After 70 years of baffling researchers, this 3,700-year-old Babylonian clay tablet’s secret was finally solved by… mathematicians! A new study by Dr. Mansfield and Dr. Norman Wildberger presents another perspective. “[Plimpton 322] contains a special pattern of numbers called Pythagorean triples,” said UNSW researcher Dr. Daniel Mansfield.
“Our research reveals that Plimpton 322 describes the shapes of right-angle triangles using a novel kind of trigonometry based on ratios, not angles and circles. It is 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.”
According to western education, Greek astronomer Hipparchus (120 years BC) is the father of trigonometry with his ‘table of chords’ in a circle.
“Plimpton 322 predates Hipparchus by more than 1,000 years. It opens up new possibilities not just for modern mathematics research, but also for mathematics education. With Plimpton 322 we see a simpler, more accurate trigonometry that has clear advantages over our own,” Dr. Wildberger said.
By “accident,” Dr. Mansfield read about Plimpton 322 while preparing for class and decided to study Babylonian mathematics’ different understandings.
“The 15 rows on the tablet describe a sequence of 15 right-angle triangles, which are steadily decreasing in inclination,” they explained.
The main body of the obverse is ruled by neat horizontal lines into 15 equally spaced rows containing sexagesimal (base 60) numbers, some of which are quite large. The vertical lines continue on the bottom and reverse, which are otherwise empty.
“The left-hand edge of the tablet is broken, and we build on previous research to present new mathematical evidence that there were originally 6 columns and that the tablet was meant to be completed with 38 rows.”
“We also demonstrate how the ancient scribes, who used a base 60 numerical arithmetic similar to our time clock, rather than the base 10 number system we use, could have generated the numbers on the tablet using their mathematical techniques.”
Archaeologists trace Plimpton 322 to the ancient Sumerian city of Larsa near present-day southern Iraq and probably written between 1822-1762 BC during the reign of the sixth king of the First Babylonian Dynasty.
Actually, this artifact was discovered by the real-life Indiana Jones! Edgar J. Banks was an archaeologist, academic, and adventurer who found it in the early 1900’s and sold it to the American publisher George Arthur Plimpton.
Plimpton donated his collection of mathematical artifacts to Columbia University in 1936, where it remains today.
- Daniel F. Mansfield & N.J. Wildberger. Plimpton 322 is Babylonian exact sexagesimal trigonometry. Historia Mathematica, published online August 24, 2017; doi: 10.1016/j.hm.2017.08.001