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The One That Broke Off

Around 1800 BCE, somewhere near Larsa in southern Iraq, someone ruled a clay tablet into four columns and filled it with fifteen rows of numbers. It is the most famous mathematical object to survive from the ancient world, and the argument about what it is for has run for eighty years. The numbers can be put in your hands. Every one of them recomputes here from two columns; the scribe's seven slips surface as arithmetic that fails; and the single digit that would settle the argument broke off the left edge, where no calculation can reach it.

Start with what is not in dispute. Columns II and III of Plimpton 322 hold two whole numbers per row. Call them s and d, the short side and the diagonal of a right triangle. Nothing on the tablet gives you the third side. You can get it anyway: subtract the squares and take the root.

Do that fifteen times and something happens that has no business happening by chance. Every single time, the root comes out a whole number. The tablet is a list of Pythagorean triples, written more than a thousand years before Pythagoras, and it says so in a way you can check yourself in the next thirty seconds.

The tablet

Numerals below are drawn from the data, not photographed: each digit is a count of ten-wedges and unit-wedges, exactly as base sixty writes it. Tap any row to take it apart. The left edge of the tablet is broken away, and the greyed leading places are the editor's restoration, not the clay.

# Column I
the ratio		 Column II
short side s		 Column III
diagonal d		 check
√(d² − s²)

Sexagesimal places are written with spaces between them, as editors of these tablets do. 1 20 25 means 1×60² + 20×60 + 25, which is 4825.

The one that broke off

Column I is a ratio, and it is the reason the tablet is argued about. Its surviving digits are a fraction. What is missing, on the broken left edge, is whether that fraction was preceded by a 1.

This is not a small ambiguity, and it is not one that more computation can fix. With the leading 1, column I reads d²/l², the square of the diagonal over the square of the long side. Without it, the same digits read s²/l², the square of the short side over the same denominator. And because d² = s² + l², those two quantities differ by exactly one, in every row, always. The digits after the point are identical under both readings. The clay that would have told us apart is gone.

Where the rows came from

Take any row and compute x = (d + s)/l. Then 1/x is exactly (d − s)/l, and the two multiply to 1 on the nose. Every row of this tablet is one half of a reciprocal pair, run through s/l = (x − 1/x)/2 and d/l = (x + 1/x)/2.

Reciprocal pairs are not a modern reconstruction imposed on the tablet. They are the first thing a trainee scribe in this time and place learned by heart, because division in base sixty was multiplication by a reciprocal. And here is the constraint that makes the reading bite: in base sixty, 1/x terminates only when x is built from the primes 2, 3 and 5 and nothing else. A scribe could write down the reciprocal of 2, 3, 4, 5, 6, 8, but never of 7.

All fifteen generating numbers are of exactly that kind. They run downward without a single break, from 2;24 in row 1 to 1;48 in row 15, and not one of them exceeds four sexagesimal places. That is not a fact about right triangles. It is a fact about what a person holding a reed stylus could actually compute.

Seven slips

Press as the clay reads above and seven cells change. They are the scribe's own mistakes, and they are more informative than the fourteen hundred correct digits around them, because a mistake carries the shape of the method that produced it.

Two of them are the same slip twice. In row 2, column I should run … 14 50 06 15; the clay has … 14 56 15. In row 8 it should run 41 33 45 14 03 45; the clay has 41 33 59 03 45. Look at what happened: 50 and 06 became 56. And 45 and 14 became 59. Two adjacent places were added together into one. That is a mistake you can only make in a place-value system where the places are written side by side with nothing between them, which is precisely what this notation is, and precisely why it needed a zero it did not yet have.

Row 13 is the one that gives the method away. Column II should read 2 41, which is 161. The clay reads 7 12 01, which is 25921. That is 161 squared, exactly. The scribe was working from squares and, in this one row, wrote down the square instead of taking its root.

Row 15 is the honest hard case, and it does not resolve. The clay reads 56 and 53. Either the short side is wrong and the triple is 28, 45, 53, or the diagonal is wrong and the triple is 56, 1 30, 1 46. Both are consistent with everything else on the tablet. Robson prefers the first, following Friberg; Neugebauer and Sachs preferred the second. The table above shows the first, because a table must show something, and that choice is an editor's, not the clay's.

Is it a trigonometric table?

This is the reading the tablet is famous for outside the field, and it was revived in 2017 to a great deal of press. In its usual form the claim is that column I is tan² or 1/cos², and that the rows are arranged so the acute angle falls by roughly one degree a line.

The angles are computable, so the second half of that claim is checkable without taking anyone's word. Here is what the fifteen rows actually do.

The steps are not a degree apart. They are not evenly anything: the smallest is under half a degree and the largest is nearly two, a spread of more than four to one, and they fall into two loose clusters rather than a progression. A table built to step through angles is not what this looks like.

There is a second objection, and it is the heavier one, because it is not about the numbers at all. Measured angle, in the sense the claim requires, is not attested in Mesopotamian mathematics of this period. Slopes were handled as ratios of run to rise, and right angles were distinguished from what Robson calls wrong angles with a tolerance of ten or fifteen degrees. Reading these fifteen rows as a trigonometric table asks the tablet to be doing something for which there is no other evidence anywhere in the corpus, in order to explain something reciprocal pairs already explain using only what we know these scribes were taught.

Two cautions on the paragraph above. The angle numbers are ours and are exact; the historical claim about attested angle measurement is Robson's, and it is an argument from the surviving record, which is a strong argument but not a proof. And the uneven steps do not by themselves refute a trigonometric purpose. They refute one common description of it.

What it probably is

Robson's reading, which is the one the evidence outside the numbers supports best, is unglamorous and much more human than a lost trigonometry. The tablet's format, its landscape shape, its column headings, and its descending first column all match a class of administrative tables written in Larsa between about 1822 and 1784 BCE. Its methods, reciprocal pairs and completing the square and dividing out regular factors, are the ordinary furniture of a scribal school. And its structure, one set-up repeated fifteen times with different well-behaved numbers, matches a teacher's problem list: a way to set the same exercise over and over and check the answers without redoing the work.

Plimpton 322 is also a repetition of the same mathematical set-up fifteen times, each with a different group of well-behaved regular numbers. It would have enabled a teacher to set his students repeated exercises on the same mathematical problem, and to check their intermediate and final answers without repeating the calculations himself.
Robson 2002, p. 117.

That is a smaller claim than a lost trigonometry, and it is worth being clear that it is still a claim. The tablet does not say what it is for. What it says is fifteen rows of numbers, and those we now have exactly.

Two things we could not close

The first is a count. Robson writes that of the fifteen reciprocal pairs, only five pairs occur in the standard list of reciprocals a scribe memorised. Counting the same way, taking a pair to occur when either member is a head number of that standard table up to a power of sixty, we get four, not five: rows 1, 6, 11 and 13. Adding the three regular numbers her Figure 9 omits does not produce a fifth. We could not reconstruct the reading of occur under which the count is five, and we are not asserting the paper is wrong; the discrepancy may be entirely in our reading of the phrase. It does not touch her argument either way, since her point is that most of the pairs are not in the standard list but were easy to derive.

The second is exhaustiveness. It is tempting to say the fifteen rows are every row available. They are not. Counting the regular numbers in the tablet's own range that terminate within three sexagesimal places, there are 28, and the scribe used 15 of them. Thirteen usable generators were left on the table, including 2;16 32 and 1;53 46 40. Whatever governed the selection, it was not simply taking everything that fit.

Show the check

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