Mesopotamian square root approximation by a sequence of rectangles (Q6546577)

Old Babylonian mathematics contains a well-known and very good sexagesimal approximation for \(\sqrt{2}\), namely 1,\,24,\,51,\,10. How this approximation was first derived is unknown and several authors have proposed explanations. Geometrically, the square root of 2 is the (length of the) diagonal of a unit square. Using the standard Old Babylonian approach to estimating diagonals gives a first approximation of 1,\,30 (\(\frac{3}{2}\)), and a second iteration reaches 1,\,25 (also recorded as a diagonal coefficient for the square). At this point a problem arises, for 1,\,25 (\(\frac{17}{12}\)) does not have a finite sexagesimal reciprocal, and so cannot be used as a divisor. In this paper, the author suggests that the path forward was to take 42,\,21,\,10 as the approximate reciprocal of 1,\,25 and then, indeed, the next iteration derives 1,\,24,\,51,\,10. While this particular approximation is not attested, there are a couple of tables showing an Old Babylonian interest in reciprocal approximations.\N\NAlong with this numerical approach, the author also gives a geometrical interpretation. The standard Old Babylonian technique of quadratic geometry would pose two unit squares next to each other, forming a \(1\times 2\) rectangle. One square is then torn in half and the half removed and placed below the other square to form a gnomon, thereby showing that the approximation 1,\,30 for the diagonal is an over-estimate. The author suggests that the 1 by 30 rectangle is stretched out to form a 1,\,30 by 20 rectangle, converting the gnomon into a rectangle, so allowing the process to be repeated to derive the 1,\,25 approximation. This scaling technique is seen in a problem text involving finding the approximate diagonal of a rectangle, although in that case the procedure only goes through one iteration. The only step not attested is the scaling of the torn-off rectangle to repeat the procedure.