In this examination of the Babylonian cuneiform "algebra" texts, based on a detailed investigation of the terminology and discursive organization of the texts, Jens Høyrup proposes that the traditional interpretation must be rejected. The texts turn out to speak not of pure numbers, but of the dimensions and areas of rectangles and other measurable geometrical magnitudes, often serving as representatives of other magnitudes (prices, workdays, etc...), much as pure numbers represent concrete magnitudes in modern applied algebra. Moreover, the geometrical procedures are seen to be reasoned to the same extent as the solutions of modern equation algebra, though not built on any explicit deductive structure.

From the reviews:

J. Høyrup

Lengths, Widths, Surfaces

A Portrait of Old Babylonian Algebra and its Kin

"A valuable addition to the literature of the ancient Babylonians, the subsequent Seleucid mathematics and their relations with other (later) mathematics. Drawing on studies of the culture in which the texts were developed, it contains extensive 'conformal translations' of the most important texts, as well as analyses of them and interpretations of the pictures that emerge."THE AUSTRALIAN MATHEMATICAL SOCIETY

"Høyrup has drawn together and updated the key insights of his intellectual journey of the past 15 years and presented them in one unified, handsomely produced volume. it contains a wealth of information and can be minded by the interested reader for years to come. It is a worthy testament to a career of deep scholarship." (Duncan J. Melville, Historia Mathematica, Vol. 33, 2006)

Autorentext

Jens Høyrup (*1943): educated as a physicist at Copenhagen University. From 1971 to 1973 he taught physics in an engineering school, and from 1973 onward he taught first in the domain of social, then human sciences, at Roskilde University, Denmark, until he retired in 2005. From 1995 until retirement he taught a course of general history of science and guided student projects broadly within this field. Much of his research has dealt with the conceptual, cultural and social history of pre-Modern mathematics.

Inhalt
I Introduction.- II A New Reading.- III Select Textual Examples.- IV Methods.- V Further Algebraic Texts.- VI Quasi-Algebraic Geometry.- VII Old Babylonian Algebra: A Global Characterization.- VIII The Historical Framework.- IX The Finer Structure of the Old Babylonian Corpus.- X The Origin and Transformations of Old Babylonian Algebra.- XI Repercussions and Influences.- Abbreviations and Bibliography.- Index of Tablets.- Index of Akkadian and Sumerian Terms and Key Phrases.- Name Index.