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Neither Sherlock Holmes nor Babylon: A reassessment of Plimpton 322. (English) Zbl 0991.01001

Since it was first interpreted in 1945, the incomplete Old Babylonian tablet Plimpton 322 has been a favourite piece in general histories of mathematics because it listed Pythagorean triples. Among historians of mathematics, there has been an ongoing debate both about the principles according to which it was constructed and regarding its purpose. As Eleanor Robson argues forcefully, the use in general histories is historically misleading, viewing the tablet as a piece of modern mathematics and not approaching it through its place within a whole mathematical culture; but even historians who should have known better have done much the same. All of this, though quite polemic in tone, is argued concretely and not in abstract terms of principle, and would be healthy reading for everybody who intends to include a chapter on Mesopotamia or any other distant culture in a forthcoming general history of Babylonian mathematics.
A valid interpretation of the text, as Robson points out, should be correlated to what else we know about Old Babylonian (not generally Babylonian) mathematics; should explain both calculations and calculational errors; should respect the dimensions of the complete tablet as these can be derived from the part we possess; should take the words of the column headings as seriously as the numbers; and respect the “grammar” of mathematical tables, which are always calculated from left to right, and always have the numbers in the left-hand columns in increasing or decreasing order.
These considerations, which have never been considered in full by earlier workers (the last one never), allow Robson to rule out widespread the explanation in terms of “generating numbers”. The alternative explanation first mentioned but discarded by Neugebauer and Sachs and later reproposed in various shapes by Bruins, Schmidt, Friberg and others that the triples were generated from pairs of reciprocals \((x, 1/x) =96\) turns out to be tenable, and allows a full reconstruction of the tablet. Robson also shows that the selection of pairs is fully explainable from pragmatic considerations (numbers not with too many places, and easily derived from the standard table of reciprocals).
In the end, an insightful and convincing explanation of the purpose of the tablet is given: it is a teacher’s aid, created as a means to construct solvable mathematical problems, probably of (geometric-)“algebraic” character and almost certainly about a figure with a diagonal (Robson proposes a right triangle, the reviewer would opt for a rectangle). The technique for creating the problems and for solving them are thus not identical, and the text links together the two strong components of Old Babylonian mathematics: the manipulation of the sexagesimal place value system by means of factorization and other numerical procedures, and the component traditionally known as ”algebra”. Such links are rare elsewhere in the record; apart from trivial linking in “algebraic” problems about “the reciprocal and its reciprocal”, only inhomogeneous cubic problems solved by means of factorization come to the reviewer’s mind. Rare are also texts that reflect not the solution but the construction of problems, and all other texts of this kind are quite simple; Plimpton 322 thus remains an extremely interesting text in the new perspective.
It may be added that comparison with the format of other texts allows Robson to conclude that the tablet was written in Larsa or its vicinity between 1822 and 1762 BCE.

MSC:

01A17 History of Babylonian mathematics
01A85 Historiography
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References:

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