Scientists Decipher Ancient Mathematical Mystery

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Discussion Ideas

The fascinating new research analyzes Plimpton 322, a 3700-year-old Babylonian clay tablet written in cuneiform. What is cuneiform?
- Cuneiform is a written language made up of different collections of wedge or triangle shapes. Cuneiform was common throughout many cultures of ancient Mesopotamia, including Sumeria, Akkadia, Assyria, and Babylonia.
  - Mesopotamia is an ancient name for the rich valley of the Tigris-Euphrates river system. The Tigris-Euphrates river system includes most of what is now Iraq, and parts of Kuwait, Iran, Turkey, and Syria.
  - Take a look at this terrific example of how the cuneiform sign for “head” evolved from the Uruk period (3000s BCE, long before Plimpton 322 was written) to around 1 CE.

The Australian researchers working on Plimpton 322 say it may be the world’s oldest trigonometry table. What is trigonometry?
- Trigonometry is a branch of mathematics dealing with the properties and relationships among angles and sides of triangles. Learn more here.

How were the Pythagorean triples in Plimpton 322 used by Babylonians?
- We’re not entirely sure! As the mathematician in the terrific video above says, scientists have known for decades that the tablet’s “unusual series of numbers proves that the Babylonians knew the Pythagorean theorem a thousand years before Pythagoras was born. But while there is agreement on what the tablet contains, there’s been no agreement on what it was used for.”
- The new research posits that, contrary to modern understanding of trigonometry, regular triangles were not the primary interest of trigonometry in ancient Babylon. Instead, “[g]eometry in ancient Babylon arose from the practical needs of administrators, surveyors, and builders. From their measurements of fields, walls, poles, buildings, gardens, canals, and ziggurats, a metrical understanding of the fundamental types of practical shapes was forged; typically squares, rectangles, trapezoids and right triangles.”

Why does the mathematician in the video say that the trigonometry in Plimpton 322 may be more accurate than the trig we use today?
- According to the video, “It all comes down to fractions. We count in ‘base 10’ [the decimal system], which only has two exact fractions: .5 and .2 … The Babylonians counted in ‘base 60’ [the sexagesimal system], the same system we use for telling time. This has many more exact fractions … By using this system, the Babylonians were able to make calculations that completely avoided any inexact numbers, thereby avoiding any errors associated with multiplying those numbers.”

If Babylonian trigonometry was more accurate, why hasn’t it survived?
- Good question. “Perhaps it went out of fashion because the Greek approach using angles is more suitable for astronomical calculations. Perhaps this understanding was lost in 1762 BCE when Larsa [where the tablet was created] was captured by Hammurabi of Babylon. Without evidence, we can only speculate.”

What are criticisms of the new research? Read through this Nat Geo News article for some help.
- The left edge of the tablet is broken, which “invites a great deal of purely mathematical speculation,” says one mathematician skeptical of the new conclusions.
- The tablet relies on a very different concept of trigonometry than the one we’re familiar with. It uses ratios (fractions), not angles.
  - “The Babylonians had a completely different conceptualisation of a right triangle. They saw it as half of a rectangle, and due to their sophisticated sexagesimal (base 60) number system they were able to construct a wide variety of right triangles using only exact ratios.”