Hints of Trigonometry on a 3,700-Year-Old Babylonian Tablet

Hints of Trigonometry on a 3,700-Year-Old Babylonian Tablet by Kenneth Chang for the NYTimes (AUG. 29, 2017)

Suppose that a ramp leading to the top of a ziggurat wall is 56 cubits long, and the vertical height of the ziggurat is 45 cubits. What is the distance x from the outside base of the ramp to the point directly below the top? (Ziggurats were terraced pyramids built in the ancient Middle East; a cubit is a length of measure equal to about 18 inches or 44 centimeters.)

Could the Babylonians who lived in what is now Iraq more than 3,700 years ago solve a word problem like this?

Two Australian mathematicians assert that an ancient clay tablet was a tool for working out trigonometry problems, possibly adding to the many techniques that Babylonian mathematicians had mastered.

“It’s a trigonometric table, which is 3,000 years ahead of its time,” said Daniel F. Mansfield of the University of New South Wales. Dr. Mansfield and his colleague Norman J. Wildberger reported their findings last week in the journal Historia Mathematica.


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The other three columns are much more intriguing. In the 1940s, Otto E. Neugebauer and Abraham J. Sachs, mathematics historians, pointed out that the other three columns were essentially Pythagorean triples — sets of integers, or whole numbers, that satisfy the equation a2 + b2 = c2.

This equation also represents a fundamental property of right triangles — that the square of the longest side, or hypotenuse, is the sum of the squares of the other two shorter sides.

That by itself was remarkable given that the Greek mathematician Pythagoras, for whom the triples were named, would not be born for another thousand years.

Why the Babylonians compiled the triples and wrote them down has remained a matter of debate. One interpretation was that it helped teachers generate and check problems for students.


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