cf the verdict of Muroi, HSJ 12 (2003), note 4: “The reader should carefully read this paper written in a non-scientific style, because there are some inaccurate descriptions of Babylonian mathematics and several mistakes in Figure 1, Tables, and transliterations.” A briefer and less polemical, but still pointless, version of the same story can be found in Robson, AMM 109 (2002).3
Odd that archeology is often still thought of as one of the humanities.
Our aim at this point is to see what traces or precursors of the Pythagorean Theorem we can find in Mesopotamia; and if there are any, whether they then migrated somehow to Egypt or even directly to Greece. We recognize that the ambiguity of the evidence and the need to pick at the scholarly scabs over it will make our telling hypermodern: one of those “design it yourself” dramas, constructed not only by the participants but by its writers and readers as well. Yet some Guido or, collegially, Guidos did come up with the stages of this insight, and then its proof; and narrowing down to a local habitation if not a name is surely less profound a task than was theirs.
From what was practiced in their schools, the Old Babylonians appear to have prized virtuosity over curiosity, technique over insight. Whether they were simply addicting students to the calculations they would need as administrators, or whether computation induced an almost erotic ecstasy (as anything chant-like and repetitive may), we find countless examples of methodological rather than critical thought, of assertions rather than explanations. Theirs was, at first glance, a culture of algorithm. Hammurabi’s Code of Laws well represents their point of view: it is a collection of particular judgments that serve as precedents for subsequent decisions. Each of their mathematical problems too is an example, none is exemplary: every figure has particular numbers attached to it, such as lengths and areas: there is no “general figure” (while a triangle in Euclid’s geometry stands for any triangle of its ilk) and certainly no notion of a “general number”—the x that, millennia later, would lift arithmetic to algebra.