Babylonian Algebra

Definition

Babylonian algebra is the body of arithmetical-algebraic technique attested in the Old Babylonian mathematical problem texts (c. 1900-1600 BCE), in which relations between unknown quantities are stated and quadratic problems are systematically posed and solved. Neugebauer characterises it as the body of techniques for solving quadratic equations for two unknowns reduced to a "normal form" in which a product and either a sum or difference of two numbers are given, with extensions reaching linear systems in several unknowns and equations equivalent to fourth, sixth, and even eighth order — though a consciously algebraic notation was never reached.

In Tradition

Neugebauer treats Babylonian algebra as the high point of Old-Babylonian mathematics and the prerequisite arithmetical-computational substrate on which Late-Babylonian mathematical astronomy could later rest. He argues that, despite the geometrical terminology of "length" and "width" used in the problem-text statements, geometrical concepts play only a secondary part: areas and lengths are added together and numbers of men and days combined, showing that the scribe's interest lies in the abstract algebraic relation, not in any geometric reality.

In Practice

For the reader of an Old-Babylonian problem text, the algebraic-technique recognition is the decisive interpretive move. Neugebauer's Chapter III §33 worked example shows the structure directly: a single controlling relation between a length and a width (xy = 10,0 in sexagesimal) combined with individual linear relations in each section leads to quadratic equations for the two unknowns; the place-value reading of the parameters yields x = 30 consistent with the surviving traces. Neugebauer stresses that the advanced algebraic level of Babylonian mathematics was itself an aid to decipherment, because a mathematically complex text leaves fewer possible interpretations of its procedure than a text combining only a few numbers by addition or subtraction. The same arithmetical-technique class supplies the foundation Neugebauer reads as anchoring the later ACT computational methods: the parameter-substitution, linear-system, and quadratic-recipe habits visible in System-A step-function zone calibrations and System-B zigzag M/m fitting both descend from the Old-Babylonian algebraic problem-text tradition documented in the Chapter II §§21-23 corpus.

Historical Origin

Attested across the Old Babylonian mathematical problem-text corpus (c. 1900-1600 BCE) and presupposed in the Seleucid-period ACT mathematical-astronomy corpus from Babylon and Uruk. Modern critical treatments: Otto Neugebauer, *The Exact Sciences in Antiquity* (1957/1969), Chapter II §§21-23, pp. 40-44 and Chapter III §33, pp. 65-66.

Further Reading

• Otto Neugebauer, The Exact Sciences in Antiquity