YBC 7289 is a Babylonian clay tablet in the Yale Babylonian Collection, notable for containing an accurate sexagesimal approximation to the square root of 2, the length of the diagonal of a unit square. It is believed to come from southern Mesopotamia from some time in the range from 1800–1600 BC. Its significance was first recognized by Neugebauer & Sachs (1945).
The tablet depicts a square with its two diagonals. One side of the square is labeled with the sexagesimal number 30. The diagonal of the square is labeled with two sexagesimal numbers. The first of these two, 1;24,51,10 represents the number 305470/216000 ≈ 1.414213, an accurate numerical approximation of the square root of two. The second of the two numbers is 42;25,35 = 42 307/720 ≈ 42.426, the result of multiplying 30 by the given approximation to the square root of two.
Because the Babylonian sexagesimal notation did not indicate which digit had which place value, an equivalent alternative interpretation is that the number on the side of the square is 30/60 = 1/2. Under this alternative interpretation, the number on the diagonal is 30547/43200 ≈ 0.70711, an accurate numerical approximation of 1/√2. As Fowler & Robson (1998) write, "Thus we have a reciprocal pair of numbers with a geometric interpretation ... The importance of reciprocals in Babylonian mathematics is well known."
Although YBC 7289 is frequently depicted (as in the photo) with the square oriented diagonally, the standard Babylonian conventions for drawing squares would have made the sides of the square vertical and horizontal, with the numbered side at the top. The small round shape of the tablet, and the large writing on it, suggests that it was a "hand tablet" of a type typically used for rough work by a student who would hold it in the palm of his hand. The student would likely have copied the sexagesimal value of the square root of 2 from another tablet, but an iterative procedure for computing this value can be found in another Babylonian tablet, BM 96957 + VAT 6598.
The tablet "demonstrates the greatest known computational accuracy obtained anywhere in the ancient world", the equivalent of six decimal digits of accuracy. Other Babylonian tablets include the computations of areas of hexagons and heptagons, which involve the approximation of more complicated algebraic numbers such as √3. The same number √3 can also be used in the interpretation of certain ancient Egyptian calculations of the dimensions of pyramids. However, the much greater numerical precision of the numbers on YBC 7289 makes it more clear that they are the result of a general procedure for calculating them, rather than merely being an estimate.
The same sexagesimal approximation to √2, 1;24,51,10, was used much later by Greek mathematician Claudius Ptolemy in his Almagest. Ptolemy did not explain where this approximation came from and it may be assumed to have been well known by his time.
Provenance and curation
It is unknown where in Mesopotamia YBC 7289 is from, but its shape and writing style makes it likely to come from southern Mesopotamia, in dates ranging from 1800BC to 1600BC. Yale University acquired it in 1909 as a donation from the estate of J. P. Morgan, who had collected many Babylonian tablets. His bequest became the Yale Babylonian Collection.
At Yale, the Institute for the Preservation of Cultural Heritage has produced a digital model of the tablet, suitable for 3D printing, so that students of the history of mathematics worldwide can interact with it without damaging the original artifact.
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