**WORLD**

What other “firsts” were documented in ancient clay tablets?

*Teachers, scroll down for a quick list of key resources in our Teachers Toolkit, including today’s simple MapMaker Interactive map.*

**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.

- 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.

- 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.

- What trigonometric patterns are found in Plimpton 322?
- Plimpton 322 may be one of the earliest documentations of the famous Pythagorean theorem, possibly the only trigonometry many adults remember. The Pythagorean theorem describes the relation between a right triangle’s hypotenuse (the side opposite the right angle) and the triangle’s other two sides. The Pythagorean equation states that the square of the hypotenuse (c) is equal to the square of the other two sides (a and b): a
^{2}+b^{2}=c^{2}.- Plimpton 322 lists
*Pythagorean triples*, positive integers that satisfy the Pythagorean equation.- The most famous Pythagorean triple is 3,4,5. Do the math!
- 3
^{2}+4^{2}=5^{2} - 9+16=25
- It checks out.

- 3
- “The surviving fragment of Plimpton 322 starts with the Pythagorean triple 119, 120, 169. The next triple is 3367, 3456, 4825. This makes sense when you realise that the first triple is almost a square (which is an extreme kind of rectangle), and the next is slightly flatter. In fact, the right-angled triangles are slowly but steadily getting flatter throughout the entire sequence.”

- The most famous Pythagorean triple is 3,4,5. Do the math!

- Plimpton 322 lists

- Plimpton 322 may be one of the earliest documentations of the famous Pythagorean theorem, possibly the only trigonometry many adults remember. The Pythagorean theorem describes the relation between a right triangle’s hypotenuse (the side opposite the right angle) and the triangle’s other two sides. The Pythagorean equation states that the square of the hypotenuse (c) is equal to the square of the other two sides (a and b): a

- 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.”

- “The Babylonians had a completely different conceptualisation of a right triangle. They saw it as

- How is trigonometry used today? Read through this article for some help.
*Geography!*Trigonometry is crucial to navigation, calculating a location in GPS, determining the height of a mountain or building, documenting the slope of a canyon, and more.*Gaming!*Game designers use trig to calculate ways a character or player can avoid obstacles.*Crime Scene Investigation*! “In criminology, trigonometry can help to calculate a projectile’s trajectory or to estimate what might have caused a collision in a car accident.”*Oceanography!*Trigonometry is a vital part of bathymetry and marine engineering—calculating how deep a submersible is, for instance.*Aviation!*Flight engineers use trigonometry when assessing speed, distance, and direction along with the speed and direction of the wind.*Archaeology!*“Trigonometry is used to divide up the excavation sites properly into equal areas of work.”*Construction and surveying*! Just as in Babylonian times, trigonometry can be used to measure plots of land, making walls parallel or perpendicular, and determining roof inclination.

**TEACHERS TOOLKIT**

*The Guardian:* Mathematical secrets of ancient tablet unlocked after nearly a century of study

*The Conversation:* Written in stone: the world’s first trigonometry revealed in an ancient Babylonian tablet

*Nat Geo:* Where was Plimpton 322 found? MapMaker Interactive map

*Nat Geo:* Ancient Tablet May Show Earliest Use of This Advanced Math

*Technology for Secondary/College Mathematics:* Pythagorean triples

(extra credit!) *Historia Mathematica:* Plimpton 322 is Babylonian exact sexagesimal trigonometry