Mathematicians Crack Mystery of Babylonian Clay Tablet ‘Plimpton 322’

Plimpton 322, the most famous of Old Babylonian tablets (1900-1600 BC), is the world’s oldest trigonometric table, possibly used by Babylonian scholars to calculate how to construct stepped pyramids, palaces and temples, according to a duo of researchers from the School of Mathematics and Statistics at the University of New South Wales (UNSW), Sydney, Australia.

Plimpton 322, one of the most sophisticated scientific artifacts of the ancient world, likely came from the ancient Sumerian city of Larsa, which was located near modern-day Tell as-Senkereh in southern Iraq.

The tablet was most likely written between 1822-1762 BC (around the time of Hammurabi, the sixth king of the First Babylonian Dynasty).

It was discovered in the early 1900s by the archaeologist, academic and adventurer Edgar J. Banks, the person on whom the fictional character Indiana Jones was based.

In the 1920s, Banks sold the tablet to the American publisher and philanthropist George Arthur Plimpton.

Plimpton bequeathed his entire collection of mathematical artifacts to Columbia University in 1936, and it resides there today in the Rare Book and Manuscript Library.

Physically Plimpton 322 is made from clay and measures 12.7 x 8.8 cm, with the left-hand edge showing clear evidence of being broken, and indeed remnants of modern glue suggests that the break occurred in recent times.

Plimpton 322’s obverse (front) is divided by 3 vertical lines into 4 columns, each with a heading, the first of which is partially obscured by damage, while the others are clearly readable.

The main body of the obverse is ruled by neat horizontal lines into 15 equally spaced rows containing sexagesimal (base 60) numbers, some of which are quite large. The vertical lines continue on the bottom and reverse, which are otherwise empty.

“Plimpton 322 has puzzled mathematicians for more than 70 years, since it was realized it contains a special pattern of numbers called Pythagorean triples,” said UNSW researcher Dr. Daniel Mansfield.

“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.”

The new study by Dr. Mansfield and his colleague, Dr. Norman Wildberger, provides an alternative to the widely-accepted view that Plimpton 322 was a scribal school text.

“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,” Dr. Mansfield said.

“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.”

The Greek astronomer Hipparchus, who lived about 120 years BC, has long been regarded as the father of trigonometry, with his ‘table of chords’ on a circle considered the oldest trigonometric table.

“Plimpton 322 predates Hipparchus by more than 1000 years. 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,” Dr. Wildberger said.

Dr. Mansfield read about Plimpton 322 by chance when preparing material for first year mathematics students at UNSW.

He and Dr. Wildberger decided to study Babylonian mathematics and examine the different historical interpretations of the tablet’s meaning.

“The 15 rows on the tablet describe a sequence of 15 right-angle triangles, which are steadily decreasing in inclination,” they explained.

“The left-hand edge of the tablet is broken and we build on previous research to present new mathematical evidence that there were originally 6 columns and that the tablet was meant to be completed with 38 rows.”

“We also demonstrate 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, could have generated the numbers on the tablet using their mathematical techniques.”

“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,” Dr. Mansfield added.

The research is published in the journal Historia Mathematica.