Plimpton 322 - Interpretation

Interpretation

In each row, the number in the second column can be interpreted as the shortest side s of a right triangle, and the number in the third column can be interpreted as the hypotenuse d of the triangle. The number in the first column is either the fraction or, where l denotes the longest side of the same right triangle. Scholars still differ, however, on how these numbers were generated.

Otto E. Neugebauer (1957) argued for a number-theoretic interpretation, pointing out that this table provides a list of (pairs of numbers from) Pythagorean triples. For instance, line 11 of the table can be interpreted as describing a triangle with short side 3/4 and hypotenuse 5/4, forming the side:hypotenuse ratio of the familiar (3,4,5) right triangle. If p and q are two coprime numbers, then form a Pythagorean triple, and all Pythagorean triples can be formed in this way or as multiples of a triple formed in this way. For instance, line 11 can be generated by this formula with p = 1 and q = 1/2. As Neugebauer argues, each line of the tablet can be generated by a pair (p,q) that are both regular numbers, integer divisors of a power of 60. This property of p and q being regular leads to a denominator that is regular, and therefore to a finite sexagesimal representation for the fraction in the first column. Neugebauer's explanation is the one followed e.g. by Conway & Guy (1996). However, as Eleanor Robson (2002) points out, Neugebauer's theory fails to explain how the values of p and q were chosen: there are 92 pairs of coprime regular numbers up to 60, and only 15 entries in the table. In addition, it does not explain why the table entries are in the order they are listed in, nor what the numbers in the first column were used for.

Buck (1980) discussed a possible trigonometric explanation: the values of the first column can be interpreted as the squared cosine or tangent (depending on the missing digit) of the angle opposite the short side of the right triangle described by each row, and the rows are sorted by these angles in roughly one-degree increments. However, Robson argues on linguistic grounds that this theory is "conceptually anachronistic": it depends on too many other ideas not present in the record of Babylonian mathematics from that time.

In contraposition with earlier explanations of the tablet, Robson (2002) claims that historical, cultural and linguistic evidence all reveal the tablet to be more likely "a list of regular reciprocal pairs." In 2003, the MAA awarded Robson with the Lester R. Ford Award for her work, stating it is "unlikely that the author of Plimpton 322 was either a professional or amateur mathematician. More likely he seems to have been a teacher and Plimpton 322 a set of exercises." Robson takes an approach that in modern terms would be characterized as algebraic, though she describes it in concrete geometric terms and argues that the Babylonians would also have interpreted this approach geometrically.

Robson bases her interpretation on another tablet, YBC 6967, from roughly the same time and place. This tablet describes a method for solving what we would nowadays describe as quadratic equations of the form, by steps (described in geometric terms) in which the solver calculates a sequence of intermediate values v₁ = c/2, v₂ = v₁2, v₃ = 1 + v₂, and v₄ = v₃1/2, from which one can calculate x = v₄ + v₁ and 1/x = v₄ - v₁.

Robson argues that the columns of Plimpton 322 can be interpreted as the following values, for regular number values of x and 1/x in numerical order:

v₃ in the first column,

v₁ = (x - 1/x)/2 in the second column, and

v₄ = (x + 1/x)/2 in the third column.

In this interpretation, x and 1/x would have appeared on the tablet in the broken-off portion to the left of the first column. For instance, row 11 of Plimpton 322 can be generated in this way for x = 2. Thus, the tablet can be interpreted as giving a sequence of worked-out exercises of the type solved by the method from tablet YBC 6967, and reveals mathematical methods typical of scribal schools of the time, and that it is written in a document format used by administrators in that period. Therefore, Robson argues that the author was likely a scribe, a bureaucrat in Larsa. The repetitive mathematical set-up of the tablet, and of similar tablets such as BM 80209, would have been useful in allowing a teacher to set problems in the same format as each other but with different data. In short, Robson suggests that the tablet would likely have been used by a teacher as a problem set to assign to students.