In the early 20th century, a small clay tablet was discovered in Iraq that has caused historians to question the origins of the Pythagorean theorem. The tablet holds 60 numbers organized in 15 rows and 4 columns that clearly show a recognition of what we know today to be Pythagorean triplets — sets of integers, or whole numbers, that satisfy the equation a2 + b2 = c2.
The piece is known as “Plimpton 322,” named for the archaeological collector George Arthur Plimpton, who donated the tablet to Columbia University, where it is currently on display in the university’s Rare Book and Manuscript Library.
Based on the style of the script inscribed upon the clay, it is dated between 1822 and 1762 B.C. That’s more than 1,000 years before the birth of Greek mathematician Pythagoras, for whom the equation was named.
Dr. Daniel Mansfield of the School of Mathematics and Statistics at the University of New South Wales Faculty of Science and narrator of the above video, has theorized that this is an example of an ancient trigonometric table. He called the tablet, “A fascinating mathematical work that demonstrates undoubted genius. The tablet not only contains the world’s oldest trigonometric table; it is also the only completely accurate trigonometric table, because of the very different Babylonian approach to arithmetic and geometry.”
There are several working theories for the purpose of this list of numbers, including a tool for teachers to generate math problems for students, and the opinion that the tablet has been broken and is impossible to recognize, since an unknown amount of information has been lost. Evidence of this can be seen in a clear break on the left side of the tablet.
Alexander R. Jones, director of the Institute for the Study of the Ancient World at New York University contended that the interpretation of this artifact as a trigonometric table is possible, however he went on to clarify that, “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.”
Dr. Mansfield disregards the doubt, and is convinced that his theory is correct.
“You don’t make a trigonometric table by accident,” Dr. Mansfield said. “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.”
In the above video, Mansfield also shows that the Babylonian tablet uses a more efficient system of calculation, with a base of 60, as opposed to our base of 10. He shows that with a base of 60, divisions will result in far fewer uneven fractions than our current system offers. He suggests that this method of counting may be beneficial to adopt in today’s mathematics.
The New York Times had some fun making up a word problem that an ancient Babylonian student may have had to decipher using Plimpton 322:
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.
And today we celebrate...