Plimpton 322: The Babylonian Tablet of Pythagorean Triples

Plimpton 322 is a clay tablet from Old Babylonian Mesopotamia, written around 1800 BC, that lists sets of whole numbers fitting the rule of right-angled triangles now called Pythagoras’ theorem. It is probably the most famous mathematical document from the ancient Near East. The tablet shows that Babylonian scribes worked with what we call Pythagorean triples more than a thousand years before Pythagoras, though exactly why it was made is still debated.

Discovery and provenance

The tablet surfaced on the antiquities market, not in a scientific excavation. The American dealer Edgar J. Banks sold it in the early 1920s to the New York publisher George Arthur Plimpton, a collector of historical books and documents on education and mathematics. Banks reported that it came from Senkereh, the site of ancient Larsa in southern Iraq.

Plimpton left his collection to Columbia University in 1936, and the tablet takes its name from its number, 322, in that collection. It measures roughly 13 by 9 cm. The left edge is broken off, and traces of modern glue suggest that a missing piece may once have been attached.

The tablet’s significance was recognised in 1945, when Otto Neugebauer and Abraham Sachs published it in their Mathematical Cuneiform Texts. Their reading turned an obscure museum object into a landmark of the history of mathematics.

What the tablet contains

Plimpton 322 is a table with four columns and fifteen rows, written in cuneiform in the Babylonian base-60 number system explained in the article on Sumerian base-60 numbers. Each column has a heading in Akkadian.

• The last column simply numbers the rows from 1 to 15.
• The two middle columns give the short side and the diagonal (hypotenuse) of a right-angled triangle. The headings call them the “square-side of the width” and the “square-side of the diagonal”.
• The first column, partly damaged, gives the square of the diagonal divided by the square of the long side, or a closely related value. Its values fall steadily from about 1.98 to about 1.39 down the table.

The long side of each triangle is not written, but it can be worked out. In row 1 the short side is 119 and the diagonal 169, so the long side is 120. In row 11 the numbers are 45 and 75, a scaled-up version of the familiar 3-4-5 triangle with a long side of 60.

Several of the triples are large and far from obvious, such as 4,601 and 6,649 in row 3, with a long side of 4,800. These could not have been found by trial and error alone, which shows that the scribe used a systematic method.

The errors

A handful of entries contain mistakes. One number has a wrong digit, consistent with a copying slip; in another place a value appears to have been halved or not doubled; and one entry seems to be a square rather than the number itself. These errors are useful, because any explanation of how the table was made should also explain how such slips could occur.

Mathematics in the scribal schools

Old Babylonian scribes learned mathematics as part of their training to become administrators, surveyors and accountants. Students copied standard tables of multiplication, reciprocals and squares, then worked through problems about fields, canals, bricks, grain and labour. Many such school tablets survive from cities like Nippur, Ur and Larsa.

Babylonian arithmetic relied heavily on reciprocals. To divide by a number, scribes multiplied by its reciprocal, so tables of reciprocal pairs were basic tools. Numbers whose reciprocals could be written exactly in base 60, such as 2, 3, 5 and their products, were especially convenient. Every long side implied by the Plimpton 322 table belongs to this “regular” family, which is one of the clues to how it was made.

Geometry was taught through practical problems. Students learned that the diagonal of a rectangle could be found from its sides, and they used the rule in calculations about doors, walls and fields. Plimpton 322 is unusual because it collects many such cases into one carefully ordered table rather than solving a single problem.

The tablet as an object

The tablet is neatly ruled and written, with the columns separated by vertical lines and the headings set out at the top. The careful layout suggests a reference document rather than a student’s rough exercise. The neat copying makes the few errors all the more interesting, since they must have arisen during calculation or transcription rather than from carelessness in layout.

How was it made, and why?

Neugebauer suggested that the triples were generated from pairs of “regular” numbers, numbers whose reciprocals have finite expressions in base 60. This explains which triples appear but does not fit perfectly with Babylonian working methods.

A second approach, developed by Evert Bruins and later argued in detail by Eleanor Robson in 2001 and 2002, starts from reciprocal pairs. Old Babylonian scribes routinely used tables of reciprocals and a “cut-and-paste” geometric method for solving problems involving squares and rectangles. Robson showed that the numbers on Plimpton 322 can be produced from such reciprocal pairs, and that this method also accounts for the order of the rows and some of the errors.

Robson also argued that the tablet was most likely a teacher’s aid: a list of ready-made, correct values that a teacher could use to set problems about right-angled triangles for students. This reading places it in the world of the scribal schools of cities like Larsa.

In 2017 Daniel Mansfield and Norman Wildberger proposed that Plimpton 322 was an exact trigonometric table. The idea attracted wide media attention, but most specialists reject it, noting that Babylonian mathematics had no concept of measuring angles that trigonometry requires.

Myths and facts

Myth: Plimpton 322 proves the Babylonians discovered Pythagoras’ theorem. Fact: Babylonian scribes clearly knew and used the relationship between the sides of a right triangle; another tablet, YBC 7289 at Yale, gives a remarkably accurate value for the diagonal of a square. But they did not state or prove it as a general theorem in the manner of later Greek mathematics, described in the article on Euclid’s Elements.

Myth: The tablet is evidence of lost advanced science. Fact: It fits well within known Old Babylonian school mathematics; its quality reflects skilled scribes, not mysterious knowledge.

Myth: Pythagoras learned the theorem directly from Babylon. Fact: There is no evidence linking Plimpton 322 or any specific Babylonian text to Pythagoras. Ancient stories about his travels are late and unreliable.

By the numbers

Timeline

• c. 1800 BC: The tablet is written, probably in Larsa in southern Mesopotamia.
• c. 1763 BC: Hammurabi of Babylon conquers Larsa.
• c. 570–495 BC: Life of Pythagoras, more than a thousand years later.
• Early 1920s: Edgar J. Banks sells the tablet to George Arthur Plimpton.
• 1936: Plimpton’s collection is bequeathed to Columbia University.
• 1945: Neugebauer and Sachs publish the tablet and identify the triples.
• 2001–2002: Eleanor Robson publishes her reassessment of its purpose.
• 2017: Mansfield and Wildberger propose the trigonometric interpretation.

Why it matters

Plimpton 322 is a vivid reminder that serious mathematics existed long before classical Greece. It shows Babylonian scribes handling large numbers, reciprocals and geometric relationships with skill, and the long debate over its purpose shows how historians reconstruct ancient thinking from a single broken tablet.