The Babylonian Clay Tablet Plimpton 322

According to a report published in Science News, two mathematicians have solved the puzzle of the Babylonian Clay Tablet Plimpton 322.

Plimpton 322, a 3,700-year-old Babylonian tablet held in the Rare Book and Manuscript Library at Columbia University in New York.

 

 

 

 

 

 

 

 

 

 

 

 

According to the Wikipedia,

Plimpton 322 is a Babylonian clay tablet, notable as containing an example of Babylonian mathematics. This tablet, believed to have been written about 1800 BC, has a table of four columns and 15 rows of numbers in the cuneiform script of the period.

This table lists two of the three numbers in what are now called Pythagorean triples. From a modern perspective, a method for constructing such triples is a significant early achievement, known long before the Greek mathematicians discovered solutions to this problem. At the same time, one should recall the tablet’s author was a scribe, rather than a professional mathematician; it has been suggested that one of his goals may have been to produce examples for school problems. There has been significant scholarly debate on the nature and purpose of the tablet.

Below is an excerpt of the article published in Science News:

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.

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.

To read the article in Science News, click here.

This article was based on the original research published in Historia Mathematica:

Daniel F. Mansfield and N. J. Wildberger, “Plimpton 322 is Babylonian exact sexagesimal trigonometry,” Historia Mathematica 44 no. 4 (November 2017): 395-419.

The original research is available in PDF format: click here.

The video below explains the significance of the world’s oldest trigonometric table.

Claude Mariottini
Emeritus Professor of Old Testament
Northern Baptist Seminary

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