Plimpton 322 and Babylonian Pythagorean Triples

Plimpton 322 is a Babylonian clay tablet, written in cuneiform around 1800 BCE, whose 15 rows of base-60 numbers correspond to Pythagorean triples: whole-number sides of right triangles such as 119, 120 and 169. It is strong evidence that Babylonian scribes knew the relationship a² + b² = c² more than a thousand years before Pythagoras.

Below you will learn to read its numbers yourself, see all fifteen rows as modern triples, and weigh the rival explanations of why it was made.

Plimpton 322 at a glance

How to read Babylonian numbers

Babylonian scribes counted in base 60 (sexagesimal), pressing numbers into wet clay with a reed stylus. Two signs were enough: a vertical wedge worth 1 and a corner wedge worth 10, grouped to make any digit from 1 to 59. Digits are written in place value like ours, but each place is worth 60 times the next.

Modern scholars transliterate the digits with commas between places, so 1,59 means 1 × 60 + 59 = 119 and 2,49 means 2 × 60 + 49 = 169. A semicolon marks where the whole part ends and the fractional part begins:

There is a catch: Old Babylonian scribes wrote no semicolon and had no sign for zero. The same wedges could mean 119, or 1 + 59/60, or 7,140, and the reader judged the scale from context, as historians must on Plimpton 322.

What the four columns say

Here are the first five rows in modern digits, one base-60 digit per group. The long side of each triangle is not on the tablet; you have to work it out.

• Column IV just numbers the lines, 1 to 15.
• Column II has a heading that mentions the "width" and gives the short side s of a right triangle (or rectangle).
• Column III has a heading that mentions the "diagonal" and gives the hypotenuse d.
• Column I is a fraction between 1 and 2: the square of the diagonal divided by the square of the missing long side l.

Because the edge is broken, some scholars drop the leading 1, which turns column I into s²/l². The two readings differ by exactly 1, since d²/l² − s²/l² = 1 is just the Pythagorean theorem divided by l². Eleanor Robson's reading of the damaged heading, roughly "the holding-square of the diagonal, from which 1 is torn out, so that the width comes up", describes exactly that step. Every implied long side (120, 3,456, 4,800 …) is a regular number, so column I always has a finite base-60 expansion.

All 15 rows of Plimpton 322

Here is every row as a modern triple, with the scribe's errors corrected. The long side l and the last two columns are our additions: p and q generate the row through Euclid's formula (s = p² − q², l = 2pq, d = p² + q²), and the angle is opposite the short side, to the nearest hundredth of a degree.

Column I falls steadily, so the triangles run from nearly isosceles (close to 45°) down to about 32°, in steps of roughly one degree. Row 11 is the 3-4-5 triangle scaled by 15 and row 15 is twice 28, 45, 53; every other row is a primitive triple. Nobody finds 12,709, 13,500, 18,541 by trial and error, so the scribe had a method. In 1964 Derek de Solla Price noticed that there are exactly fifteen pairs of regular numbers p and q with no common factor, q below 60 and p/q between 9/5 and 12/5, and sorted by size they give the tablet's rows in order.

The scribe's six mistakes

Most scholars count six errors, which are useful clues to how the numbers were computed:

• Row 2, column I: the digits 50 06 are run together as 56.
• Row 2, column III: 3 12 01 instead of 1 20 25 (4,825), not explained by any single slip.
• Row 8, column I: the two digits 45 14 are replaced by their sum, 59.
• Row 9, column II: 9 01 (541) is written for 8 01 (481).
• Row 13, column II: 7 12 01 (25,921) for 2 41 (161). Since 161² = 25921, the scribe copied the square, not its root.
• Row 15: 56 and 53 do not belong together. Either 53 should be 1 46 (106) or 56 should be 28.

What was Plimpton 322 for?

Nobody knows for certain. The four main interpretations answer two separate questions: how the numbers were calculated, and why anyone wanted them.

1. Number theory (Neugebauer and Sachs, 1945)

Otto Neugebauer and Abraham Sachs, who first published the tablet's mathematics in 1945, suggested the scribe chose regular numbers p and q and computed the sides with a rule like Euclid's formula, as a study of whole-number solutions of s² + l² = d². Critics such as R. Creighton Buck and Eleanor Robson reply that this leaves column I, the order of the rows and the errors unexplained, and that pure number theory looks out of place among the practical problem texts of the period.

2. Reciprocal pairs and cut-and-paste geometry

The explanation favored by many specialists today starts from a single number x and its reciprocal 1/x, the kind of pair listed in the reciprocal tables that every scribe learned. The idea goes back to Evert Bruins in 1949 and was developed by Jöran Friberg and Eleanor Robson, among others. From the pair you compute

which are the short side and diagonal of a right triangle whose long side is 1. For row 1, x = 2; 24 = 12/5 and 1/x = 0; 25 = 5/12:

Scale up by 120 to clear the fractions and you have 119, 120 and 169. The geometry behind it is the Babylonian "cut-and-paste" method for completing the square, which appears in other problem texts of the same period.

This reading explains column I (the big square: 1 plus the small square), the order of the rows (decreasing x), the headings and several of the errors. Two other Old Babylonian tablets, MS 3052 and MS 3971 in the Schøyen Collection, compute a rectangle's width and diagonal starting from a reciprocal pair, which strongly supports the idea.

3. A teacher's list (Robson, 2001–2002)

Robson also asked who would make such a table. The tablet's format matches administrative documents from Larsa, so she argued that its author was a trained scribe and that the table was most likely a teacher's list of numbers for setting right-triangle problems with exact answers. Her 2002 article on it won the Mathematical Association of America's Lester R. Ford Award.

4. A trigonometric table (Mansfield and Wildberger, 2017)

Neugebauer and Sachs had noticed that column I is the square of a trigonometric ratio (the secant, or the tangent without the leading 1; the two readings differ by the Pythagorean identity sec²θ = 1 + tan²θ). In 2017 Daniel Mansfield and Norman Wildberger argued in Historia Mathematica that the tablet is "exact sexagesimal trigonometry", a table of exact side ratios built without angles, and the publicity around the paper called it superior in some ways to modern trigonometry.

Historians were skeptical. The trigonometric observation was decades old; there is no evidence that Old Babylonian scribes measured angles at all; a tablet with six errors can hardly be "completely accurate", as the publicity claimed; and the reciprocal-pair method already explains the numbers with techniques attested elsewhere. Robson had earlier called the trigonometric reading "conceptually anachronistic", and the writer Evelyn Lamb summed up the objections in Scientific American. Trigonometry is a fair modern way to describe the numbers, as the angle column shows, but probably not how the scribe thought of them.

Where things stand

YBC 7289: the square root of 2 tablet

YBC 7289, in the Yale Babylonian Collection, is a small round school tablet from roughly 1800–1600 BCE. It shows a square with both diagonals; one side is labeled 30, and two numbers run along a diagonal.

The first number is the ratio of a square's diagonal to its side, which by the theorem is √2 for any square (a square is two 45-45-90 triangles):

The true value is √2 ≈ 1.4142136, so the scribe is too small by about 0.0000006, roughly one part in 2.4 million. The second number, 42;25,35, is 30 times the first: the diagonal of a square with side 30, about 42.4264. Since place value is not marked, the side could also mean 30/60 = 1/2, making the second number 0; 42, 25, 35 ≈ 0.70711 ≈ 1/√2. Either way, the scribe was applying the diagonal rule, and the same value was still in use in Ptolemy's Almagest almost two thousand years later. No fraction can be exactly right, because √2 is irrational.

What the tablet tells us about the theorem's history

Plimpton 322 contains no proof and states no general rule. But fifteen exact triples with five-digit sides, produced by a consistent method, are hard to explain unless the author knew that the square on the diagonal equals the squares on the two sides. Other tablets agree: IM 67118, excavated in 1962 near Baghdad, finds the sides of a rectangle from its area and diagonal, then checks the answer with the rule. So historians date the theorem to at least the Old Babylonian period, about 1,200 years before Pythagoras. The wider story, including India, China and Egypt's much weaker evidence, is on the history of the Pythagorean theorem page and the Egyptian rope stretchers page.