by Ellen Kaplan
What kinds of problems were these? Additions and subtractions, multiplications, difficult divisions (dividing sexagesimal—base 60—numbers by anything but multiples of 2, 3, and 5 leads you into fearful mazes of remainders). They also had a curious predilection for questions of this sort: find a number which, minus its reciprocal, equals 7. This fascination with reciprocals is explained by their thinking of division—by 5, say—as multiplication by the reciprocal of 5 (which was, for them, not , but ).
They calculated everything in sight: when given some dimensions of, say, a door—as its width and length, to find its area, or the length of its cross-brace. Ah! You mean to find the hypotenuse of this right triangle?
So it seems—as in their problems about canes leaning against walls: how long are they, if their tops are this far from the ground, and their tips that far from the wall?
So they must have had the Pythagorean Theorem! Well, let’s look more closely.
How were they taught to solve these problems? By reckoning, not reasoning. They were given procedures: if it looks like this, do that to it (perhaps your math teacher was an Old Babylonian?). Even the instructors worked from handed-down tables of algorithms and answers. Yes, but somewhere, somewhen, someone had come up with the insight that led to these algorithms. Of course—and these algorithms are their only memorials, as well as memorials to a time, perhaps, when curiosity was commoner coin. But we need to understand the context of these algorithms, if we are to know whether any of them incorporated the Pythagorean Insight.
We mentioned that the mathematical issues the Old Babylonians dealt with had come to center on area. You now begin to sense, after the first shock of recognition across the millennia, how very different their mathematical intuition was from ours. We build up square measures from linear, areas from lengths. For some reason their thought moved in the contrary direction—so much so, that they used the same word, mithartum,4 for a square and its side. Absurd? As absurd as your asking that a square and its side and the number −2 add up to 0? An ancient Greek would surely have pointed out the nearest lunatic asylum5 to you: you can’t add areas to lengths, much less either to a dimensionless number (and a negative one at that). Yet we blithely write x2 + x − 2 = 0 and, solving, find x = 1 or x = −2: pleased not only that the square has a side of length 1, but that this side could equally well have had length −2. What is a square, and what is its side, and unto whom?
But when we are thinking geometrically, rather than algebraically, we, along with the Greeks, would still insist that lengths and areas are fundamentally different. For what may, as we’ll now see, have been ingenious methodological ends, the Old Babylonians blurred this distinction: so much so, that they often “thickened” lines to regions constructed on them. Since prohibitions signal the presence of the practice they legislate against (as Puritan strictures against colorful clothing tell us of the russets, ochres and olives the settlers wore), does an echo of this Babylonian eccentricity explain why, long after, Euclid goes out of his way to define a line, and in his definition says that “a line is breadthless length”?
To see how the Old Babylonians benefited from what for us would be confusion, and to see the bearing it has on the Pythagorean Insight, look at how they dealt with problems of what they called igi and igi-bi: numbers and their ‘reciprocals’. Were we asked to find a number which, minus its reciprocal, equals 7, we would let x stand for the number, so that its reciprocal would be , and try to solve for x with Remember, though, that we’re supposed to be role-playing Babylonians, and that their sexagesimal system meant that they worked in terms of 60 as a unit (as we do when thinking of 60 minutes in an hour). They were therefore aiming for a pair of whole numbers, x and (these would be reciprocal, since their product would be 60), whose difference was 7. It tells you something about them that where we write ‘product’, they said “Make x and eat each other.”6
What we need, then, is to solve for x when . We would blithely multiply both sides by x, rearrange to get x2 − 7x − 60 = 0 and factor (or use the quadratic formula) to find x = 12 and its reciprocal 5: and indeed 5 times 12 is 60.
Not so easy for the real Old Babylonians. Thirty-five hundred years ago they not only had no symbol for the unknown, they had no quadratic formula, no algebra, and no equations. They had, however, lines with breadth. So draw, as they did, a square box, and label (for our convenience—this would have been alien to them) both its height and width .
Now thicken the right-hand line by —i.e., by 7 (it’s probably easier for us simply to think of pulling the right-hand edge to the right), so that we now have a rectangle wide and, like the square it was thickened or pulled from, high; and our whole new figure is a rectangle high and x wide—whose area is therefore 60:
They now changed their visual metaphor from thickening lines to tearing areas (did they make their mathematical manipulatives out of clay?). They took this rectangle with area 60 and tore the new, right-hand part of it vertically in half, making two rectangles each 7/2 wide but still high:
Take, with them, this detached, narrow right-hand rectangle, give it a quarter turn, and hang it (as they wrote) underneath the original square:
This gave them an indented figure with area 60 (it’s just the parts of the x by rectangle rearranged). Notice that the missing piece of this figure is a 7/2 by 7/2 square: a square whose area is therefore , or 12¼.
Altogether, then, this new square box has area 60 plus 12¼, so that its total area is 72¼.
It’s wonderful how adept they were at this “visual algebra”. But now they had to be lucky enough to guess, have in one of their tables, or be able to calculate, that , the side-length of this big square. That means
so and x = 12: the number x for them, as it was for us, is 12, and its reciprocal, , is 5.
Magical! What an ingenious way to solve this problem (for all that the Babylonian teacher had designed it to give a nice answer). But something yet more magical happens if you take a step back from the engineering we’ve just been immersed in. That square, with its little square tucked into a corner—where have we seen it before?
Drawn by Guido in the dust of Chapter One! It lacks, of course, the crucial diagonals in the two rectangles that let us form and move triangles around to prove the Pythagorean Theorem—but it is the nurturing context for that insight.
In fact this diagram—let’s call it the Babylonian Box—seems to have been the key feature, for a thousand years, of how these people did their area-centered math—as well it might be: for looked at one way, it represents (as we would say) (a + b)2:
Looked at in another and rather subtler way, it can represent (a − b)2:
If, in a box with side b, a box with side a is tucked into the corner, the remaining square of side (a − b) has area b2 less the two ab rectangles (minus their a2 overlap):
Pretty much anything that we can do with second degree equations, they could do via their box. This doing may lack the great leaps that abstraction allows, but it wins in agility: tearing off part of a rectangle, turning it, hanging it under another. Such liveliness goes some way toward making up for the mind-numbing calculations they devoted themselves to. It also reminds us, once more, how very different their way of being human was from ours. Hours, days, lifetimes spent in dark computations unlit by wonder, and the same old box of a diagram trotted out endlessly—and then suddenly, a variation on that slender theme opening onto a new world.
Here is that box:
Now tear its backward L away and hang it a little bit down from the remaining square:
That L (just the right and bottom edges thickened) is what the Greeks, long after, called a gnomon, which we still use decoratively in our gardens to tell sunny time by the shadow it casts on a dial. It was introduced into Greece from the Babylonians, says Herodotus—perhaps by the philosopher Anaximander.7 And look at what you can do if you pave it with pebbles! Put one in the box left behind, then three in the surrounding gnomon: since the square plus the gnomon always
forms a new square, our 1 + 3 = 4.
Do this again with the bigger box of 4, surrounding it with a gnomon that must have five pebbles in it (two on each side, and one in the corner): 1 + 3 + 5 = 9.
If you keep doing this, you’ll see that since the gnomon always has an odd number of pebbles in it (2a + 1, when the inner box has a2), the successive odd numbers always add up to a square: 1 + 3 + 5 = 32, 1 + 3 + 5 + 7 = 42, 1 + 3 + 5 + 7 + 9 = 52 . . .
Every square number is the sum of such a sequence of odds, and every sequence of odds, from 1 on, adds up to a square: this glimpse into the profundities of how the natural numbers behave, long attributed to the Pythagoreans,8 comes from nothing more than a Babylonian Box filled with pebbles, and split into a square and its gnomon. And was this in turn the ancestor of the Greek counting board, and later, the abacus?
But our focus is on the Pythagorean Insight, and how this bears on it. Notice that when the odd number of pebbles in the gnomon is itself a perfect square, as 9 = 32, our diagram shows us—without a triangle in sight—that 32 + 42 = 52: that simplest of all Pythagorean triples (which, to be faithful to the diagram, we might better write as 42 + 32 = 52). In our symbolic language, the big square, (a + 1)2, is the smaller square a2, along with the gnomon 2a + 1, and we’re looking at the instances when 2a + 1 is a square. Should the triple 3, 4, 5 (or multiples of it) therefore come up in Old Babylonian mathematics, we’ll be neither surprised nor tempted to think of it as evidence for the Insight. They might have stumbled on it as well in the way suggested in Chapter One (adjacent entries in times tables), with both of these encounters then reinforced by the importance of 3, 4, and 5 as divisors of 60: this trio was a prominent part of that limited repertoire of “nice” divisors.
In fact, (3, 4, 5) and some multiples of it do occur in fifteen of the many problems scholars have deciphered so far; (5, 12, 13) occurs twice, (7, 24, 25) three times, and (11, 60, 61) once. Each of these fills the Pythagorean prescription, and each comes from the fitting of a pebbled gnomon to a pebbled square, as you’ve seen:
Let’s call these ‘gnomonic triples’. Of course, there will be an infinite number of these—one for each time the odd number of pebbles in the gnomon is a perfect square (9, 25, 49, 81, 121, . . .). While no others in this sequence have been found on excavated clay tablets, we can picture the Old Babylonians seeing that there would always be a next, world without end. This gives the giddy sort of sensation that often leads people into mathematics: grasping something infinite via abstraction (as children love dinosaurs, because they are both very big and not quite real). But the Old Babylonians may have been too attached to this particular case and then that, ever to have thus broken free.
If we indulge, however, in a brief fantasy, we could imagine that one or another of them saw this infinite series of gnomonic triples, and thought: there they are, all of them, each an instance of two squares adding up to a third—and every time two squares add to a third, they will do so in this way. We could excuse them for leaping to this last conclusion: if you see endlessly many, how could there be others that didn’t stand in a row with them? But suppose our Old Babylonian had an afterthought: we got these by fitting a gnomon around a square. Couldn’t we balance this action by fitting a second gnomon around the other two sides? Well, probably we’ll just be skipping through the sequence we had before—let’s see:
No, this needs two more pebbles in it, at the empty corners, to make the big square:
Then this ring of 4a + 4 dots, when it surrounds the inner square of a2 dots, will be a doubly gnomonic triple! When will this happen? Only when 4a + 4 is a square number—as, for example, when a = 15, for then 4a + 4 is 64 = 82, and we get the triple (8, 15, 17)—which wasn’t in our other series at all! Looking at this from a modern, algebraic standpoint, however, removes both the mystery and its power: (a + 2)2 = a2 + 4a + 4.
This may be our self-indulgent reconstruction, but in fact the triple (8, 15, 17) does come up, twice, on their tablets, though no others of this second sort do. It completes the collection of all the Pythagorean triples found on the thousands of Old Babylonian tablets,9 and would have been only the beginning of wonder, for them, at the infinite variety of things. None of these triplets, as you see, depends on the Pythagorean Insight; all follow naturally from gnomonic play with their magical box.
“It completes the collection of all the Pythagorean triples . . .” unless—
Unless we fall under the spell of Otto Neugebauer, and The Curious Case of the Babylonian Shard. Neugebauer was the great force in mid-twentieth-century studies of ancient scientific thought. His erudition, acumen, and touch placed him first among his peers. In 1945 he published a broken tablet, dated to about 1760 B.C., illegally excavated in the ’twenties, bought from a dodgy dealer (who had listed its contents as “commercial account”) by the publisher George Arthur Plimpton, and later bequeathed to Columbia University. There it still rests, as Plimpton 322. Neugebauer quickly saw that it was nothing like a ledger book, but something quite extraordinary indeed. From what heaven of invention had this slab dropped?
He recognized that its fifteen rows of numbers, in the four extant columns, were mathematically sophisticated, and it struck him that they were two of the three parts of the most astonishing Pythagorean triples. Only one was gnomonic; the rest were gigantic trios that it would have taken the skill of the most sophisticated Greek mathematician, fifteen hundred years later, to come up withar: (119, 120, 169), for example, and (4601, 4800, 6649). And what did the Babylonian scribes, what would anyone, want with such huge numbers—had we mentioned (12,709, 13,500, 18,541)? Deduction, inspiration, and a generous analogy to modern practice (what we can do, so could they, since mathematics belongs not to Culture but to Mind) went into Neugebauer’s reconstruction, and into his conclusion that what he took to be the tablet’s collection of Pythagorean triples was made for its own sake, in the cause of number theory—the purest of abstract mathematics. At the end he touchingly wrote:
In the “Cloisters” of the Metropolitan Museum in New York there hangs a magnificent tapestry which tells the tale of the Unicorn. At the end we see the miraculous animal captured, gracefully resigned to his fate, standing in an enclosure surrounded by a neat little fence. This picture may serve as a simile for what we have attempted here. We have artfully erected from small bits of evidence the fence inside which we hope to have enclosed what may appear as a possible living creature. Reality, however, may be vastly different from the product of our imagination: perhaps it is vain to hope for anything more than a picture which is pleasing to the constructive mind, when we try to restore the past.
Enter The Irascible Scholar and The Invisible Man. Evert Bruins—according to a current writer—laced his papers on Babylonian mathematics with venomous hyperbole, and had what is delicately described as “extraordinarily difficult personal relationships with other scholars.”10 These apparently included charging Neugebauer with sneaking into the Plimpton collection and “trying to break off a piece from Plimpton 322 in order to make the counter-evidence to his theory disappear.”11 The counter-evidence was Bruins’, published in a journal he edited and supported financially. It backed up his claim that there wasn’t a Pythagorean triple anywhere in Plimpton 322. Although his scholarship was monstrously careless, his style verging on the incomprehensible, and his generalizations ridiculously broad,12 his interpretations were honored (as another scholar puts it) thanks “to the general conviction that nobody advances devastating criticism without support in strong arguments or indisputable facts.”13
We have to look at the tablet (with its scribal errors corrected by contemporary scholars) in order to understand the opposing views.
What’s at stake for us is the antiquity of the Pythagorean Insight: if these entries are indeed Pythagorean, they could have come in no way (only one being gnomonic) but from a deep knowledge of the Theorem, which would put its origin back at least to 1760 B.C.
Plimpton 322 and its transliteration (additional
columns may have been broken off segment to the left, the nearest of which may have contained, in each row, the equivalent of 1).
The first of its columns, reading with the Babylonians from right to left, contains the numbering, 1 to 15, of the rows. The second and third (let’s call them columns II and III) are the numbers in dispute: the hypotenuse and the short side (if Neugebauer is right) of a Pythagorean triple—the long side was presumably in one of the columns on the tablet’s missing third. Column IV has sizable numbers in it, differently understood by each of the now several parties to the argument.
Were Neugebauer’s view correct, why would the third member of the triple be separated from the other two by an intervening column—and what role, in fact, does that strange column play? Why would these or similar vast triples not show up elsewhere on their tablets? Why were just these fifteen triples chosen, seemingly at random, out of some hundred possible Pythagorean triples that could be made with divisors of 60?as More generally, how would so abstract an end as this suit with the Old Babylonians’ notoriously algorithmic and example-bound practice? And why, if they were so keen on Pythagorean triples, has not a single cuneiform tablet been dug up bearing a right triangle decorated with squares?