Babylonian Trig

Did the Babylonians develop tables for doing the math of trigonometry centuries before Hipparchus made the kinds of tables that two millennia of trig students might recognize?

It’s an intriguing possibility that’s currently making the rounds of science, math and archaeological news sites. The following video features the University of New South Wales’ Daniel Mansfield making the argument that an ancient clay tablet that was already notable may be more significant than anyone previously realized.

At this point, it’s not a slam dunk that the new interpretation of the inscriptions on the clay tablet represents the Babylonian equivalent of trigonometric tables.

If the new interpretation is right, P322 would not only contain the earliest evidence of trigonometry, but it would also represent an exact form of the mathematical discipline, rather than the approximations that estimated numerical values for sines and cosines provide, notes Mathieu Ossendrijver, a historian of ancient science at Humboldt University in Berlin. The table, he says, contains exact values of the sides for a range of right triangles. That means that—as for modern trigonometric tables—someone using the known ratio of two sides can use information in the tablet to find the ratios of the two other sides.

What’s still lacking is proof that the Babylonians did in fact use this table, or others like it, for solving problems in the manner suggested in the new paper, Ossendrijver says. And science historian Jöran Friberg, retired from the Chalmers University of Technology in Sweden, blasts the idea. The Babylonians “knew NOTHING about ratios of sides!” he wrote in an email to Science. He maintains that P322 is “a table of parameters needed for the composition of school texts and, [only] incidentally, a table of right triangles with whole numbers as sides.” But Mansfield and Wildberger contend that the Babylonians, expert surveyors, could have used their tables to construct palaces, temples, and canals.

Mathematical historian Christine Proust of the French National Center for Scientific Research in Paris, an expert on the tablet, calls the team’s hypothesis “a very seductive idea.” But she points out that no known Babylonian texts suggest that the tablet was used to solve or understand right triangles. The hypothesis is “mathematically robust, but for the time being, it is highly speculative,” she says. A thorough search of other Babylonian mathematical tablets may yet prove their hypothesis, Ossendrijver says. “But that is really an open question at the moment.”

Indeed it is, but then, that’s how science is supposed to work!