Babylonian tablet Plimpton 322

Credit: UNSW/Andrew Kelly

Trigonometry is the branch of mathematics dealing with specific functions of length and angles and their application to calculations of triangles.  Ancient Greek astronomers like Hipparchus, Menelaus, and Ptolomy transformed trigonometry into an ordered science. But there is great doubt now about whether the Greeks actually invented trigonometry. The Babylonian tablet Plimpton 322 now located at Columbia University, was recovered from an unknown place in the Iraq desert.  It was discovered by the archaeologist, diplomat and antiquities dealer Edgar Banks in the early 1900’s, the person on whom the fictional character Indiana Jones was based on.  The tablet was written originally sometime around 1800 BCE based on its writing style.

The Babylonian number system,  was remarkably sophisticated written in wet clay with a stylus, where the number 1 `written’ with a single `stroke’  and the numbers 2 through 9 were written by combining multiples of a single stroke:

The number 10 was written in a single character  and the numbers 20 to 50 were written with multiples of this character:


More on the number system here.  Despite understanding the number system used by the Babylonians, the 3,700-year-old broken clay tablet has baffled mathematicians for more than 70 years, because it contains part of a list of Pythagorean triples, that is to say integers w, l, d with w+ l2 = dwhich form the sides of right triangles with d being the hypotenuse. The interesting thing is these calculations on the tablet were done a 1,000 years before Pythagoras’s right triangle theorem.

A team from the University of New South Wales in Sydney (UNSW) believe that the four columns and 15 rows of cuneiform on the ancient clay tablet represent the world’s oldest and most accurate working trigonometric table.  Speculation is that the calculations were used to engineer how to build temples, palaces and pyramids. Perhaps these sophisticated calculations explain how the fabled Hanging Gardens of Babylon, written of by Greek historians as one of the seven wonders of the ancient world could have been built in such ancient times?

The new study, by Dr. Daniel Mansfield and his UNSW colleague Associate Professor Norman Wildberger was published in the journal Historia Mathematica.  Dr. Mansfield, described Plimpton 322 as “a fascinating mathematical work that demonstrates undoubted genius” – with viable application today since the base 60 used in calculations by the Babylonians offered many more accurate fractions than the contemporary base 10.

Dr. Mansfield added: “With Plimpton 322 we see a simpler, more accurate trigonometry that has clear advantages over our own.”

“A treasure-trove of Babylonian tablets exists, but only a fraction of them have been studied yet.”

“The mathematical world is only waking up to the fact that this ancient but very sophisticated mathematical culture has much to teach us.”