Ancient Babylonian Tablet May Hold Earliest Examples of Trigonometry

A new analysis of a long-studied Babylonian tablet suggests that trigonometry, the subject so many of us struggled through in high school, may actually be a lot older than previously thought.

The small clay tablet, which dates back to the year 1800 B.C.E., is dubbed Plimpton 322 after George Arthur Plimpton, a New York publisher who purchased it in the 192o's.  He donated the tablet with its scrawled rows of numbers to Columbia University in 1936—where it still remains today, researchers of the new study Daniel Mansfield and Norman Wildberger write for The Conversation.

In the decades since its discovery, researchers have debated about the meaning of those numbers, reports Carl Engelking for Discover magazine. In his 1945 book, mathematician and historian Otto Neugebauer first suggested that Plimpton 322 represents a glimpse at early trigonometry, a field of math concerning the relationship of the sides and angles in triangles. The numbers on the tablet represented Pythagorean triples in Neugebauer's mind, which are sets of three numbers that can be used to solve the Pythagorean theorem (a²+b²=c²), writes Engelking.

Later researchers, such as mathematical historian Eleanor Robson, threw cold water on that idea, arguing that Plimpton 322 was more simply a teaching aid. Robson argued that the chosen numbers didn't seem to align with groundbreaking research.

Science historians have long regarded the creator of trigonometry to be the Greek astronomer Hipparchus and his contemporaries. They are believed to develop the system around the second century C.E. to precisely calculate the movement of the zodiac signs in the sky.

But in the new study, published in the journal Historia Mathematica, Mansfield and Wildberger lend some credence to Neugebauer's thinking, reports Ron Cowen for Science Magazine. The key is to get a new angle on the tablet's numbers.

Instead of the traditional method of trigonometry based on the angles of triangles, Cowen reports, Plimpton 322 actually uses calculations based on the ratios of the lengths of sides of right triangles, rather than relationships based on their angles. And instead of the base-10 system of numbers used today, the study suggests that the Babylonian tablet uses a base-60 system (similar to how we count time).

Using this tablet and its system of numbers, the Babylonians could precisely calculate figures to a whole number more accurately than we could today with traditional trigonometry, Mansfield and Wildberger argue. The write:

"The sexagesimal system is better suited for exact calculation. For example, if you divide one hour by three then you get exactly 20 minutes. But if you divide one dollar by three then you get 33 cents, with 1 cent left over. The fundamental difference is the convention to treat hours and dollars in different number systems: time is sexagesimal and dollars are decimal."

"It opens up new possibilities not just for modern mathematics research, but also for mathematics education," Wildberger says in a statement. "With Plimpton 322 we see a simpler, more accurate trigonometry that has clear advantages over our own."

The tablet could have had practical use in surveying or construction, writes Sarah Gibbens for National Geographic, allowing builders to take the heights and lengths of buildings and calculate the slope of a roof.

Other mathematicians urge caution in the latest Plimpton 322 interpretation, writes Cowen at Science. Babylonian mathematics expert Jöran Friberg is skeptical that the culture had any knowledge of ratios advanced enough to create this form of math, while mathematical historian Christine Proust says there is no evidence in other surviving texts that tablets like this could have been used in the way the authors suggest.

Meanwhile, mathematician Donald Allen tells Gibbens that it's hard to really know whether Mansfield and Wildberger's theory is right because they had to recreate a broken section of the tablet, making any conclusion "conjecture."

However, the Australian mathematicians hope to see more research done on the insights that the Babylonians might have for modern-day people, as they write for The Conversation.

"We are only beginning to understand this ancient civilization, which is likely to hold many more secrets waiting to be discovered."

Ben Panko | READ MORE

Ben Panko is a staff writer for Smithsonian.com