Science

Archeologists decipher ancient Babylonian trigonometry tablet

Plimpton 322, a 3,700-year old clay tablet, has been found to be the world's oldest and most accurate trigonometric table
Plimpton 322, a 3,700-year old clay tablet, has been found to be the world's oldest and most accurate trigonometric table
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UNSW's Daniel Mansfield, holding the Plimpton 322 tablet
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UNSW's Daniel Mansfield, holding the Plimpton 322 tablet
Plimpton 322, a 3,700-year old clay tablet, has been found to be the world's oldest and most accurate trigonometric table
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Plimpton 322, a 3,700-year old clay tablet, has been found to be the world's oldest and most accurate trigonometric table

It's long been accepted that the ancient Greeks were responsible for developing the mathematical concept of trigonometry, but a new discovery indicates they weren't the first to figure it out after all. Scientists from the University of New South Wales (UNSW) have shed light on the purpose of a mysterious clay tablet containing a trigonometric table created by the Babylonians a full 1,000 years earlier than Pythagoras – and it's more accurate than our techniques today.

The tablet – a small clay slab known as Plimpton 322 – was discovered in the early 20th century in southern Iraq, and currently resides in the Rare Book and Manuscript Library at Columbia University in New York. It's inscribed with numbers in four columns and 15 rows, using a base 60 system similar to the one we use to measure time. These numbers are arranged in what we now call Pythagorean triples, sets of three numbers that can form the equation a2 + b2 = c2.

"Plimpton 322 has puzzled mathematicians for more than 70 years, since it was realised it contains a special pattern of numbers called Pythagorean triples," says Daniel Mansfield, co-author of the study. "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."

For decades, the commonly-held answer to the riddle was that Plimpton 322 was used by teachers to check their students' answers to geometric problems. But the UNSW researchers noticed similarities between the numbers on the tablet and rational trigonometry, an emerging branch of mathematics.

"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," says Mansfield. "It is a fascinating mathematical work that demonstrates undoubted genius. 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."

UNSW's Daniel Mansfield, holding the Plimpton 322 tablet
UNSW's Daniel Mansfield, holding the Plimpton 322 tablet

Long before the advent of electronic calculators, trigonometric tables were the easiest way to solve certain geometric problems. Essentially, users can take a known ratio of the sides of a right-angle triangle and use it to figure out the two unknown ratios. This technique was long attributed to the Greek astronomer Hipparchus, who developed the circular "table of chords" in around 120 BCE. But Plimpton 322 has been dated back to between 1822 and 1762 BCE.

"Plimpton 322 predates Hipparchus by more than 1,000 years," says Norman Wildberger, co-author of the study. "It opens up new possibilities not just for modern mathematics research, but also for mathematics education. With Plimpton 322 we see a simpler, more accurate trigonometry that has clear advantages over our own."

The key to the tablets' accuracy is the base 60 arithmetic, which the researchers say allowed the ancient Babylonians to divide numbers more neatly than our base 10 system can handle. Their studies indicate that the tablets' 15 rows describe a series of 15 right-angle triangles of decreasing inclination. It's broken along the left-hand side, and the team says it most likely used to contain an extra two columns, and a total of 38 rows.

"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," says Mansfield.

This finding indicates the ancient Babylonians were mathematically far more advanced than we might give them credit for, and the scientists say that studying other artefacts could reveal similar secrets.

"A treasure-trove of Babylonian tablets exists, but only a fraction of them have been studied yet," says Wildberger. "The mathematical world is only waking up to the fact that this ancient but very sophisticated mathematical culture has much to teach us."

The research was published in the journal Historia Mathematica, and the team describes the find in the video below.

Source: UNSW

Ancient Babylonian tablet - world's first trig table

7 comments
Chris74
the question is how did the Arabian people forget this where ancient Greeks passed the knowledge down generation to generation? why did the Arabians forget cuniform, ancient persian, and Sanskrit where ancient greek was never forgotten? looking at history i can only imagine it has something to do with the Mongolian Invasion of Arabia and the end of the golden age of Islam and enslavement of Arabs and the erasing of knowledgeable people. Makes you wonder what role Hebrews played in it all since their language remains where all others have not.
bql
middle east became arabic only after muslim invasions, which were carried out by arabs. before that it was caucasian. look up pictures theres still some blue eyed fair haired muslims living there that havent been wiped or race mixed out.
StWils
There are lots of examples of societies advancing significantly and then stalling and collapsing. Typically a series of injuries crack and break an otherwise previously stable society. As an example one of the south American cultures had a massive irrigation system and hybridized Quinoa, Corn, and Squash, had an elaborate literary tradition and still collapsed from several successive stresses long before the Spanish arrived. I think a bigger miracle is that someone now has found and de-cyphered the few surviving scraps. Go look up the evolving work on the mechanical astronomical computer found on the shore of Antikytera, Greece. A multi-discipline team only recently figured out how to line up the marks with our stars as they appeared some thing like 2500 years ago.
Bob
I always wondered if using the base 10 number system held back progress over the years. While it seems handy, would another system have worked better? Computers work on base 2, 8, 16 and it has always been a problem converting accurately back to base 10. Maybe if early man had eight or twelve fingers and toes, would history have been different? Just wondering.
MarkGatti
more evidence we have devolved since Adam , whole 'we come from cavemen' idea is just nothing but a fading fairy tale . and, how fragile a good, society is, its not just about tech.and knowledge , really not a good idea to mess around with social experiments . We do know the medes and persians , and then Alexander ,and Rome all came and invaded . if its taken us 70 years to decipher when researchers are wanting to , its unlikely people with a 'conquering' mind set would even bother , though alexander did tend to leave established government structures in place . Come to think, Genghis came through too , and he obliterated any thing he didn't understand , till later on in life .
MattII
@MarkGatti, Adam is the fantasy, cavemen are the fact. Not that the Babylonians were actually cavemen though, because cavemen don't get to be major players in the Bronze Age, which the Babylonians were.
onestagetospace
I find it strange that this is presented as a new finding. I first heard about this in the beginning of the 90'ies, when I was twelve and learned to use the 60-base, 9-base, 12-base, 24-base and 64-base counting systems used in antiquity, in my classical education, each with their own purpose and cultural background. Every classical scholar knows that all of these systems were used contemporaneously on all continents and far predated the classical Greeks (except Australia, as far as I was thought, no racist intentions). Maybe modern mathematics education in school should again include classical languages because it exposes you to a wide variety of basic scientific ideas and global cultures. As the saying goes, nohil nove sub sole; or the more prosaic Russian version: The new is the old that has been forgotten.