An approximation of the square root of 2
Artifact: Clay tablet
Period: Old Babylonian
Current location: Yale Babylonian Collection (YBC 7289)
Text genre, language: arithmetical
A diagram representing a square with its two diagonals is drawn on the obverse of the round tablet YBC 7289. Numbers in sexagesimal place value notation are written along one of the sides and one of the diagonals. Traces of a rectangle with one diagonal and numbers can be distinguished on the reverse. On the obverse, the number 30 is noted along one side, and the numbers 184.108.40.206 and 42.25.35 are noted below one diagonal. The number 220.127.116.11 corresponds to the diagonal of the square whose side is 1, and the number 42.25.35 corresponds to the diagonal of the square whose side is 30.
The square of the number 1;24.51.10 is 1;18.104.22.168.1.40, that is, a number extremely close to 2 in sexagesimal place value notation. (For comparing the values, it is convenient, but anachronistic, to consider absolute values by using marks such as “;”. However, such marks do not exist in cuneiform notations).
As we see, the number 22.214.171.124 noted on tablet YBC 7289 reflects an impressive approximation of the square root of 2. This number is also attested in a list of coefficients dated to the same period, YBC 7243, where it appears in line 10 and is defined as the coefficient to be used to calculate the diagonal of a square: “126.96.36.199 diagonal, equal side”. In other mathematical texts, the approximation of the square root of 2 commonly used is 1.25, a much less accurate value.
The method that was used by ancient scribes to found such an approximation of the square root of 2 is unknown. Several attempts to reconstruct this method were undertaken, among them that of Neugebauer and Sachs, Fowler and Robson (see bibliography below).
Edition(s): Neugebauer, Otto and Abraham J. Sachs. 1945. Mathematical Cuneiform Texts, p. 42
- Fowler, D.H. and Robson, E.R. (1998). 'Square root approximations in Old Babylonian mathematics: YBC 7289 in context.' Historia Mathematica 25, 366-378.
- Høyrup, Jens (2002), Lengths, Widths, Surfaces. A Portrait of Old Babylonian Algebra and Its Kin, Berlin & Londres: Springer, p. 261-265.