Study of Ancient Tablet Leads to Revision of Trigonometry’s History

The 3,700-year-old Babylonian tablet Plimpton 322 at the Rare Book and Manuscript Library at Columbia University in New York. (UNSW/Andrew Kelly)

(CN) – An ancient tablet originally discovered by the real-life inspiration for Indiana Jones offers insight into the advanced mathematical understanding of the Babylonians, according to a new study.

The report, published Thursday in the journal Historia Mathematica, shows that the Babylonians invented trigonometry – the study of triangles – more than 1,000 years before the Greeks, and reveals the ancient civilization’s level of mathematical sophistication.

Known as Plimpton 322, the small tablet has been the subject of debate among scholars over its role in Babylonian society. However, the new study finds the relic could have been used to construct temples, palaces and canals.

“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,” Daniel Mansfield, a researcher at the University of New South Wales in Sydney, Australia, and co-author of the study, said in a statement.

“Our research reveals that Plimpton 322 describes the shapes of right-angled triangles using a novel kind of trigonometry based on ratios, not angles and circles. It is a fascinating mathematical work that demonstrates undoubted genius.”

Mathematicians previously discovered that the tablet contains a special pattern of numbers called Pythagorean triples, which consist of three positive, whole numbers that can be used to determine the hypotenuse, or diagonal side, opposite a right angle in triangles. Plimpton 322 includes large Pythagorean triples, such as the triple 119, 120 and 169, which is referenced in the first row of the tablet.

The tablet is the world’s oldest trigonometric table, which allows a person to use one known ratio of the side of a right-angled triangle to calculate the two unknown ratios. In addition, the team notes that Plimpton 322 features the only completely accurate trigonometric tablet, owing to the Babylonians’ unique approach to geometry and arithmetic.

UNSW Sydney scientist Dr. Daniel Mansfield with the 3,700-year-old Babylonian tablet Plimpton 322 at the Rare Book and Manuscript Library at Columbia University in New York. (UNSW/Andrew Kelly)

“This means it has great relevance for our modern world. Babylonian mathematics may have been out of fashion for more than 3,000 years, but it has possible practical applications in surveying, computer graphics and education,” Mansfield said. “This is a rare example of the ancient world teaching us something new.”

Mansfield learned about Plimpton 322 by chance while preparing material for mathematics students at UNSW. Along with his colleague, UNSW associate professor Norman Wildberger, Mansfield decided to study the tablet after noticing parallels to Wildberger’s book “Divine Proportions: Rational Trigonometry to Universal Geometry.”

“This Old Babylonian trigonometry has close links to the modern theory of rational trigonometry,” Wildberger told Courthouse News in an email. “Modern mathematics education could be made a lot simpler by moving away from an angle-based trigonometry towards such a ratio-based trigonometry as we find in Plimpton 322.”

The tablet has four columns and 15 rows of numbers that describe a sequence of 15 right-angled triangles written on it using a base-60, or sexagesimal, system – similar to our clocks. Modern mathematics use a base-10 system.

“In base 60, there are many more fractions that can be converted to finite sexagesimal expressions, in fact whenever the denominator is built from the factors 2, 3 and 5,” Wildberger said. “This means greater potential for exact computations.”

While the Greek astronomer Hipparchus, who lived more than 2,100 years ago, has long been considered the father of trigonometry – using a “table of chords” on a circle, regarded as the oldest trigonometric table – the new findings update his position in the evolution of mathematics. The study also shows that the Babylonians had a vastly separate and distinct approach to trigonometry.

“The ratios are completely different in character,” Wildberger said. “The Old Babylonian scribe created very special integral right triangles and sought to explain the relations between ratios of them, exactly. Hipparchus’ table of chords introduces angles and relates ratios of sides to angles, necessarily approximately. Angles do not appear in the Old Babylonian scheme of things.”

The left-hand edge of the tablet is broken, leading scientists to wonder how many ratios were lost as a result of the tablet’s deterioration. However, the UNSW team was able to build on previous research to offer new evidence that there were originally six columns and that the tablet was supposed to be completed with 36 rows.

Besides the physical limitations of the partial tablet, the team also had to recalibrate their views on ancient mathematics.

“Perhaps the biggest challenge was to overcome our prejudice that modern mathematics must be superior to such ancient mathematics,” Wildberger said. “Only gradually did it dawn on us that this was the work of a genius, speaking to us in an unfamiliar cultural and arithmetical context.”

In addition to adjusting the timeline of the development of trigonometry, the tablet could also have modern applications.

“We think there are rich further avenues of investigation, not just for mathematics education and modernizing trigonometry, but also in thinking about the Old Babylonian sexagesimal system and its potential power for modern computations,” Wildberger said.

Researchers believe Plimpton 322 comes from the ancient Sumerian city of Larsa and has been dated between 1822 and 1762 B.C.

The tablet was originally discovered in the early 1900s in southern Iraq by Edgar Banks, an archaeologist and antiquities dealer and the real-life inspiration for the fictional character Indiana Jones.

 

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