A 3,700-year-old Babylonian clay tablet discovered by the real Indiana Jones has been revealed as the world’s oldest – and most accurate – trigonometric table.
Mathematicians believe that the tablet, known as Plimpton 322, may have been used by ancient mathematical scribes to calculate how to build palaces, temples and build canals.
The new research shows the Babylonians beat the Greeks to the invention of trigonometry – the study of triangles – by more than 1,000 years.
The tablet was discovered in the early 1900s in what is now southern Iraq by archaeologist, diplomat and antiquities dealer Edgar Banks, the person on whom the fictional character Indiana Jones was based.
It has four columns and 15 rows of numbers written on it in the cuneiform script of the time using a base 60, or sexagesimal, system.
Dr Daniel Mansfield, of the University of New South Wales (UNSW) in Australia, said: “Plimpton 322 has puzzled mathematicians for more than 70 years, since it was realised it contains a special pattern of numbers called Pythagorean triples.
“The huge mystery, until now, was its purpose – why the ancient scribes carried out the complex task of generating and sorting the numbers on the tablet.
“Our research reveals that Plimpton 322 describes the shapes of right-angle triangles using a novel kind of trigonometry based on ratios, not angles and circles. It is a fascinating mathematical work that demonstrates undoubted genius.”
He added: “The tablet not only contains the world’s oldest trigonometric table; it is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry.
“This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3000 years, but it has possible practical applications in surveying, computer graphics and education.
“This is a rare example of the ancient world teaching us something new.”
The new study, by Dr Mansfield and his UNSW colleague Associate Professor Norman Wildberger was published in the journal Historia Mathematica.
A trigonometric table allows people to use one known ratio of the sides of a right-angle triangle to determine the other two unknown ratios.
The Greek astronomer Hipparchus has long been regarded as the father of trigonometry, with his “table of chords” on a circle considered the oldest trigonometric table.
But Dr Wildberger said: “Plimpton 322 predates Hipparchus by more than 1,000 years.
“It opens up new possibilities not just for modern mathematics research, but also for mathematics education.”
He 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.”
He said The 15 rows on the tablet describe a sequence of 15 right-angle triangles, which are steadily decreasing in inclination.
The left-hand edge of the tablet is broken and the UNSW researchers built on previous research to present new mathematical evidence that there were originally six columns and that the tablet was meant to be completed with 38 rows.
They also showed how the ancient scribes, who used a base 60 numerical arithmetic similar to our time clock, rather than the base 10 number system we use today, could have generated the numbers on the tablet using their mathematical techniques.
The researchers also provided evidence that discounts the widely-accepted view that the tablet was simply a teacher’s aid for checking students’ solutions of quadratic problems.
Dr Mansfield added: “Plimpton 322 was a powerful tool that could have been used for surveying fields or making architectural calculations to build palaces, temples or step pyramids.”
The tablet, which is thought to have come from the ancient Sumerian city of Larsa, has been dated to between 1822BC and 1762BC. It is now in the Rare Book and Manuscript Library at Columbia University in New York.