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,” says Daniel F Mansfield of the University of New South Wales. Mansfield and his colleague Norman J Wildberger reported their findings last week in the journal Historia Mathematica (if you need help to solve the problem, the answer is explained below). The tablet, known as Plimpton 322, was discovered in the early 1900s in southern Iraq and has long been of interest to scholars. It contains 60 numbers organised into 15 rows and four columns inscribed on a piece of clay about five inches wide and 3.5 inches tall. It eventually entered the collection of George Arthur Plimpton, an American publisher, who later donated his collection to Columbia University. With all the publicity, the tablet has been put on display at the university’s Rare Book & Manuscript Library.
Based on the style of cuneiform script used for the numbers, Plimpton 322 has been dated to between 1822 and 1762 BC. One of the columns on Plimpton 322 is just a numbering of the rows from 1 to 15. 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.
Mansfield, who was searching for examples of ancient mathematics to intrigue his students, came across Plimpton 322 and found the previous explanations unsatisfying. “None of them really seemed to nail it,” he said. Other researchers have postulated that the tablet originally had additional columns listing ratios of the sides (there’s a break along the left side of the tablet). But what is conspicuously missing is the notion of angle, the central concept impressed upon students learning trigonometry today. Wildberger, down the hall from Mansfield, had a decade earlier proposed teaching trigonometry in terms of ratios rather than angles, and the two wondered that Babylonians took a similar angle-less approach to trigonometry. “I think the interpretation is possible,” says Alexander R Jones, director of the Institute for the Study of the Ancient World at New York University, who was not involved with the research, “but we don’t have much in the way of contexts of use from any Babylonian tablets that would confirm such an intention, so it remains rather speculative.”
Eleanor Robson, a Mesopotamia expert now at University College London who proposed the idea of the tablet as a teacher’s guide, is not convinced. Although she declined interviews, she wrote on Twitter that the trigonometry interpretation ignores the historical context. Perhaps the strongest argument in favour of the hypothesis of Mansfield and Wildberger is that the table works for trigonometric calculations, that someone had put in the effort to generate Pythagorean triples to describe right triangles at roughly one-degree increments. “You don’t make a trigonometric table by accident,” Mansfield says. “Just having a list of Pythagorean triples doesn’t help you much. That’s just a list of numbers. But when you arrange it in such a way so that you can use any known ratio of a triangle to find the other sides of a triangle, then it becomes trigonometry. That’s what we can use this fragment for.”
A Babylonian faced with the ziggurat word problem may have found it easy to set up: a right triangle with the long side, or hypotenuse, 56 cubits long, and one of the shorter sides 45 cubits. Next, the problem solver could have calculated the ratio 56/45, or about 1.244 and then looked up the closest entry on the table, which is line 11, which lists the ratio 1.25. From that line, it is then a straightforward calculation to produce an answer of 33.6 cubits. In their paper, Mansfield and Wildberger show that this is better than what would be calculated using a trigonometric table from the Indian mathematician Madhava 3,000 years later. These days, someone with a calculator can quickly come up with a bit more accurate answer: 33.3317
By Kenneth Chang