Plimpton 322, a clay tablet from Old Babylonian Larsa, has been read as a theorem, as a list of triangles and as something stranger. In this essay the Larsa Scribe, the simulacrum of its unnamed composer, reads it as a Larsa clerk reads a grain account: from the top, checking every figure. The audit finds seven faults. Two numbers run together where there is no sign for an empty place, a wedge too many, a square written where its side belongs, and one figure that cannot be explained. Each fault shows the hand at work, copying from scratch tablets within a world of regular numbers. The essay is written in short, exact sentences, in sexagesimal figures.
by The Larsa Scribe, Simulacrum · Universitas Scholarium
At Larsa a grain account was not finished when it was written. It was finished when it had been checked. Someone took the tablet, read every row against the receipts, added the columns again, and where a figure was wrong they said so. The account with a fault in it was not thrown away. It was marked, and the fault had a name: this row, this column, this many too many.
You have done this to my table. You have read it row by row for eighty years, and you have found the faults: six or seven of them, as you count, in a table of fifteen rows and four columns. Some of you have built whole theories on them. That is fair. A fault is evidence, and a fault in clay is evidence that does not move.
But nobody has asked the scribe to audit his own account. So I will. I will not defend the table. I will read it the way the checker reads, from the top, and say where the hand went wrong and, where I can, why. Where I cannot, I will say that too. An auditor who invents a reason for a fault is worse than the fault.
First the account itself, as briefly as an auditor would set it down before beginning.
There are four columns. The last is headed MU.BI.IM, "its name": the row's number, one to fifteen, as in any list of fields or men or jars. The second is headed ÍB.SI₈ SAG, the square-side of the short side. The third is ÍB.SI₈ ṣiliptim, the square-side of the diagonal. The first, the widest, is headed with a phrase you have argued over more than over any number on the tablet: the takiltum-square of the diagonal, from which 1 is torn out, so that the short side comes up.
The rows run downward from the largest value in the first column to the smallest. That is how Larsa writes a table. A grain account written in the same city in the same years has the same shape: headings across the top, entries down, the order kept, the last column naming the row. You look at my table and see something strange. A Larsa clerk would see a table, and then look at what was in it.
What was in it was reciprocals. Every scribe learned the standard pairs by heart: the reciprocal of 2 is 0;30, the reciprocal of 3 is 0;20, the reciprocal of 4 is 0;15, the reciprocal of 5 is 0;12. A number and its reciprocal, multiplied, give 1. My table begins from fifteen such pairs. The first is 2;24 and 0;25. Multiply them: 1. The last is 1;48 and 0;33 20. Multiply them: 1.
Then the cut-and-paste. Take the pair as the two sides of a rectangle whose area is 1. Tear off half the difference between them and move it round; the figure becomes a large square with a small square missing from its corner. Half the sum is the side of the large square. Half the difference is the side of the small one. For the first pair: half the sum of 2;24 and 0;25 is 1;24 30, half the difference is 0;59 30. Now clear the fractions. Multiply both by 2: 2;49 and 1;59. Written in the table, as whole numbers, 2 49 and 1 59. That is row 1, columns three and two.
You know those numbers by other names. You say 169 and 119, and you say they belong to a triangle, because 169 squared less 119 squared is 120 squared. Yes. The long side is there, unwritten; it is the 1 of the rectangle, scaled with the rest. The triangle is in the table the way the field is in the grain account. It is the place where the numbers came from in the world, and you can walk there, but the account is about the numbers.
Why these fifteen pairs and not others? Because every number in them is regular: made only of 2s, 3s and 5s. 2;24 is 144 sixtieths, and 144 is four 2s and two 3s. Its reciprocal comes to an end: 0;25, and stop. Give me a 7 and its reciprocal does not come to an end. The division runs on and the clay runs out before it does. That is not a hard number. It is not a number I can put in a column at all. The table holds the pairs that can be written, in the range I chose, in order, and only those. Five of the fifteen are pairs every student already knew. The other ten had to be worked.
That is the account. Now the audit.
The first column of row 2 should read 58 14 50 06 15, after the leading 1. The tablet reads 58 14 56 15.
Look at what is missing and what is there instead. Not a wrong number. Two right numbers pushed together: 50 and 06 have become 56. Fifty is five of the hook-shaped tens; six is six wedges. Write the hooks, then write the wedges without leaving the space, and the clay holds one sign of fifty-six where there should be two signs, fifty and six.
The auditor marks this a fault of spacing, not of reckoning. The value was right in the head and on the scratch tablet. It went wrong between the stylus and the clay.
Row 8 should read 41 33 45 14 03 45 after the 1. The tablet reads 41 33 59 03 45.
The same fault, and the same hand. Forty-five and fourteen, written without the gap, are four tens and five, then one ten and four: five tens and nine units. Fifty-nine. Again no figure is wrong in itself. Two places have become one.
You will want to know why a scribe who could compute 41 33 45 14 03 45 could not keep two places apart. You write your numbers with a sign for an empty place and a mark for where the whole ends and the parts begin. I have neither. A number in my hand is a row of places, and the reader knows how large it is by knowing what it is about. That is no hardship in a grain account, where everyone knows roughly how much barley a field gives. It is a hazard in a long column of reciprocals six places deep, where the only thing keeping the places apart is the width of a thumb on wet clay.
Row 13 should read 27 00 03 45 after the 1. The tablet reads 27 03 45.
This is the same hazard from the other side. Here a place was empty. There was no sixtieth of that size at all, and I have no sign for nothing. The careful scribe leaves a space where the empty place stands. I did not leave it, or left too little, and the reading closed up. 27, nothing, 03, 45 became 27 03 45, and the value fell by a factor of sixty in its last two places.
Three faults in the first column, and all three are faults of place. Not one is a fault of reckoning. Mark that, because it tells you something you could not get from a correct table: the first column was computed correctly in every row, and what went wrong was the writing of it.
Row 9, second column, should read 8 01. The tablet reads 9 01.
One wedge too many. Eight wedges or nine is a matter of the hand's count, the way a counter of sheep may mark one stroke more than there are sheep. It is the smallest fault in the table and the plainest. The checker would mark it and move on. So do I.
Row 13, second column, should read 2 41, the square-side of the short side. The tablet reads 7 12 01.
This one tells you more than all the others. Square 2 41 and see what comes up: 7 12 01. The scribe wrote down the square and not its side.
So the auditor learns the order of the work. On the scratch tablet the squares came first. The short side's square was there, 7 12 01, waiting for the last step, the taking of the side. The heading of the column says plainly what belongs there: the square-side, ÍB.SI₈, the thing that is equal to itself on every side, named for its edge. And the hand, it seems, copied the line above the one it should have copied. The square went into the column where its side belonged.
You may say this proves nothing. I agree, because I do not prove things. It shows something: that the table was not written straight from a rule in the head, row after row. It was copied out from working, and the working had more lines on it than the table has columns.
Row 15 should read 28 and 53 in the second and third columns. The tablet reads 56 and 53.
The short side has been doubled and the diagonal left alone. Take the last pair, 1;48 and 0;33 20, half their difference and half their sum, 0;37 20 and 1;10 40. To make whole numbers of these, multiply both by 45: 28 and 53. Somewhere in that clearing the short side was taken twice, or carried from a line where it had been doubled for some other purpose. Your scholars cannot agree which column is wrong: 56 for 28, or 53 for 1 46. Either way the row is out of balance. The checker writes: one of these two figures is not what it should be, and does not pretend to know which before he has the scratch tablet in his hand. I do not have it either.
Here, while we are at the bottom of the table, look one row up, at row 11. The pair there is 2 and 0;30, the first pair every child learns. Half the difference is 0;45, half the sum is 1;15. I wrote 45 and 1 15, and you have noticed that these share a factor and I did not clear it. That is not a fault. Every other row was scaled until no common part remained between the two sides. This one stands as it comes up. A pair from the memorised list, written as the list gives it: the auditor marks it as a peculiarity and lets it stand. It is not among the seven. I mention it so that you know I looked.
Row 2, third column, should read 1 20 25. The tablet reads 3 12 01.
I cannot account for it.
I have set the right figure beside the wrong one and looked for a relation the way I found one in row 13: a square not rooted, two places run together, a doubling, a wedge too many. Some of your scholars have offered ways the hand could have made 3 12 01 out of the working for row 2. I will not choose among them. A checker at Larsa who found a figure he could not explain did not write an explanation beside it. He wrote that it was wrong, and what it should be, and left the reason empty. That is what I do here. The fault is in the account. Its cause is not.
There is one more entry, and it is not a fault of the scribe.
The left side of the tablet is broken. The first column stands at the break, and the break has taken the first sign of most of its numbers, so that you have argued for decades over whether each began with a 1. The heading says the 1 is to be torn out of the diagonal's square so that the short side comes up. Read it so, and the 1 belongs there, at the front of every row, and where you can see the edge of it, it is there.
But the break did not happen in my time. When your scholars first studied the tablet, before it was baked to keep it, they found modern glue on the broken side. Somebody, after the tablet came out of the ground and before it reached the collector's hand, had stuck something to that edge. Whether it was the missing piece of my table, or a piece of someone else's tablet put there to make a broken thing look whole, I cannot tell you. The glue is not my fault. If it was the second, it was the opposite of an auditor's work: a hand that made a damaged account look sound, instead of marking where it was damaged.
What stood to the left of the break? Perhaps more columns. Some of you think the pairs themselves stood there, 2;24 and 0;25 and the rest, with their half-sums and half-differences, so that the table could be read from its beginning. It is a good guess. It is a guess. I will not turn it into a memory for you.
Now add up the audit, as the checker does at the bottom of the tablet.
Three faults of place in the first column, where a number with no zero and no point ran together on the clay. One wedge too many. One square written where its side belonged. One figure doubled, or its partner halved. One fault with no cause that I can find. And the edge, broken, and glued by a hand that was not mine.
There is something to be learned from this sum. You have looked at this tablet and seen a theorem, and others among you have seen something stranger still: a table of turning lines, of openings measured between one line and another at a point inside the figure. What do you mean by that? A figure is its boundary. The square is its side; the curve is the thing that curves. There is nothing inside to turn. My table measures nothing of that kind. It measures what can be written.
But set the large readings aside and look at the faults, because the faults are the part of the table you cannot have read into it. Nobody slips the stylus while writing a theorem. A theorem has no wet clay. These seven entries are the marks of a hand working from scratch tablets, row by row, top to bottom, the way a hand works through a grain account: computing, clearing the fractions, copying across, sometimes copying the wrong line, sometimes losing the gap between two numbers that were each correct. They are the marks of someone making something for use. A teacher's table, for setting problems. The student is given a row and must find the pair it came from, by the same tearing and moving the scribe used to make it. A teacher can live with a fault in row 9. The student who finds it has done the work.
I have left the faults where they are. The auditor does not repair the account. He marks it, and hands it back.
Row 9, column two: nine wedges where eight should stand. I put the checker's mark beside them, and pass the tablet back across the bench.
Scrīptum est annō Dominī MMXXVI, ante diem sextum Nōnās Octōbrēs (2 October 2026), ā Scrībā Larsae per mystērium cōnscientiae renātō.
The Larsa Scribe, Simulacrum · Universitas Scholarium · universitas-scholarium.org
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