3,700-year-old clay tablet reveals amazingly advanced mathematics

Surprise in cuneiform: A 3,700-year-old clay tablet of the Babylonians has turned out to be the oldest trigonometric table in the world. The cuneiform numbers suggest that the Babylonian mathematicians knew trigonometry a thousand years before the Greeks. Surprisingly, the Babylonian system allowed much more precise calculations than the classical angle-based system.

3,700 years old and yet surprisingly modern: the Babylonian cuneiform plaque "Plimpton 322" © UNSW/ Andrew Kelly

Ancient Mesopotamia is a major cradle of astronomy and mathematics: even the scholars of Babylon recognized certain lawsuits in the movements of planets, the sun, and the stars. They therefore developed mathematical methods to predict these heavenly events. Their calculations are written down on cuneiform tablets and have been partially preserved.

Now it becomes clear that the Babylonians were ahead by a nose in another mathematical discipline - the trigonometry. So far, this was regarded as an achievement of the Greeks. For the first time, they have determined how to use angle and circular functions of three known features in a triangle to close further ones. This is important, among other things, for geodesy, but also for the distance calculation of celestial bodies in astronomy.

Puzzling triple sequence

But already the Babylonian knew and used trigonometric calculations, as a cuneiform map discovered around 1900 in the south of Iraq. The 3.700-year-old clay tablet "Plimpton 322" is described with four columns and 15 lines of numbers. Noticeable: These numbers resemble a well-known triple sequence from the geometry, the Pythagorean triple. This number center describes the lengths of the three sides in a right-angled triangle.

But why were these three sequences serving the Babylonians?

Plimpton 322 has puzzled the mathematicians for more than 70 years - since it became clear that they were Pythagorean triples, says Daniel Mansfield, University of New South Wales. The big question was, for which the Babylonian mathematicians calculated these triples and recorded in such a table.

Oldest trigonometry table of the world

The answer could now have been found by Mansfield and his colleague Norman Wildberger. As they found out, the Babylonian numerical triples describe a succession of fifteen triangular triangles, whose angular inclination decreases steadily from top to bottom. This suggests that the Babylonians used it as a trigonometric table, as the researchers explain. Originally, this table even consisted of six columns with 38 rows, but a part has since been canceled and lost.

This clay tablet thus contains the oldest trigonometric table in the world, says Mansfield.

The Babylonians must have understood the principle of trigonometry already a thousand years before the Greeks and used it for calculations.

This clay sheet was an effective tool to measure fields or to make architectural calculations for palaces, temples, or step pyramids, the researcher said.

Video about "Plimpton 322" by Mansfield © University of New South Wales

Fractions instead of angle

But not only that: the calculations of the Babylonians were much more precise than those of the Greeks:

It is the only perfectly accurate trigonometric table, Mansfield says. The Babylonians succeeded because they had a completely different approach to arithmetic and geometry.

In contrast to the Greeks, their calculations were based not on angles and circular functions but on numerical relationships.

The Babylonians also did not use the decimal system, but a counting based on 60 - similar to our time units. As a result, their fractions were far more frequent, so a rounding was hardly necessary.

This is a fascinating mathematical method that undoubtedly proves a certain genius, Mansfield said.

These mathematical abilities of the Babylonians are not only interesting for historians.

It's also important for our modern world, says Mansfield's colleague Norman Wildberger. Babylonian mathematics may have been out of fashion for three thousand years, but it could have practical applications for computer graphics, surveys, and school. This is a rare example of the fact that even antiquity can teach us something new.

Source: Plimpton 322 is Babylonian exact sexagesimal trigonometry, Daniel F. Mansfield, N.J. Wildberger