Jens Hoyrup 
Lengths, Widths, Surfaces [PDF ebook] 
A Portrait of Old Babylonian Algebra and Its Kin

In this examination of the Babylonian cuneiform "algebra" texts, based on a detailed investigation of the terminology and discursive organization of the texts, Jens Hoyrup 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.