The tablet, known as Plimpton 332, was discovered in the early 1900s in Southern Iraq by the American archaeologist and diplomat Edgar Banks, who was the inspiration for Indiana Jones.

The true meaning of the tablet has eluded experts until now but new research by the University of New South Wales, Australia, has shown it is the world’s oldest and most accurate trigonometric table, which was probably used by ancient architects to construct temples, palaces and canals.

However unlike today’s trigonometry, Babylonian mathematics used a base 60, or sexagesimal system, rather than the 10 which is used today. Because 60 is far easier to divide by three, experts studying the tablet, found that the calculations are far more accurate.

“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,” said Dr Daniel Mansfield of the School of Mathematics and Statistics in the UNSW Faculty of Science.

“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.

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

A trigonometric table allows a user to determine two unknown ratios of a right-angled triangle using just one known ratio. But the tablet is far older than Hipparchus, demonstrating that the Babylonians were already well advanced in complex mathematics far earlier.

The tablet, which is thought to have come from the ancient Sumerian city of Larsa, has been dated to between 1822 and 1762 BC. It is now in the Rare Book and Manuscript Library at Columbia University in New York.

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

“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.

“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.”

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 but the researchers believe t there were originally six columns and that the tablet was meant to be completed with 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,” added Dr Mansfield.

The new study is published in Historia Mathematica, the official journal of the International Commission on the History of Mathematics.

]]>The tablet, discovered in the early 1900s and first interpreted in 1945, has long fascinated mathematics scholars, but they were puzzled by its description of triangles, which researchers recently linked to a type of trigonometry.

These ancient mathematical inscriptions predate the earliest known evidence of trigonometry — thought to have originated around 120 B.C. with Greek astronomer Hipparchus — by approximately 1,000 years, the researchers reported in a new study.

This finding suggests that the Babylonians, not the ancient Greeks, were the first to study trigonometry — the mathematics of triangles — perhaps using it in architectural calculations for constructing pyramids, temples and palaces, the study authors wrote. [The 7 Most Mysterious Archaeological Finds on Earth]

The tablet, which measures 5 inches (12.7 centimeters) long and 3.5 inches (8.8 cm) wide, is known as Plimpton 322, named for its owner, American philanthropist George Arthur Plimpton, who purchased the artifact in 1922 from archaeologist and antiquities dealer Edgar Banks.

Banks — the real-life inspiration for the adventure-seeking archaeologist movie character Indiana Jones — discovered the clay object in Iraq. Similarities in its writing style to that on other Babylonian tablets enabled experts to date it to between 1822 B.C. and 1726 B.C., around the time that King Hammurabi ruled the Babylonian Empire.

Experts interpreted the 15 rows of characters written in four columns on the tablet as descriptions of 15 triangles forming right angles, with their angles of inclination decreasing incrementally, the study authors wrote.

About 70 years ago, researchers determined that the notations on the tablet represented a special numerical pattern known as Pythagorean triples, a grouping of three positive integers, study co-author David Mansfield, a researcher with the School of Mathematics and Statistics at the University of New South Wales in Sydney, said in a statement.

“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,” Mansfield said.

**A new angle**

Trigonometry analyzes the relationships between the sides and angles of triangles; it is intrinsic to geometry and plays an important role in other branches of mathematics. The study authors expanded on prior research suggesting that Plimpton 322 was broken and incomplete, and they determined that there were originally six columns of figures on the tablet. Relationships between numbers in the completed table would have represented a novel type of trigonometry — one that relied on ratios instead of angles and circles, according to the study.

Thousands of years ago, mathematicians in Babylonia used a base 60 numerical system rather than the base 10 system that forms the foundation of modern arithmetic. In the study, the authors used the ancient base 60 system to demonstrate how scribes would have arrived at the numbers that were chiseled on Plimpton 322.

“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,” Mansfield said.

The simplicity and accuracy of this once-lost form of Babylonian trigonometry “has clear advantages” over modern trigonometry, study co-author Norman Wildberger, an associate professor in the School of Mathematics and Statistics at the University of New South Wales in Sydney, said in the statement.

Archaeologists have uncovered numerous tablets produced during the time of the Babylonian Empire, but very few of them have been examined in detail. This study’s findings hint that these understudied artifacts from a long-dead empire could hold exciting discoveries, not only for understanding the history of mathematics but also for enhancing how math is studied today, Wildberger explained.

“It opens up new possibilities not just for modern mathematics research, but also for mathematics education,” he said. “The mathematical world is only waking up to the fact that this ancient but very sophisticated mathematical culture has much to teach us.”

The findings were published online today (Aug. 24) in the journal Historia Mathematica.

source: Historia Mathematica

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