3 · River civilisations science and technology

Babylonian mathematics

about 1800 BC Larsa, Mesopotamia (now Iraq)

In short

The Babylonians counted in base 60, and from them we have inherited the 60 minutes in an hour and the 360 degrees in a circle. They could solve difficult problems, such as working out the area of a field or the square root of a number. A famous tablet, called Plimpton 322, contains a table of numbers linked to right-angled triangles. It was written more than a thousand years before Pythagoras was born.

Why it matters Every time we look at a clock we are using a Babylonian idea without knowing it.

Source: INAF Media, “Il teorema di Pitagora al tempo dei Babilonesi” (in Italian)

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Illustration: Babylonian mathematics (Larsa, Mesopotamia (now Iraq)) — Illustration created with AI for Storiascopio

Where: Larsa, Mesopotamia (now Iraq)

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The story

A tablet bought by a collector

At the start of the twentieth century an American publisher from New York, George Arthur Plimpton, collected ancient manuscripts. Around 1922 he bought from an antiquities dealer a clay tablet, broken at one corner, as big as the palm of a hand, written in cuneiform. He later gave it to Columbia University, where it is kept today under the label “Plimpton 322”. For years nobody paid it any attention. In 1945 two historians of mathematics, Otto Neugebauer and Abraham Sachs, studied it and realised that it contained a table of very particular numbers. Since then scholars have not stopped discussing it.

The tablet, written probably around 1800 BC and perhaps coming from Larsa, in the south of Iraq, has fifteen rows of numbers. According to Neugebauer and Sachs, these numbers are linked to right-angled triangles, that is, to triples of numbers like 3, 4, 5 or 5, 12, 13 for which the square of the longest side equals the sum of the squares of the other two. If so, the Babylonians knew that property more than a thousand years before Pythagoras. Other scholars, such as Eleanor Robson, maintain that the tablet is above all a list of numbers prepared by a teacher to build problems for pupils. The question remains open, but everyone agrees on one thing: behind those numbers lies a sophisticated mathematics.

Counting in base 60

The Babylonians did not count as we do. Our system is base ten: after nine, we move to ten. Theirs was base sixty, called sexagesimal. We do not know for certain why they chose it: perhaps because 60 divides exactly by many numbers (2, 3, 4, 5, 6, 10, 12, 15, 20, 30), and is very handy for dividing into equal parts. A curious hypothesis is that it comes from a way of counting on the fingers: with the thumb of one hand you touch the twelve finger-joints of the other four fingers, and with the other hand you keep count of the rounds.

Above all, the Babylonian system was “positional”: the value of a digit depends on the place it stands in, as in our 305, where the 3 is worth three hundred. This makes calculations with enormous or tiny numbers easy, and it was a fundamental invention. For a long time, however, there was no zero: where today we write 305, the scribes left an empty space, and only much later, around the 3rd century BC, did they invent a sign to mark the empty place.

What they could do

Babylonian scribes, trained in the schools of Nippur, Ur and Larsa, used ready-made tables: of multiplications, of reciprocals (the numbers to divide by), of squares and roots. With these tools they solved practical problems, such as calculating the area of a field, the volume of a heap of earth or a canal, the number of bricks for a wall, or how long it takes for a sum of money to double at a certain interest.

They could also solve quadratic equations, with methods that resemble those studied today. Another famous tablet, YBC 7289, kept at Yale University, shows a square with its diagonals and a number written in base 60 that gives the square root of 2 to an accuracy of six decimal places: 1.41421296. Whoever calculated this number, more than 3,600 years ago, had no calculator.

The sky, time and the clock

The Babylonians were also great observers of the sky. For centuries they noted down the movements of the Moon and planets, eclipses and the risings of the stars, and learned to predict them by calculation. Their numbers were taken up by Greek astronomers, in particular by Hipparchus in the 2nd century BC and Ptolemy in the 2nd century AD, who adopted the sexagesimal system for fractions. From here comes a legacy we use every day: the hour divided into 60 “minute parts”, that is, minutes, and the minute into another 60 “second” parts (in Latin partes minutae secundae). The 24-hour day, on the other hand, is of Egyptian origin, as we have seen.

The division of the circle into 360 degrees also comes from that tradition. Why 360? Perhaps because the year lasts about 360 days, perhaps because 360 is a multiple of 60 and has a great many divisors. Scholars are not sure of the reason. But the Greeks took it from the Babylonians, and from them it has reached us.

Doubts and caution

We must be careful not to exaggerate. The Babylonians did not write proofs like those of the Greeks: they said “do this and you get that”, with numerical examples, and did not explain why the method worked. Historians debate whether they really had a theory or only very good recipes. And the texts we have are few and damaged: our picture of Babylonian mathematics could change with new discoveries.

Why it happened

Babylonian mathematics was born from practical needs. Mesopotamian cities had harvests to record, fields to measure, canals to dig, temples to build, goods to sell and debts to calculate. Those who managed these affairs, the scribes, had to know how to do precise sums.

A second cause was school. Scribes trained for years in the “tablet houses”, where they learned the tables by heart and solved problems of increasing difficulty. In the schools the habit arose of inventing problems that were not practical too, for the pleasure of solving them, and from this a more abstract mathematics developed.

A third cause was astronomy. Priests watched the sky to foresee omens and to fix the calendar of festivals. For this they needed angular measures and long calculations, and the base-60 system was perfect for fractions.

What changed

The most visible legacy is that of time and angles: the 60 minutes of the hour, the 60 seconds of the minute, the 360 degrees of the circle. Every time we look at a clock or a compass we use, without knowing it, an idea from four thousand years ago.

The positional system, together with the calculation tables, was a model for later mathematicians: the Greeks knew it through the astronomers, the Indians developed our decimal system (see the card on Indo-Arabic numerals), and the Arabs passed it on to Europe. Without the first idea of giving a digit a value by its position, modern calculation would be much harder.

Finally, the discovery of Plimpton 322 and the other tablets has changed the history of mathematics: it was not born only in Greece, but has older roots, in Mesopotamia. Today's scholars try to read these tablets in a new way, with modern techniques such as three-dimensional digital images.

Key people and places

People

George Arthur Plimpton: American publisher and collector (1855–1936) who bought the tablet and then gave it to Columbia University.
Otto Neugebauer: Austrian-American mathematician and historian (1899–1990), the greatest scholar of Babylonian mathematics and astronomy. In 1945 he explained Plimpton 322 with Abraham Sachs.
Eleanor Robson: British historian of mathematics, who in 2001 proposed that Plimpton 322 was a school text for building problems.
Hipparchus: Greek astronomer of the 2nd century BC who helped to spread in Greece the division of the circle into 360 degrees, taken from the Babylonians.

Places

Larsa (southern Iraq): A city in the south of Mesopotamia that was the capital of a kingdom rival to Babylon, the probable place of origin of Plimpton 322.
Nippur (Iraq): A sacred city with one of the most important scribal schools, where thousands of exercises have been found.
Columbia University (New York): The American university where the Plimpton 322 tablet is kept.

Good to know

Plimpton 322 was bought around 1922 by a New York publisher, who gave it to a university: for decades nobody understood its value.
The Babylonians calculated the square root of 2 to six decimal places, between about 1800 and 1600 BC.
For a long time they had no zero: they left an empty space, and only around the 3rd century BC did they invent a sign for it.
Some tablets contain problems about interest: for example, how long it takes for a sum of money to double.
The division of the circle into 360 degrees and of the hour into 60 minutes is a legacy, passed through the Greeks, of Babylonian mathematics.